Department of Mathematics and Statistics, Binghamton University seminars:colloquium

Colloquium 2014-2015

2020-01-28T13:27:44-04:00

Colloquium 2014-2015

February 19, 4:30pm
Speaker: Jonathan Williams (University of Georgia)
Topic: A new approach to general smooth 4-manifolds

Abstract:
Some consider smooth 4-manifolds to be a mature field, which typically means its approachable yet nontrivial problems have become scarce. This is mainly due to a lack of tools. In this talk I will present a new way to depict any smooth, closed oriented 4-manifold that opens the doors to two of the most successful tools from 3-manifolds: pseudoholomorphic curves and discrete groups.

February 26, 4:30pm
Speaker: Niels Martin Moeller (Princeton)
Topic: Gluing of Geometric PDEs - Obstructions vs. Constructions for Minimal Surfaces & Mean Curvature Flow Solitons

Abstract:
For geometric nonlinear PDEs, where no easy superposition principle holds, examples of (global, geometrically/topologically interesting) solutions can be hard to come about. In certain situations, for example for 2-surfaces satisfying an equation of mean curvature type, one can generally “fuse” two or more such surfaces satisfying the PDE, as long as certain global obstructions are respected - at the cost (or benefit) of increasing the genus significantly. The key to success in such a gluing procedure is to understand the obstructions from a more local perspective, and to allow sufficiently large geometric deformations to take place. In the talk I will introduce some of the basic ideas and techniques (and pictures) in the gluing of minimal 2-surfaces in a 3-manifold. Then I will explain two recent applications, one to the study of solitons with genus in the singularity theory for mean curvature flow (rigorous construction of Ilmanen's conjectured “planosphere” self-shrinkers), and another to the non-compactness of moduli spaces of finite total curvature minimal surfaces (a problem posed by Ros & Hoffman-Meeks). Some of this work is joint w/ Steve Kleene and/or Nicos Kapouleas.

April 2, 4:30pm
Speaker: Toke Knudsen (SUNY Oneonta)
Topic: Rationales of Ancient Mathematical Methods

Abstract:
The Śulbasūtras, generally dated to 800-200 BCE, are a group of texts that provide the mathematical methods necessary for carrying out various rituals of ancient India. The texts do not seek to convince the reader that a particular formula is correct, but rather focus on providing the reader with working methods. As such, it is often not clear exactly how the authors of the texts arrived at their mathematical results. We will explore some of the mathematical statements of the texts and modern attempts at reconstructing the rationale behind them.

April 16, 4:00pm
Speaker: William Wild (SUNY Buffalo)
Topic: Eye of the Beholder: Is it Possible to Agree on What Makes an Exam Difficult?
Abstract: The level of difficulty at which students are assessed on exams can vary considerably across course sections, raising issues for both academic equity and student proficiency. This may arise when instructors vary from one another in the level of difficulty at which they intend to test. Even when instructors intend to test at the same level, however, judgments regarding what constitutes a “basic” or “difficult” question can vary widely.

This discussion will present a framework that enables instructors to rate problem difficulty in an objective manner. Used a priori, it facilitates improved control over exam design, helping instructors make purposeful and accurate choices about the difficulty profile they wish to construct. Used post hoc, it provides insight into the factors driving the exam outcomes, and by implication, student learning. The tool has validated empirically over several years use in introductory level Physics and Calculus courses.

This discussion may be of interest to:
- Instructors: seeking to gauge the level of proficiency to expect from their students, or, seeking to better understand the factors driving class performance on a given exam
- Curriculum developers: interested in achieving a greater consistency across course offerings.
- Learning Outcomes Assessors: interested in articulating outcomes that measure not only the type of student proficiency, but the level as well.

April 23, No Colloquium. Special event:
Hilton Memorial Lecture at 3pm in Science II, Room 140.
Speaker: Ralf Spatzier (University of Michigan)

April 30, 2:50pm
Dean's Lecture in Geometry and Topology.
Speaker: Karsten Grove (Notre Dame University)
Topic: Symmetry, Positive Curvature and Beyond

Abstract: Although constituting a vast extension of ancient Spherical Geometry, the beautiful class of positively curved (Riemannian) spaces is like the “Tip of the Iceberg” among all (Riemannian) spaces. Accordingly, non-symmetric positively curved spaces are known only in a few sporadic dimensions, and yet only a few obstructions to their existence are known.

In this talk, we will describe the current state of affair of the subject including tools and methods, with emphasis on the impact symmetries have had on the development during the last few decades.

May 7, 4:30pm
Speaker: Victoria Sadovskaya (Penn State)
Topic: Linear cocycles over hyperbolic systems and their periodic data

Abstract:
We consider a hyperbolic diffeomorphism f of a manifold M. A linear cocycle over f is an automorphism of a vector bundle over M that projects to f. An important example comes from the differential of f or its restriction to an invariant sub-bundle of the tangent bundle. For a trivial bundle, a linear cocycle can be viewed as a GL(d,R)-valued function on the manifold. We discuss what conclusions can be made about cocycles based on their behavior at the periodic points of f. In particular, we consider the questions when two cocycles are cohomologous and when a cocycle is conformal or isometric.

Fall 2014

October 9
Speaker: Su Yang (Chinese Academy of Sciences)
Topic: On the classification of certain 5-manifolds with fundamental group Z

Abstract:
In this talk I will give the classification of 5-manifolds with fundamental group Z and whose second homotopy group is finitely generated abelian group. As an application we obtain a criterion for 5-manifolds with fundamental group Z being a fiber bundle over the circle. The classification is also applied to classify certain knotted 3-spheres in the 5-sphere. This is a joint work with M. Kreck.

December 1
Time: 1:10 pm
Location: Science 3, Room 214
Speaker: Ruriko Yoshida (University of Kentucky)
Topic: KDETrees: Nonparametric Estimation of Phylogenetic Tree Distributions

Abstract:
While the majority of gene histories found in a clade of organisms are expected to be generated by a common process (e.g. the coalescent process), it is well-known that numerous other coexisting processes (e.g. horizontal gene transfers, gene duplication and subsequent neofunctionalization) will cause some genes to exhibit a history quite distinct from those of the majority of genes. Such “outlying” gene trees are considered to be biologically interesting and identifying these genes has become an important problem in phylogenetics.

In this talk we propose a nonparametric method of estimating distributions of phylogenetic trees, with the goal of identifying trees which are significantly different from the rest of the trees in the sample. Our method compares favorably with a similar recently-published method, featuring an improvement of one polynomial order of computational complexity (to quadratic in the number of trees analyzed), with simulation studies suggesting only a small penalty to classification accuracy. Application of our implemented software KDETrees to a set of Apicomplexa genes identified several unreliable sequence alignments which had escaped previous detection, as well as a gene independently reported as a possible case of horizontal gene transfer.

This is joint work with G. Weyenberg, P. Huggins, C. Schardl, and D. Howe.

December 1
Time: 3:30 pm
Location: WH 100E
Speaker: Angelica Cueto (Columbia University)
Topic: Non-Archimedean Combinatorics

Abstract:
Non-Archimedean analytic geometry, as developed by Berkovich, is a variation of classical complex analytic geometry for non-Archimedean fields such as p-adic numbers. Solutions to a system of polynomial equations over these fields form a totally disconnected space in their natural topology. The process of analytification adds just enough points to make them locally connected and Hausdorff. The resulting spaces are technically difficult to study but, notably, their heart is combinatorial: they can be examined through the lens of tropical and polyhedral geometry.

I will illustrate this powerful philosophy through complete examples, including elliptic curves, the tropical Grassmannian of planes of Speyer-Sturmfels, and a compactification of the well-known space of phylogenetic trees of Billera-Holmes-Vogtmann.

This talk is based on joint works with M. Haebich, H. Markwig and A. Werner.

December 2
Time: 4:30 pm
Location: WH 100E
Speaker: Ruriko Yoshida (University of Kentucky)
Topic: Markov bases for the toric homogeneous Markov Chain models

Abstract:
Discrete time Markov chains are often used in statistical models to fit the observed data from a random physical process. Sometimes, in order to simplify the model, it is convenient to consider time-homogeneous Markov chains, where the transition probabilities do not depend on the time $T$. While under the time-homogeneous Markov chain model it is assumed that the row sums of the transition probabilities are equal to one, under the toric homogeneous Markov chain (THMC) model the parameters are free and the row sums of the transition probabilities are not restricted.

In order for a statistical model to reflect the observed data, a goodness-of-fit test is applied. For instance, for the time-homogeneous Markov chain model, it is necessary to test if the assumption of time-homogeneity fits the observed data. In 1998, Diaconis-Sturmfels developed a Markov Chain Monte Carlo method (MCMC) for goodness-of-fit test by using Markov bases. A Markov basis is a set of moves between elements in the conditional sample space with the same sufficient statistics so that the transition graph for the MCMC is guaranteed to be connected for any observed value of the sufficient statistics. In algebraic terms, a Markov basis is a generating set of a toric ideal defined as the kernel of a monomial map between two polynomial rings. In algebraic statistics, the monomial map comes from the design matrix (configuration) associated with a statistical model.

In this talk we will consider a Markov basis and a Groebner basis for the toric ideal associate with the design matrix defined by the THMC model with $S \geq 2$ states without initial parameters for any time $T \geq 3$. First we will show the upper bound of the Markov degree, the degree of a minimal Markov base, of the THMC model with $S = 3$ for $T \geq 3$. In order to compute the upper bound, we use the model polytope — the convex hull of the columns of the design matrix. Here we will show the model polytope has only 24 facets for $T \geq 5$ and a complete description of the facets for $T \geq 3$. Finally, we will show a condition when the THMC with any $S \geq 2$ states for $T \geq 3$ have a square-free quadratic Groebner basis and Markov basis. One such example is the embedded discrete Markov chain (jump chain) of the Kimura three parameter model.

This is joint work with Davis Haws (IBM), Abraham Martin del Campo (IST Austria), and Akimichi Takemura (University of Tokyo).

December 3
Time: 5:00 pm (Please note the postponed time.)
Location: WH 100E
Speaker: Lucy Xia (Princeton University)
Topic: Robust Sparse Quadratic Discrimination

Abstract:
We propose a novel Rayleigh quotient based sparse quadratic dimension reduction method – named QUADRO – for analyzing high dimensional data. Unlike in the linear setting where Rayleigh quotient optimization coincides with classification, these two problems are very different under nonlinear settings. One major challenge of Rayleigh quotient optimization is that the variance of quadratic statistics involves all fourth cross-moments of predictors, which are infeasible to compute for high-dimensional applications and may accumulate too many stochastic errors. This issue is resolved by considering a family of elliptical models. Moreover, for heavy-tail distributions, robust estimates of mean vectors and covariance matrices are employed to guarantee uniform convergence in estimating nonpolynomially many parameters, even though the fourth moments are assumed. Computationally, we propose an efficient linearized augmented Lagrangian method to solve the constrained optimization problem. Theoretically, we provide explicit rates of convergence in terms of Rayleigh quotient under both Gaussian and general elliptical models. Thorough numerical results on both synthetic and real datasets are also provided to back up our theoretical results.

December 4
Time: 4:30 pm
Location: WH 100E
Speaker: Robert Haslhofer (Courant Institute, New York University)
Topic: Mean curvature flow

Abstract:
A family of hypersurfaces $M_t\subset R^{n+1}$ evolves by mean curvature flow (MCF) if the velocity at each point is given by the mean curvature vector. MCF can be viewed as a geometric heat equation, deforming surfaces towards optimal ones. If the initial surface $M_0$ is convex, then the evolving surfaces $M_t$ become rounder and rounder and converge (after rescaling) to the standard sphere $S^n$. The central task in studying MCF for more general initial surfaces is to analyze the formation of singularities. For example, if $M_0$ looks like a a dumbbell, then the neck will pinch off preventing one from continuing the flow in a smooth way. To resolve this issue, one can either try to continue the flow as a generalized weak solution or try to perform surgery (i.e. cut along necks and replace them by caps). These ideas have been implemented in the last 15 years in the deep work of White and Huisken-Sinestrari, and recently Kleiner and I found a streamlined and unified approach (arXiv: 1304.0926, 1404.2332). In this lecture, I will survey these developments for a general audience.

December 5
Time: 3:30 pm
Location: WH 100E
Speaker: Max Wakefield (US Naval Academy)
Topic: Coloring Partitions and Configuration spaces

Abstract:
There is a deep interplay between the combinatorics (matroid), algebra (cohomology or rational model), and geometry (complement) of a subspace arrangement (finite collection of subspaces in a vector space). For example if the subspaces are complex and complex codimension 1 (hyperplanes) then the Betti numbers are exactly the (unsigned) Whitney numbers of the first kind on the intersection lattice. Subspace arrangements of the braid arrangement can be enumerated by partitions. It turns out that the Whitney numbers of these subspace arrangements can be found by looking at a generalized chromatic polynomial of the associated partitions. Unfortunately, these Whitney numbers do not give the Betti numbers of the complement and finding a closed formula for these Betti numbers is not known. However, using tools from rational homotopy theory we can show that certain classes of these arrangements are rationally formal and non-formal. At the end we will construct a new differential graded algebra which presents a kind of model for the collection of all k-equal arrangements (configuration spaces where k-1 points can collide) which gives hints at a nice presentation for the cohomology and the Betti numbers.

December 5
Time: 5:00 pm (Please note the postponed time.)
Location: WH 100E
Speaker: Quefeng Li (Princeton University)
Topic: Robust Estimation of High-Dimensional Mean Regression

Abstract:
Data subject to heavy-tailed errors are commonly encountered in various scientific fields, especially in the modern era with explosion of massive data. To address this problem, procedures based on quantile regression and Least Absolute Deviation (LAD) regression have been developed in recent years. These methods essentially estimate the conditional median (or quantile) function. They can be very different from the conditional mean functions when distributions are asymmetric and heteroscedastic. How can we efficiently estimate the mean regression functions in ultra-high dimensional setting with existence of only the second moment? To solve this problem, we propose a penalized Huber loss with diverging parameter to reduce biases created by the traditional Huber loss. Such a penalized robust approximate quadratic (RA-quadratic) loss will be called RA-Lasso. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, our results reveal that the RA-lasso estimator produces a consistent estimator at the same rate as the optimal rate under the light-tail situation. We further study the computational convergence of RA-Lasso and show that the composite gradient descent algorithm indeed produces a solution that admits the same optimal rate after sufficient iterations. As a byproduct, we also establish the concentration inequality for estimating population mean when there exists only the second moment. We compare RA-Lasso with other regularized robust estimators based on quantile regression and LAD regression. Extensive simulation studies demonstrate the satisfactory finite-sample performance of RA-Lasso.

December 8
Time: 5:00 pm
Location: WH 100E
Speaker: Daniel Sewell (University of Illinois at Urbana-Champaign)
Topic: Latent space models for dynamic networks

Abstract:
Dynamic networks are used in a variety of fields to represent the structure and evolution of the relationships between entities. We present a model which embeds longitudinal network data as trajectories in a latent Euclidean space. A Markov chain Monte Carlo algorithm is proposed to estimate the model parameters and latent positions of the actors in the network. The model yields meaningful visualization of dynamic networks, giving the researcher insight into the evolution and the structure, both local and global, of the network. The model handles directed or undirected edges, easily handles missing edges, and lends itself well to predicting future edges. Further, a novel approach is given to detect and visualize an attracting influence between actors using only the edge information. We use the case-control likelihood approximation to speed up the estimation algorithm, modifying it slightly to account for missing data. We apply the latent space model to data collected from a Dutch classroom, and cosponsorship network collected on members of the U.S. House of Representatives, illustrating the usefulness of the model by making insights into the networks.

December 10
Time: 5:00 pm
Location: WH 100E
Speaker: Zuofeng Shang (Purdue University)
Topic: Nonparametric Bernstein-von Mises Phenomenon: A Tuning Prior Perspective

Abstract:
Statistical inference on infinite-dimensional parameters in Bayesian framework is investigated. The main contribution of our work is to demonstrate that nonparametric Bernstein-von Mises theorem can be established in a very general class of nonparametric regression models under the novel tuning priors. Surprisingly, this type of prior connects two important classes of statistical methods: nonparametric Bayes and smoothing spline at a fundamental level. The association with smoothing spline facilitates both theoretical analysis and applications for nonparametric Bayesian inference. For example, the selection of a proper tuning prior can be easily done through generalized cross validation, which can be well implemented by existing R packages.

This is a joint work with Guang Cheng (Purdue).

December 11
Time: 2:50 pm
Location: WH 100E
Speaker: Mark Hagen (University of Michigan)
Topic: Coarse, flexible, and rigid structures in geometric group theory

Abstract:
Geometric group theory is partly the study of groups arising naturally in geometry and topology: this includes fundamental groups of interesting spaces (e.g. 3-manifold groups) or groups of symmetries (isometries, homeomorphisms, etc.) of interesting spaces (e.g. mapping class groups). Geometric group theory is also the study of groups as geometric objects in their own right.

This talk deals with three viewpoints from which a group can be analyzed, and the interplay between these. First, the source of many questions in geometric group theory is the topological viewpoint, in which spaces are distinguished up to homeomorphism, homotopy equivalence, etc. Once one has isolated the fundamental group of one's space, and found generators, the natural geometry becomes “coarse”, and things are generally true up to a relation called “quasi-isometry”. Often, it is desirable to realize the group as a group of automorphisms of some very specific, rigid combinatorial structure; this is the third viewpoint. I will discuss examples of how each of these approaches can naturally lead to and interact with the others.

I will conclude with a brief discussion of very recent joint work with J. Behrstock and A. Sisto, in which we define a class of spaces that includes mapping class groups, many cubical groups (which will have been defined), and most 3-manifold groups, and build tools to study the coarse geometry of such spaces from a common perspective.

December 12
Time: 5:00 pm
Location: WH 100E
Speaker: Michael Dobbins (Center for Geometry and its Applications, Pohang University of Science and Technology)
Topic: A Point in a $nd$-Polytope is the Barycenter of $n$ Points in its $d$-Faces.

Abstract:
In this talk I show that it is always possible to find $n$ points in the $d$-dimensional faces of a $nd$-dimensional convex polytope $P$ so that their center of mass is a target point in $P$. Equivalently, the $n$-fold Minkowski sum of the polytope's $d$-skeleton is the polytope scaled by $n$. This verifies a conjecture by Takeshi Tokuyama.

December 15
Time: 1:45 pm
Location: WH 100E
Speaker: Michael Jablonski (University of Oklahoma)
Topic: Non-compact, homogeneous Einstein spaces.

Abstract: For over a century, Einstein metrics have remained of core interest in modern geometry. On a homogeneous space the Einstein condition reduces to a collection of polynomials and so, in principal, such spaces should be easy to understand and classify. However, the reality is much more complicated and no classification exists in either the compact or non-compact settings. In this talk, we present the current state of knowledge on the classification of non-compact, homogeneous Einstein spaces.

Colloquium 2015-2016

2020-01-28T13:27:07-04:00

Colloquium 2015-2016

Unless stated otherwise, colloquia are scheduled for Thursdays 4:30-5:30pm in WH-100E with refreshments served from 4:00-4:25 pm in WH-102.

Organizers: Michael Dobbins, Andrey Gogolev and Vladislav Kargin

October 15, 4:30pm
Speaker: Wushi Goldring (University of Washington in Saint Louis)
Topic: The limits of algebraicity: The case of automorphic representations

Abstract:
This talk will explore the general theme I like to call “algebraicity” — that objects which seem purely analytic in nature actually admit a deep algebraic interpretation — in the context of the Langlands correspondence. The latter predicts that a specific, distinguished subset of all automorphic representations should have deep algebraic properties, such as having algebraic Hecke eigenvalues, admitting a system of associated Galois representations and, ultimately, corresponding to a motive in the sense of Grothendieck. Two approaches have been used to verify Langlands' prediction: (1) Finding and exploiting a direct link with algebraic geometry and (2) Using Langlands' Functoriality Principle. I will discuss the possibilities and limitations of the two approaches and report on recent work on each approach. The results using the geometric approach are joint work with Jean-Stefan Koskivirta, see arXiv:1507.05032; closely related results were obtained independently around the same time by Pilloni-Stroh, and in a more restricted setting announced also by Boxer.

October 22, 4:30pm
Speaker: Gary Greaves (Tohoku)
Topic: Equiangular Lines in Euclidean Space

Abstract:
Given some dimension $d$, what is the maximum number of lines in $\mathbb R^d$ such that the angle between any pair of lines is constant? (Such a system of lines is called “equiangular”.) This classical problem was initiated by Haantjes in 1948 in the context of elliptic geometry. In 1966, Van Lint and Seidel showed that graphs could be used to study equiangular line systems.

Recently this area has enjoyed a renewed interest due to the current attention the quantum information community is giving to its complex analogue. I will give an introduction to the area and report on some new developments in the theory of equiangular lines in Euclidean space. Among other things, I will present a new construction using real mutually unbiased bases (orthonormal bases such that the angle between two elements of different bases is $arccos(\frac{1}{\sqrt{d}})$, as well as improvements to two long-standing upper bounds for equiangular lines in dimensions 14 and 16.

October 29, 4:30pm
Speaker: Alice Medvedev (City College of New York)
Topic: Model Theory and Algebraic Dynamics.
(Joint work with Thomas Scanlon.)

Abstract: Model Theory, a branch of mathematical logic, has arcane abstract definitions of geometric concepts like “dimension.” They specialize to interesting and useful notions in many settings around arithmetic geometry and algebraic number theory. The goal of the talk is to present the intuitive meaning of these notions through the relatively simple example of coordinate-wise polynomial discrete dynamical systems.

Consider a (discrete) dynamical system $F(x, y, z) := ( f(x), g(y), h(z) )$ for polynomials $f$, $g$, and $h$, acting on the three-dimensional space over complex numbers. What subsets $S$ are invariant under $F$, in the sense that $F(S)$ is a subset of $S$? In particular, what algebraic sets, that is solution sets of systems of polynomial equations, are invariant under $F$?

I will describe the tools from modern model theory, a branch of mathematical logic, that reduce this question to understanding composition of one-variable polynomials, and an old theorem of Ritt that supplies this understanding.

November 12, 4:30pm
Speaker: Xiangjin Xu (Binghamton University)
Topic: Sharp Li–Yau type estimates and new heat kernel estimates on negative curved manifolds

Abstract: In this talk, we firstly discuss some background and history related to the Gaussian type upper bound of the heat kernel and the sharp Li–Yau type estimates for the positive solution $u(x,t)$ of the heat equations $u_t-\Delta u=0$ on a complete manifold. Then we obtain some new almost sharp Li–Yau type Harnack inequalitieson a complete manifold with $Ricci(M)\ge-k$. As applications, new parabolic Harnack inequalities are derived, and monotonicity of Perelman type entropy for the heat kernel and the positive solutions are achieved. And we are able to obtain the Gaussian type upper bound of the heat kernel on a complete manifold with $Ricci(M)\ge-k$. At the end, we discuss some open questions related to the sharp Li–Yau type estimates. Part of talk is joint with Junfang Li.

December 2, 4:40pm
Speaker: Dehan Kong (University of North Carolina at Chapel Hill)
Topic: High-dimensional Matrix Linear Regression Model
Time: 4:40–5:40pm

Abstract: We develop a high-dimensional matrix linear regression model (HMLRM) to correlate matrix responses with high-dimensional scalar covariates when coefficient matrices have low-rank structures. We propose a fast and efficient screening procedure based on the spectral norm to deal with the case that the dimension of scalar covariates is ultra-high. We develop an efficient estimation procedure based on the nuclear norm regularization, which explicitly borrows the matrix structure of coefficient matrices. We systematically investigate various theoretical properties of our estimators, including estimation consistency, rank consistency, and the sure independence screening property under HMLRM. We examine the finite-sample performance of our methods using simulations and a large-scale imaging genetic dataset collected by the Alzheimer's Disease Neuroimaging Initiative study.

December 4, 4:40pm
Speaker: Ni Zhao (Fred Hutchinson Cancer Research Center)
Topic: Statistical Inference Methods for High Dimensional “Omics” Data
Time: 4:40–5:40pm

Abstract: Recent advances in high-throughput biotechnology have enabled multiple platform “omics” profiling of biological samples. In this talk, I will present two statistical inference methods for testing the association between high-dimensional “omics” data and a phenotype of interest. The first method is designed to analyze large-scale methylation changes using high-resolution CpG data [Zhao et al, Genetic Epidemiology, 2015]. The second method is a powerful likelihood ratio test via the composite kernel machine regression in Genome Wide Association Studies [Zhao et al, Biometrics, Prepare for Submission]. The method tests the association between multiple SNPs with the phenotype, considering possible gene-environment interaction. The utility of the methods will be illustrated within the Norwegian Mother and Child Cohort Study: a large prospective cohort study of pregnant Norwegian women and their children.

December 7, 4:40pm
Speaker: Jyotishka Datta (Duke University and SAMSI)
Topic: Priors for Sparse High-Dimensional Discrete or Continuous Data
Time: 4:40–5:40pm

Abstract: Sparse signal detection has been one of the most important challenges in the analysis of large-scale data-sets arising from many different disciplines, e.g. Genomics, finance and astronomy. In this talk, I will focus on two key aspects of inference on a high-dimensional sparse mean vector: (1) how to provide theoretical justifications for existing methods that perform strongly, and (2) how to use this theoretical insight to develop new approaches that can outperform the current methods in the `ultra-sparse' regime. In the first half of the talk, I will discuss multiple testing optimality for continuous data, and prove Oracle properties of the popular `Horseshoe’ prior [1]. I will then develop a novel prior called the ‘Horseshoe+’ prior [2] that sharpens the ‘Horseshoe’ prior’s signal detection abilities. I will illustrate that the Horseshoe+ prior outperforms the existing methods both in theory and practice and correctly identifies the `differentially expressed' genes from microarray data. In the second half, I will briefly discuss inference on high dimensional sparse count data which is fundamentally different from the high-dimensional Gaussian case. I will present the ‘Gauss-Hypergeometric’ prior for sparse Poisson means [3], motivated by the growing interest in analyzing sparse count data and end with an application to detect mutational hotspots in whole exome sequencing data.

References:

[1] Datta, J. and Ghosh, J. K. (2013). Asymptotic properties of Bayes risk for the horseshoe prior. Bayesian Analysis, 8(1):111–131.

[2] Bhadra, A., Datta, J., Polson, N. G., and Willard, B. (2015). The Horseshoe+ Estimator of Ultra-Sparse Signals. arXiv preprint arXiv:1502.00560.

[3] Datta, J. and Dunson, D. B. (2015). Priors for High-Dimensional Sparse Poisson Means. arXiv preprint arXiv:1510.04320. (Biometrika, under revision)

January 25, 4:40pm
Speaker: Tianyi Zheng (Stanford University)
Topic: Random walk parameters and the geometry of groups
Time: 4:40–5:40pm

Abstract: The first characterization of groups by an asymptotic description of random walks on their Cayley graphs dates back to Kesten’s criterion of amenability. I will first review some connections between the random walk parameters and the geometry of the underlying groups. I will then discuss a flexible construction that gives solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy and return probability and Hilbert compression functions of groups of exponential volume growth. Based on joint work with Jeremie Brieussel.

January 28, 4:30pm
Speaker: David Renfrew (University of Colorado)
Topic: Spectral properties of large Non-Hermitian Random Matrices
Time: 4:30–5:30pm

Abstract: The study of the spectrum of Non-Hermitian random matrices with independent, identically distributed entries was introduced by Ginibre and Girko. I will present two generalizations of the iid model when the independence and identically distribution assumptions are relaxed and discuss applications to modeling Neural Networks.

January 29, 4:40pm
Speaker: Sanjeena Dang (University of Guelph)
Topic: Model-based clustering: some recent work and biological applications
Time: 4:30–5:30pm

Abstract: With advances in high throughput technologies, massive amounts of data can be generated in an increasingly shorter period of time. Challenges and approaches to dealing with high-dimensional data such as RNA-Seq data, microbiome data, and microarray data will be discussed in a model-based clustering context. The talk will provide an overview of different frameworks for clustering increasingly complex biological data using both mixtures of Gaussian distributions and mixtures of non-Gaussian distributions. A number of families of mixture models will be considered and the talk will conclude with a discussion of future trends.

February 1, 3:30-4:30pm
Speaker: Subhro Ghosh (Princeton University)
Topic: Rigidity phenomena in random point sets
Time: 3:30-4:30pm

Abstract: In several naturally occurring (infinite) random point processes, we establish that the number of the points inside a bounded domain can be determined, almost surely, by the point configuration outside the domain. This includes key examples coming from random matrices and random polynomials. We further explore other random processes where such “rigidity” extends to a number of moments of the mass distribution. The talk will focus on particle systems with such curious “rigidity” phenomena, and their implications. We will also talk about applications to natural questions in stochastic geometry and harmonic analysis.

February 2, 4:30-5:30pm
Speaker: Martin Slawski (Rutgers University)
Topic: High-dimensional regression with sign constraints and quantized observations
Time: 4:30-5:30pm

Abstract: In this talk, I will present several results on two specific aspects in the area of high-dimensional statistics and compressed sensing. The first part of the talk is concerned with the potential usefulness of sign constraints in sparse high-dimensional linear regression, which, along with some extensions, is put into the broader context of regularization-free estimation. The second part of the talks is dedicated to the trade-off between the number of samples and the bit depth per sample in compressed sensing with quantized observations. I will outline an analysis of a popular approach to signal recovery due to Plan and Vershynin in this regard, with the conclusion that a small number of bits (one or two) is optimal. If time permits I will also provide a brief overview on the related problem of similarity estimation for massive data sets based on quantized random projections.

February 4, 4:40-5:40pm
Speaker: Cheng Li (Duke University)
Topic: PIE: Simple, Scalable and Accurate Posterior Interval Estimation
Time: 4:40-5:40pm

Abstract: Scalable Bayes for big data is a rapidly growing research area, but most existing methods are either highly complex to implement for practitioners, or lack theoretical justification for uncertainty quantification. Bayesian methods quantify uncertainty through posterior and predictive distributions. For massive datasets, it is difficult to efficiently estimate summaries of these distributions, such as posterior quantiles and credible intervals. In small scale problems, posterior sampling algorithms such as Markov chain Monte Carlo (MCMC) remain the gold standard, but they face major problems in scaling up to big data. We propose a very simple and general Posterior Interval Estimation (PIE) algorithm to evaluate the posterior distributions of one-dimensional (1-d) functionals, which are typically the focus in many applications. The PIE algorithm consists of three steps. First, full data are partitioned into computationally tractable subsets. Second, sampling algorithms such as MCMC are run in parallel across every subset. Finally, PIE approximates the full posterior by simply averaging posterior quantiles estimated from each subset. This allows standard Bayesian algorithms such as MCMC to be trivially scaled up to big data. We provide strong theoretical guarantees for PIE on its posterior uncertainty quantification, and compare its empirical performance with variational Bayes and the recent WASP algorithm for mixed effects models and nonparametric Bayesian mixture models.

April 7, 4:30-5:30pm
Speaker: David Spivak (MIT)
Topic: Calculating steady states of nonlinear dynamical systems using matrix arithmetic

Abstract: Open dynamical systems are mathematical models of machines that take input, change their internal state, and produce output. For example, one may model anything from neurons to robots in this way. Several open dynamical systems can be arranged in series, in parallel, and with feedback to form a new dynamical system—this is called compositionality—and the process can be repeated in a fractal-like manner to form more complex systems of systems.

I will discuss a technique for calculating the steady states of an interconnected system of systems, in terms of the steady states of the component dynamical systems. The steady states, or equilibria, are organized into “steady state matrices” which generalize bifurcation diagrams. I'll show that the compositionality structure of dynamical systems fits with familiar operations on matrices: serial, parallel, and feedback compositions correspond to multiplication, Kronecker product, and partial trace operations on matrices. Thus we can calculate the steady states of a system of dynamical systems by doing matrix arithmetic on the individual steady state matrices. This talk will be aimed at an undergraduate level.

April 15, 3:30-4:30pm
Dean's Lecture in Geometry.
Speaker: Stephen Preston (Brooklyn College CUNY)
Title: The Geometric Approach to Partial Differential Equations

Abstract:
I will give a survey of those PDEs that can be viewed as geodesics on infinite-dimensional spaces. Vladimir Arnold observed in 1966 that an ideal fluid can be viewed as a geodesic on the group of volume-preserving diffeomorphisms of a domain, and he computed some of the sectional curvatures, showing that many of them are negative. Since then many other equations have found interpretation as geodesics, including the equation for inextensible whips, the Korteweg-de Vries equation, and other conservative PDEs. I will describe some of the finite-dimensional models (including the equations for a rigid body) along with the general aspects of finite-dimensional Riemannian geometry and what still works in infinite dimensions. Finally I will show how a new one-dimensional model of the Euler equation shares many of the same properties and also ties into the Teichmuller theory in complex analysis.

May 3
Dean's Lecture in Geometry.
Speaker: Melvyn Nathanson (CUNY)
Title: Every Finite Subset of an Abelian group is an Asymptotic Approximate Group

Abstract:
If $A$ is a nonempty subset of an additive abelian group $G$, then the $h$-fold sumset is \[hA = \{x_1 + \cdots + x_h : x_i \in A_i \text{ for } i=1,2,\ldots, h\}.\]
We do not assume that $A$ contains the identity, nor that $A$ is symmetric, nor that $A$ is finite. The set $A$ is an $(r,\ell)$-approximate group in $G$ if there exists a subset $X$ of $G$ such that $|X| \leq \ell$ and $rA \subseteq XA$. The set $A$ is an asymptotic $(r,\ell)$-approximate group if the sumset $hA$ is an $(r,\ell)$-approximate group for all sufficiently large $h.$ It is proved that every polytope in a real vector space is an asymptotic $(r,\ell)$-approximate group, that every finite set of lattice points is an asymptotic $(r,\ell)$-approximate group, and that every finite subset of every abelian group is an asymptotic $(r,\ell)$-approximate group.

May 10
Dean's Lecture in Geometry.
Speaker: Alexandru Buium (University of New Mexico)
Title: The differential geometry of Spec Z

Abstract:
The aim of this talk is to show how one can develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers Z will play the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold will be played by the prime numbers.
The role of partial derivatives of functions with respect to the coordinates will be played by the Fermat quotients of integers with respect to the primes. The role of metrics (respectively 2-forms) will be played by symmetric (respectively antisymmetric) matrices with coefficients in Z. The role of connections (respectively curvature) attached to metrics or 2-forms will be played by certain adelic (respectively global) objects attached to matrices as above. One of the main conclusions of our theory will be that Spec Z is ``intrinsically curved;” the study of this curvature will then be one of the main tasks of the theory.

Colloquium 2016-2017

2020-01-28T13:29:42-04:00

Colloquium 2016-2017

Spring 2017

February 23, 4:30 pm
Speaker: Ofer Zeitouni (NYU / Weizmann Institute of Science)
Topic: Extrema of logarithmically correlated fields:
Branching random walks, Gaussian free fields, cover times and random matrices.

Abstract: TBA

March 16, 4:30 pm
Speaker: Ivan Corwin (Columbia University)
Topic: A drunk walk in a drunk world

Abstract: A simple symmetric random walk jumps up or down with equal probability. What happens if its jump probabilities are instead taken themselves to be random in space and time (e.g. uniformly distributed on 0% to 100%)? In this talk (based on joint work with Guillaume Barraquand) I will describe the effect of this random environment on a random walk, and elucidate a new connection to the world of quantum integrable systems and the Kardar-Parisi-Zhang universality class and stochastic PDE. No prior knowledge of any of these areas will be expected.

May 4, 4:30 pm, Dean's Lecture
Speaker: Guozhen Lu (University of Connecticut)
Topic: Sharp geometric and functional inequalities and applications to geometry and PDEs

Abstract: Sharp geometric and functional inequalities play an important role in applications to geometry and PDEs. In this talk, we will discuss some important geometric inequalities such as Sobolev inequalities, Hardy inequalities, Hardy-Sobolev inequalities Trudinger-Moser and Adams inequalities, Gagliardo-Nirenberg inequalities and Caffarelli-Kohn-Nirenberg inequalities, etc. We will also brief talk about their applications in geometry and nonlinear PDEs. Some recent results will also be reported.

This talk is intended to be for the general audience.

Fall 2016

December 13, 4:40 pm
Speaker: Florian Sprung (U. Of Illinois)
Topic: Analytic ideas in the theory of elliptic curves and the Birch and Swinnerton-Dyer conjecture

Abstract:The Birch and Swinnerton-Dyer conjecture, one of the millenium problems, is a bridge between algebraic invariants of an elliptic curve and its (complex analytic) L-function. In the case of low ranks, we prove this conjecture up to the finitely many bad primes and the prime 2, by proving the Iwasawa main conjecture in full generality. The ideas in the proof and formulation also lead us to new and mysterious phenomena. This talk assumes no specialized background in number theory.

December 12, 4:40 pm
Speaker: Jenya Sapir (U. Of Illinois)
Topic: Geodesics on surfaces

Abstract: Let S be a hyperbolic surface. We will give a history of counting results for geodesics on S. In particular, we will give estimates that fill the gap between the classical results of Margulis and the more recent results of Mirzakhani. We will then give some applications of these results to the geometry of curves. In the process we highlight how combinatorial properties of curves, such as self-intersection number, influence their geometry.

December 12, 3:00 pm
Speaker: Cary Malkiewich (U. Of Illinois)
Topic: The Euler characteristic and the Reidemeister trace

Abstract: Suppose that M is a closed connected manifold of dimension at least three, and that f is a continuous map from M to itself. Can f be deformed to a map without fixed points? When f is the identity, the Euler characteristic chi(M) is the complete obstruction, meaning the fixed points can be removed precisely when chi(M) = 0. When f is not the identity, the complete obstruction is instead a more sophisticated invariant called the Reidemeister trace of f.

In this talk we will consider the Reidemeister trace, not just for manifolds, but for a very general class of cell complexes. At this level of generality, it becomes highly nontrivial to relate the Reidemeister trace back to the Euler characteristic, but such a relationship would have far-reaching consequences. We will give a precise conjecture about the two, generalizing an earlier conjecture of Geoghegan. Finally, we will outline two recent results that provide new evidence for this conjecture.

December 9, 4:30 pm
Speaker: Hung P. Tong-Viet (Kent State University)
Topic: Degrees of representations of alternating groups

Abstract: Group theory is the study of symmetry and group representation theory is the study of a group via its actions on various vector spaces over the field of comp lex numbers. Group representations carry lots of information about the groups and so group representation theory has numerous applications in other areas of mathematics such as probability, cryptography, number theory as well as in chemistry and physics. Group representation theory, founded by F.G. Frobenius $1 20$ years ago, is still an active research area with many interesting and long-standing open conjectures. Simple groups, a concept introduced by Galois in 1832, are the building blocks of all finite groups. With the completion of the classification of finite simp le groups, many important conjectures in group representation theory have been solved or reduced to simple groups.

Bertram Huppert conjectured in $2000$ that all nonabelian simple groups are determined by the set of the degrees of their complex irreducible representations up to an abelian direct factor. This conjecture is the best possible and is related to the famous Brauer's Problem $2$ which asked the following: which fini te groups are uniquely determined up to isomorphism by the structure of their group algebras? In this talk, I will review some basic concepts in group representation theory, discuss some interesting results on Brauer's Problem $2$ and will outline the proof of Huppert's conjecture for alternating groups, one of the most important family of simple groups. This is joint work with Christine Bessenrodt and Jiping Zhang.

December 8, 4:30 pm
Speaker: Lionel Levine (Cornell University)
Topic: Circles in the sand

Abstract: I will describe the role played by an Apollonian circle packing in the scaling limit of the abelian sandpile on the square grid Z^2. The sandpile solves a certain integer optimization problem. Associated to each circle in the packing is a locally optimal solution to that problem. Each locally optimal solution can be described by an infinite periodic pattern of sand, and the patterns associated to any four mutually tangent circles obey an analogue of the Descartes Circle Theorem. Joint work with Wesley Pegden and Charles Smart.

December 7, 4:40 pm
Speaker: Chen Le (University of Kansas)
Topic: Stochastic heat equation: intermittency and densities.

Abstract: Stochastic heat equation (SHE) with multiplicative noise is an important model. When the diffusion coefficient is linear, this model is also called the parabolic Anderson model, the solution of which traditionally gives the Hopf-Cole solution to the famous KPZ equation. Obtaining various fine properties of its solution will certainly deepen our understanding of these important models. In this talk, I will highlight several interesting properties of SHE and then focus on the probability densities of the solution. In a recent joint work with Y. Hu and D. Nualart, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to SHE with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders.

December 6, 4:40 pm
Speaker: Steven Frankel (Yale)
Topic: Hyperbolicity in dynamics and geometry

Abstract: A dynamical system, such as a flow or transformation, is called hyperbolic if it tends to stretch and contract the underlying space in different directions. This behavior often appears in systems that are sufficiently mixing – think of the striations that appear when stirring a drop of milk into coffee – and it lends these systems a kind of rigidity that can be useful in understanding their long-term behavior.

We will look at a number of hyperbolic dynamical systems, including Anosov and pseudo-Anosov maps and transformations, and illustrate some of the uses and consequences of their hyperbolic behavior. In addition, we will see that the dynamics of a hyperbolic system can often be understood in terms of a simpler, lower-dimensional dynamical system that lies “at infinity,” the universal circle. This plays an important role in the proof of Calegari’s conjecture, which relates the dynamical hyperbolicity of a flow with the geometric hyperbolicity of its underlying space: It says that any flow on a closed hyperbolic 3-manifold whose orbits are coarsely comparable to geodesics is equivalent, on the large scale, to a pseudo-Anosov flow.

December 5, 4:40 pm
Speaker: Gang Zhou (CalTech)
Topic: On Singularity Formation Under Mean Curvature Flow

Abstract: In this talk I present our recent works, jointly with D.Knopf and I.M.Sigal, on singularity formation under mean curvature flow. By very different techniques, we proved the uniqueness of collapsing cylinder for a generic class of initial surfaces. In the talk some key new elements will be discussed. A few problems, which might be tackled by our techniques, will be formulated.

December 2, 4:40 pm
Speaker: Ling Xiao (Rutgers)
Topic: Translating Solitons in Euclidean Space

Abstract: Mean curvature flow may be regarded as a geometric version of the heat equation. However, in contrast to the classical heat equation, mean curvature flow is described by a quasilinear evolution system of partial differential equations, and in general the solution only exists on a finite time interval. Therefore, it's very typical that the flow develops singularities.

Translating solitons arise as parabolic rescaling of type II singularities. In this talk, we shall outline a program on the classification of translating solitons. We shall also report on some recent progress we have made in the joint work with Joel Spruck.

December 1, 4:30 pm
Speaker: Ilya Vinogradov (Princeton University)
Topic: Sequences modulo one: convergence of local statistics

Abstract: The study of randomness of fixed objects is an area of active research with many exciting developments in the last few years. We will discuss recent results about sequences in the unit interval specializing to directions in affine lattices, \sqrt n modulo 1, and directions in hyperbolic lattices. Theorems about these sequences address convergence of moments as well as rates of convergence, and their proofs showcase a beautiful interplay between dynamical systems and number theory.

November 17, 4:30 pm
Speaker: Todd Fisher (Brigham Young University)
Topic: Entropy for smooth systems

Abstract: Dynamical systems studies the long-term behavior of systems that evolve in time. It is well known that given an initial state the future behavior of a system is unpredictable, even impossible to describe in many cases. The entropy of a system is a number that quantifies the complexity of the system. In studying entropy, the nicest classes of smooth systems are ones that do not undergo bifurcations for small perturbations. In this case, the entropy remains constant under perturbation. Outside of the class of systems, a perturbation of the original system may undergo bifurcations. However, this is a local phenomenon, and it is unclear when and how the local changes in the system lead to global changes in the complexity of the system. We will state recent results describing how the entropy (complexity) of the system may change under perturbation for certain classes of systems.

November 3, 4:30 pm
Speaker: William Menasco (SUNY at Buffalo)
Topic: The Kawamuro Cone and the Jones Conjecture

October 20, 4:30 pm
Speaker: Ted Chinburg (University of Pennsylvania)

Title: Capacity theory and cryptography

Abstract: This talk is about an unexpected connection between cryptography and the theory of electrostatics. RSA cryptography is based on the presumed difficulty of factoring a given large integer N. In the 1990's, Coppersmith showed how one could quickly determine whether there is a factor of N which is within N^{1/4} of a given number. Capacity theory originated in studying how charged particles distribute themselves on an object. I will discuss how an arithmetic form of capacity theory can be used to show that one cannot increase the exponent 1/4 in Coppersmith's method. This is joint work with Brett Hemenway, Nadia Heninger and Zach Scherr.

Colloquium 2017-2018

2020-01-28T13:30:52-04:00

Colloquium 2017-2018

Spring 2018

February 13, 4:15 pm
Speaker: Craig Guilbault (Winsconsin - Milwaukee)
Topic: Infinite boundary connected sums with applications to aspherical manifolds

Abstract: A boundary connected sum $Q_1\natural Q_2$ of $n$-manifolds is obtained by gluing $Q_1$ to $Q_2$ along $\left( n-1\right) $-balls in their respective boundaries. Under mild hypotheses, this gives a well-defined operation that is commutative, associative, and has an identity element. In particular (under those hypotheses) the boundary connected sum $\natural _{i=1}^{k}Q_{i}$ of a finite collection of n-manifolds is topologically well-defined. This observation fails spectacularly when we attempt to generalize it to countable collections. In this talk I will discuss a pair of reasonable (and useful) substitutes for a well-definedness theorem for infinite boundary connected sums. An application of interest in both manifold topology and geometric group theory examines aspherical manifolds with exotic, i.e., not homeomorphic to $\mathbb{R}^{n}$, universal covers. We will describe examples different from those found in the classical papers by Davis and Davis-Januszkiewicz. Much of this work is joint with Ric Ancel and Pete Sparks.

February 15, 4:15 pm
Speaker: Ya Su (TAMU)
Topic: Nonparametric Bayesian Deconvolution of a Symmetric Unimodal Density

Abstract: We consider nonparametric measurement error density deconvolution subject to heteroscedastic measurement errors as well as symmetry about zero and shape constraints, in particular unimodality. The problem is motivated by genomics applications, where the observed data are estimated effect sizes from a regression on multiple genetic factors, as occurs in genome-wide association studies and in microarray applications. We exploit the fact that any symmetric and unimodal density can be expressed as a mixture of symmetric uniforms densities, and model the mixing density using a Dirichlet process location-mixture of Gamma distributions. We do the computations within a Bayesian context, describe a simple scalable implementation that is linear in the sample size, and show that the estimate of the unknown target density is consistent. Within our application context of regression effect sizes, the target density is likely to have a large probability near zero (the near null effects) coupled with a heavy-tailed distribution (the actual effects). Simulations show that unlike standard deconvolution methods, our Constrained Bayesian method does a much better job of reconstruction of the target density. An application to a genome-wide association study to predict height shows similar results.

February 20, 4:15 pm
Speaker: Theodore Voronov (Manchester UK and Notre Dame)
Topic: Supergeometry: from super de Rham theory and the Atiyah-Singer index theorem to microformal geometry

Abstract: Supergeometry is, roughly, the geometry associated with $\mathbb{Z}_2$-graded algebra. In particular, for an odd element $Q$ of a Lie superalgebra, the two options, $Q^2\neq 0$ and $Q^2=0$, lead to “supersymmetry” and to “homological vector fields”, respectively.

The “super” notions were originally discovered as a language for describing fermions and bosons in quantum theory on an equal footing. They received their name from supersymmetric models where bosons and fermions are allowed to mix. Their mathematical roots can be traced in classical differential geometry, algebraic topology and homological algebra.

In the talk, I will introduce the basic ideas and describe some interesting results and links with other areas of mathematics. Among them: super de Rham theory and its connection with Radon transform and Gelfand's general hypergeometric equations; universal recurrence relations for super exterior powers and application to Buchstaber-Rees theory of (Frobenius) “n-homomorphisms”; analytic proof of the Atiyah-Singer index theorem; homological vector fields as a universal language for deformation theory and bracket structures (such as homotopy Lie algebras, Lie algebroids, etc.) in mathematics and gauge systems in physics. An intriguing recent result (which started from a counterexample to a conjecture by Witten) concerns volumes of classical supermanifolds such as superspheres, super Stiefel manifolds, projective superspaces, etc. Upon some universal normalization, formulas for these “super” volumes turned out to be analytic continuations of formulas for ordinary manifolds. Another recent development is “microformal geometry”. This, roughly, is a theory that replaces ordinary maps between manifolds by certain “thick morphisms”, which induce non-linear pullbacks on functions, with remarkable properties. This is motivated by application to homotopy Poisson structures; but in general, it suggests a non-linear extension of the fundamental “algebra/geometry duality”. I hope to be able to tell about that as well.

March 15, 4:15 pm
Speaker: Steve Marron (UNC)
Topic: OODA of Tree Structured Data Objects Using Persistent Homology

Abstract: The field of Object Oriented Data Analysis has made a lot of progress on the statistical analysis of the variation in populations of complex objects. A particularly challenging example of this type is populations of tree-structured objects. Deep challenges arise, whose solutions involve a marriage of ideas from statistics, geometry, and numerical analysis, because the space of trees is strongly non-Euclidean in nature. Here these challenges are addressed using the approach of persistent homologies from topological data analysis. The benefits of this data object representation are illustrated using a real data set, where each data point is the tree of blood arteries in one person's brain. Persistent homologies gives much better results than those obtained in previous studies.

March 22, 4:15 pm
Speaker: Stefan Steinerberger (Yale)
Topic: Four (confusing) Miracles in Analysis and Number Theory

Abstract: CANCELLED

March 26, 4:15 pm
Speaker: Jaehong Jeong [KAUST (King Abdullah University of Science and Technology), Saudi Arabia]
Topic: A Stochastic Generator of Global Monthly Wind Energy with Tukey g-and-h Autoregressive Processes

Abstract: Quantifying the uncertainty of wind energy potential from climate models is a very time-consuming task and requires a considerable amount of computational resources. A statistical model trained on a small set of runs can act as a stochastic approximation of the original climate model, and be used to assess the uncertainty considerably faster than by resorting to the original climate model for additional runs. While Gaussian models have been widely employed as means to approximate climate simulations, the Gaussianity assumption is not suitable for winds at policy-relevant time scales, i.e., sub-annual. We propose a trans-Gaussian model for monthly wind speed that relies on an autoregressive structure with Tukey g-and-h transformation, a flexible new class that can separately model skewness and tail behavior. This temporal structure is integrated into a multi-step spectral framework that is able to account for global nonstationarities across land/ocean boundaries, as well as across mountain ranges. Inference can be achieved by balancing memory storage and distributed computation for a data set of 220 million points. Once fitted with as few as five runs, the statistical model can generate surrogates fast and efficiently on a simple laptop, and provide uncertainty assessments very close to those obtained from all the available climate simulations on a monthly scale. This is joint work with Yuan Yan, Stefano Castruccio, and Marc G. Genton.

Fall 2017

September 28, 4:30 pm
Speaker: Paul F. Velleman (Cornell)
Topic: Six Impossible Things Before Breakfast: Integrating Randomization Methods in the Introductory Statistics Course

Abstract: The traditional introductory statistics course generally proceeds smoothly until the point where we have to admit to our students that the statistics they’ve been finding in their homework problems aren’t really the answer; they are only an answer. They can believe that. Then we tell them that those answers may be random, but they aren’t haphazard. In particular, if we gather the answers for all possible samples we can model them. They might accept that even though they can’t see why it should be true. Then we claim to be able to estimate the parameters of those models and propose to use them for inference. Then, to top it all off, we admit that we were lying when we said the model for the mean was Normal, and that when the standard deviation is estimated (that is, almost always) or we’re doing a regression, the model isn’t Normal at all but only similar to the Normal.

Many students find all those results uncomfortable. They are not used to thinking that way. The Red Queen encountered by Alice may have been able to believe six impossible things before breakfast, but it is challenging to ask that of our students.

With computer technology, we can spread out these results across the first several weeks of the course to make it easier for students to understand and accept them. And then, by introducing bootstrap methods, we can carry these ideas into the discussion of inference.

I will discuss a syllabus that does just that and demonstrate some free software that supports the approach.

October 5, 4:15 pm
Speaker: Alan Edelman (MIT)
Topic: Novel Computations with Random Matrix Theory and Julia
This speaker's visit is part of the Dean's Speaker Series in Statistics and Data Science.

Abstract: Over the many years of reading random matrix papers, it has become increasingly clear that the phenomena of random matrix theory can be difficult to understand in the absence of numerical codes to illustrate the phenomena. (We wish we could require that all random matrix papers that lend themselves to computing include a numerical simulation with publicly available code.) Of course mathematics exists without numerical experiments, and all too often a numerical experiment can be seen as an unnecessary bother. On a number of occasions, however, the numerical simulations themselves have an interesting twist of their own. This talk will illustrate a few of those simulations and illustrate why in particular the Julia computing language is just perfect for these simulations. Some topics we may discuss:

“Free” Dice
Tracy Widom
Smallest Singular Value
Jacobians of Matrix Factorizations

(joint work with Bernie Wang)

October 19, 4:15 pm
Speaker: Stefan Steinerberger (Yale)
Topic: Three (confusing) Miracles in Analysis and Number Theory

Abstract: I will discuss three different topics at the intersection of Analysis and Number Theory:

improved versions of classical inequalities for functions on the Torus whose proof requires Number Theory,
mysterious interactions between the Hardy-Littlewood maximal function and transcendental number theory (I have a proof but I still don't understand what's going on) and
a complete mystery in an old integer sequence of Stanislaw Ulam ($300 prize for an explanation).

November 16, 4:15 pm
Speaker: Slawomir Solecki (Cornell)
Topic: Logic and homogeneity of the pseudoarc

Abstract: Fraïssé theory is a method of classical Model Theory of producing canonical

limits of certain families of finite structures. For example, the random graph is the Fraïssé limit of the family of finite graphs. It turns out that this method can dualized, with

the dualization producing projective Fraïssé theory, and applied to the study of compact

metric spaces. The pseudoarc is a remarkable compact connected space; it is the generic, in

a precise sense, compact connected subset of the plane or the Hilbert cube. I will explain

the connection between the pseudoarc and projective Fraïssé limits.

November 30, 4:15 pm
Speaker: Dong Xia (Columbia U.)
Topic: Quantum State Tomography via Structured Density Matrix Estimation.

Abstract: The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. Quantum state tomography (QST) refers to the estimation of an unknown density matrix through specifically designed measurements on identically prepared copies of quantum systems. The dimension of the associated density matrix grows exponentially with the size of quantum system. This talk is on the efficient QST when the underlying density matrix possesses structural constraints.

The first part is on the low rank structure, which has been popular in the community of quantum physicists. We develop minimax lower bounds on error rates of estimation of low rank density matrices, and introduce several estimators showing that these minimax lower bounds can be attained up to logarithmic terms. These bounds are established over all the Schatten norms and quantum Kullback-Leibler divergence. This is based on a series of work with Vladimir Koltchinskii.

The second part is built upon decomposable graphical models for quantum multi-qubits system. The goal is to reduce the sample complexity required for quantum state tomography, one of the central obstacles in large scale quantum computing and quantum communication. By considering the decomposable graphical models, we show that the sample complexity is allowed to grow linearly with the system size and exponentially with only the maximum clique size. This is based on a joint work with Ming Yuan.

December 4, 4:40 pm
Speaker: Xuening Zhu (Penn State U.)
Topic: Network Vector Autoregression

Abstract: We consider here a large-scale social network with a continuous response observed for each node at equally spaced time points. The responses from different nodes constitute an ultra-high dimensional vector, whose time series dynamic is to be investigated. In addition, the network structure is also taken into consideration, for which we propose a network vector autoregressive (NAR) model. The NAR model assumes each node’s response at a given time point as a linear combination of (a) its previous value, (b) the average of its connected neighbors, © a set of node-specific covariates, and (d) an independent noise. The corresponding coefficients are referred to as the momentum effect, the network effect, and the nodal effect respectively. Conditions for strict stationarity of the NAR models are obtained. In order to estimate the NAR model, an ordinary least squares type estimator is developed, and its asymptotic properties are investigated. We further illustrate the usefulness of the NAR model through a number of interesting potential applications. Simulation studies and an empirical example are presented.

December 7, 4:15 pm
Speaker: Ziwei Zhu (Princeton U.)
Topic: Estimation of principal eigenspaces with decentralized and incomplete data

Abstract: Modern data sets are often decentralized; they are generated and stored in multiple sources across which the communication is constrained by bandwidth or privacy. Besides, the data quality often suffers from incompletion. This talk focuses on estimation of principal eigenspaces of covariance matrices when data are decentralized and incomplete. We first introduce and analyze a distributed algorithm that aggregates multiple principal eigenspaces through averaging the corresponding projection matrices. When the number of data splits is not large, this algorithm is shown to achieve the same statistical efficiency as the full-sample oracle. We then consider the presence of missing values. We show that the minimax optimal rate of estimating the principal eigenspace has a phase transition with respect to the observation probability, and this rate can be achieved by the principal eigenspace of an entry-wise weighted covariance matrix.

Colloquium 2018-2019

2020-01-28T13:33:00-04:00

Colloquium 2018-2019

Spring 2019

February 4, 4:40 pm (This is a Monday and a different starting time)
Speaker: Fangfang Wang (University of Wisconsin at Madison)
Topic: Statistical Modelling of Multivariate Time Series of Counts

Abstract: In this presentation, I will talk about a new parameter-driven model for non-stationary multivariate time series of counts. The mean process is foPace will pick up now. rmulated as the product of modulating factors and unobserved stationary processes. The former characterizes the long-run movement in the data, while the latter is responsible for rapid fluctuations and other unknown or unavailable covariates. The unobserved stationary vector process is expressed as a linear combination of possibly low-dimensional factors that govern the contemporaneous and serial correlation within and across the count series. Regression coefficients in the modulating factors are estimated via pseudo maximum likelihood estimation, and identification of common factor(s) is carried out through eigen-analysis on a positive definite matrix that aggregates the autocovariances of the count series at nonzero lags. The two-step procedure is fast to compute and easy to implement. Appropriateness of the estimation procedure is theoretically justified, and simulation results corroborate the theoretical findings in finite samples. The model is applied to time series data consisting of the numbers of National Science Foundation funding awarded to seven research universities from January 2001 to December 2012. The estimated parsimonious and easy-to-interpret factor model provides a useful framework for analyzing the interdependencies across the seven institutions.

February 7, 4:00 pm (Please note the earlier time)
Speaker: Yue Zhao (University Leuven)
Topic: The normal scores estimator for the high-dimensional Gaussian copula model

Abstract: The (semiparametric) Gaussian copula model consists of distributions that have dependence structure described by Gaussian copulas but that have arbitrary marginals. A Gaussian copula is in turn determined by an Euclidean parameter $R$ called the copula correlation matrix. In this talk we study the normal scores (rank correlation coefficient) estimator, also known as the van der Waerden coefficient, of $R$ in high dimensions. It is well known that in fixed dimensions, the normal scores estimator is the optimal estimator of $R$, i.e., it has the smallest asymptotic covariance. Curiously though, in high dimensions, nowadays the preferred estimators of $R$ are usually based on Kendall's tau or Spearman's rho. We show that the normal scores estimator in fact remains the optimal estimator of $R$ in high dimensions. More specifically, we show that the approximate linearity of the normal scores estimator in the efficient influence function, which in fixed dimensions implies the optimality of this estimator, holds in high dimensions as well.

February 8, 4:15pm (This is a Friday)
Speaker: Hai Shu (University of Texas MD Anderson Cancer Center)
Topic: Extracting Common and Distinctive Signals from High-dimensional Datasets

Abstract: Modern biomedical studies often collect large-scale multi-source/-modal datasets on a common set of objects. A typical approach to the joint analysis of such high-dimensional datasets is to decompose each data matrix into three parts: a low-rank common matrix that captures the shared information across datasets, a low-rank distinctive matrix that characterizes the individual information within the single dataset, and an additive noise matrix. Existing decomposition methods often focus on the orthogonality between the common and distinctive matrices, but inadequately consider a more necessary orthogonal relationship among the distinctive matrices. The latter guarantees that no more shared information is extractable from the distinctive matrices. We propose decomposition-based canonical correlation analysis (D-CCA), a novel decomposition method that defines the common and distinctive matrices from the L2 space of random variables rather than the conventionally used Euclidean space, with a carefully designed orthogonal relationship among the distinctive matrices. The associated estimators of common and distinctive signal matrices are asymptotically consistent and have reasonably better performance than state-of-the-art methods in both simulated data and the analyses of breast cancer genomic datasets from The Cancer Genome Atlas and motor-task functional MRI data from the Human Connectome Project.

February 13, 4:15 pm (This is a Wednesday)
Speaker: Guifang Fu (Utah State University)
Topic: A New Statistical Framework to Identify Influential Genetic and Environmental Variables Associated with Shape Variation

Abstract: The tremendous diversity of shape is widespread in nature and embodies both a response to and a source of evolution and natural selection. Genes are reported to have an important role in controlling phenotypic variation in shape, and many species exhibit morphological plasticity which allows their shape to adapt to environmental cues. In this talk, I will introduce a new statistical framework to quantify the relative importance of all explanatory variables in terms of the strength of their association with shape variation. The shape is inputted as an image and then described as a multivariate vector or a high dimensional curve. There are unique challenges in variable selection for high dimensional data. Additional challenges arise when modeling multiple correlated components as one unit rather than isolating them one by one, which greatly decreases the prediction error. I will introduce a novel Bayesian multivariate variable selection (BMVS) approach that investigates a large-scale candidate pool to identify influential variables associated with the multivariate shape vector. We integrate the estimation of covariance-related parameters and all regression parameters into one framework through a rapidly updating MCMC procedure. The BMVS approach has been proven to satisfy the strong selection consistency property under certain conditions. We use three simulations to demonstrate that the BMVS approach is empirically accurate, robust, and computationally viable. Numerical comparison indicates that BMVS outperforms some existing approaches such as canonical correlation analysis and multivariate Lasso. We apply the BMVS approach to two rice-related GWAS datasets: the first with 3,254 SNPs related to rice shape, and the second with 36,901 SNPs related to three flowering-time phenotypes. The presented BMVS approach is flexible and can be employed in a wide variety of applications. At the conclusion of the presentation, I will list several future collaboration opportunities that extend from the shape research.

February 14, 4:15 pm
Speaker: Steve Ferry (Binghamton University)
Topic: The “lost tribes” of manifolds (Gromov's joke)

Abstract: This work is joint with John Bryant. In his 1994 ICM talk Shmuel Weinberger, inspired by work of Edwards, Quinn, Cannon, and Bryant-F.-Mio-Weinberger, conjectured the existence of a new collection of spaces with many of the properties of topological manifolds. The authors have constructed spaces in dimensions $n \ge 6$ satisfying many parts of Weinberger's conjecture. Our spaces are finite dimensional and locally contractible. They have the local and global separation properties of topological manifolds, satisfying Alexander duality both locally and globally. They are homogeneous, meaning that for every x and y in a component of one of these spaces there is a homeomorphism carrying x to y. In dimensions $\ge 6$, the h- and s-cobordism theorems hold for these topologically exotic manifolds manifolds.

February 28, 4:15 pm, in the Atrium of Old Champlain Hall
Speaker: Hermann Nicolai (Max Planck Institute for Gravitational Physics, Potsdam, Germany)
Topic: Symmetry and Unification – can physics be unified into a single formula?

Abstract: Attempts to unify the known laws of Nature have a long history. Since the mid-seventies, these attempts have been reinforced with ongoing efforts to reconcile Quantum Mechanics and Einstein's theory of General Relativity into a single unified theory of quantum gravity. In this talk I will review the motivation as well as some more recent developments at an introductory level and discuss prospects for the future.

April 4, 3:00 pm LH 9. - Peter Hilton Memorial Lecture and Dean's Speaker Series in Geometry, Geometric Analysis, and Topology
Speaker: Shmuel Weinberger (University of Chicago)
Title: How hard is algebraic topology? Between the constructive and the non.

Abstract: In algebraic topology one studies geometric problems and problems of constructing and deforming highly nonlinear functions by means of algebra. If one knows that two maps are homotopic (i.e. can be deformed to one another) because a certain calculation says they both lie in the trivial group, then what has one learned? (A striking example of this is Smale's turning the sphere inside out, which now can be seen after much highly nontrivial effort, on youtube.) The question I shall discuss is how hard is it to understand what the algebraic topologists tell us.

April 9, 1:15 pm (SPECIAL DAY, PLACE AND TIME) WH-100E, joint with Combinatorics Seminar
Speaker: Gregory Warrington (University of Vermont)
Topic: Gerrymandering

Abstract: http://seminars.math.binghamton.edu/ComboSem/abstract.201904war.html

—–
===== Fall 2018 =====
October 18, 4:15 pm
Speaker: Jingchen (Monika) Hu (Vassar College)
Topic: Teaching Undergraduate Upper Level Statistics Courses through a Shared/Hybrid Model

Abstract: At liberal arts colleges, with smaller number of students taking statistics, offering advanced level courses can be difficult. Under the Liberal Arts Consortium for Online Learning (LACOL) Upper Level Math Project, I taught an elective course (Bayesian Statistics) through a shared/hybrid model in Fall 2017. Lectures were given in classroom at Vassar with Vassar students present. Each lecture was recorded and shared with both Vassar students and students from other campuses taking the course as an independent study course with a local faculty liaison. I would love to share my experience and thoughts, focusing on 1) what material to move online and how to do so, and 2) how to build up a cross-campus learning community.

December 6, 4:15 pm
Speaker: C. Sastry Aravinda (Binghamton University)
Topic: Harish-Chandra – The Mathematician and the artist

Abstract: This will be a non-technical talk on the life of Harish-Chandra mainly focussing on his singular evolution as one of the most impactful mathematicians of the last century. The talk should be accessible to a wide audience.

—–

Colloquium 2019-2020

2022-04-28T20:41:44-04:00

Colloquium 2019-2020

Unless stated otherwise, colloquia are scheduled for Thursdays 4:15-5:15pm in WH-100E with refreshments served from 4:00-4:15 pm in WH-102.

Organizers: Vladislav Kargin, Cary Malkiewich, Anton Schick, and Adrian Vasiu

Spring 2020

Friday January 31, 4:30-5:30pm, WH-100E (NOTE SPECIAL DATE AND TIME)
Speaker: Daniel Studenmund (University of Notre Dame)
Topic: Hidden symmetries of groups

Abstract: Many infinite discrete groups fail to have nice properties only because of obstructions that disappear on passage to a finite-index subgroup. Examples of such properties include superrigidity of representations of linear groups and nilpotence of groups of polynomial growth. The collection of all finite-index subgroups of a fixed group Gamma has algebraic and geometric structures that can reflect properties of Gamma. We will discuss some of these structures, including the abstract commensurator of Gamma and commensurator growth of Gamma.

Thursday February 6, 4:15-5:15pm
Speaker: Selim Sukhtaiev (Rice University)
Topic: Anderson localization for disordered quantum graphs

Abstract: Disorder is one of the central topics in modern science. In this talk, we will discuss a mathematical treatment of a particular disordered system modeling localization of quantum waves in random media. The model in question was introduced by P. W. Anderson in his Nobel prize winning work in physics which led to a rich mathematical theory of random Schrodinger operators. We will show that the transport properties of several natural Hamiltonians on metric trees with random branching numbers are suppressed by disorder. This phenomenon is called Anderson localization.

Friday February 7, 4:30-5:30pm (NOTE SPECIAL DATE AND TIME)
Speaker: Li Chen (University of Connecticut)
Topic: On several functional inequalities for Markov semigroups and their applications

Abstract: Markov semigroups lie at the interface of analysis, PDEs, probability and geometry. Markov semigroup techniques, from both analytic and probabilistic viewpoints, have important applications in the study of functional inequalities coming from harmonic analysis, PDEs and geometry.

In this talk, we discuss regularization properties of heat semigroups and their applications to the study of Sobolev type inequalities, isoperimetric inequalities and $L^p$ boundedness of Riesz transforms in different geometric settings. Fractal examples without differential structures are emphasized. Besides, we also discuss sharp and dimension-free $L^p$ bounds of singular integral operators via the martingale transform method.

Monday February 10
Speaker: Benjamin Schmidt (Michigan State University)
Topic: Preserve one, preserve all: Aleksandrov's problem in the context of Riemannian spaces.

Abstract: A classical theorem of Beckman and Quarles asserts that a function $F$ from a Euclidean space of dimension at least two to itself and having the property that $||F(x)-F(y)||=1$ whenever $||x-y||=1$ is necessarily an isometry. Aleksandrov has been credited with the problem of determining those metric spaces having this “preserve one distance, then preserve all distances” property.

Examples show that Riemannian manifolds need not have this property. However, it is expected that self-functions of complete Riemannian manifolds that preserve a sufficiently small distance are isometries. I'll formulate a precise conjecture and will discuss supporting results proved jointly with Meera Mainkar.

Thursday March 19, 4:15-5:15pm, WH-100E
Speaker: Kathryn Mann (Cornell University)
Topic: TBA

Abstract: TBA

Thursday April 23, 4:15-5:15pm, WH-100E
Speaker: Thomas Hartman (Cornell University)
Topic: TBA

Abstract: TBA

PETER HILTON MEMORIAL LECTURE
Thursday April 30, 3:00-4:00pm, LH009 (NOTE SPECIAL TIME AND LOCATION)
Speaker: Robert Gompf (University of Texas at Austin)
Topic: Exotic Smooth Structures on $\mathbb R^4$

Abstract: One of the most surprising discoveries in 4-manifold topology was the existence of smooth manifolds homeomorphic, but not diffeomorphic, to Euclidean 4-space. For fundamental reasons, this phenomenon can only occur in 4 dimensions. We will survey the subject, from its origin to recent developments regarding symmetries of such manifolds.

Archive:

2014-2015

2015-2016

2016-2017

2017-2018

2018-2019

Colloquium 2020-2021

2022-04-28T20:42:48-04:00

Colloquium 2020-2021

Unless stated otherwise, colloquia are scheduled for Thursdays 4:15-5:15pm in WH-100E with refreshments served from 4:00-4:15 pm in WH-102.

Organizers: Vladislav Kargin, Cary Malkiewich, Anton Schick, and Adrian Vasiu

Spring 2021

A Special Event: Professor Ken Ono (University of Virginia), who is a Phi Beta Kappa Visiting Scholar this year, is virtually visiting our department on March 11-12. He will present three lectures via zoom, a Math Club Talk for undergraduate and graduate students with interest in mathematics, a Colloquium Talk for our faculty and graduate students, and a Public Lecture intended for a general audience. Details of these talk are given below. The talks are sponsored by our local Phi Beta Kappa Chapter. Many thanks to Professor Alex Feingold for organizing this.

Thursday March 11, 2:50-3:50pm, THIS IS THE MATH CLUB TALK Zoom Link
Speaker: Ken Ono (University of Virginia)
Topic: What is the Riemann Hypothesis, and why does it matter?

Abstract: The Riemann hypothesis provides insights into the distribution of prime numbers, stating that the nontrivial zeros of the Riemann zeta function have a “real part” of one-half. A proof of the hypothesis would be world news and fetch a $1 million Millennium Prize. In this lecture, Ken  Ono will discuss the mathematical meaning of the Riemann hypothesis and why it matters. Along the way, he will tell tales of mysteries about prime numbers and highlight new advances.

Thursday March 11, 4:30-5:30pm, THIS IS THE COLLOQUIUM TALK Zoom Link
Speaker: Ken Ono (University of Virginia)
Topic: Gauss’ Class Number Problem

Abstract: In 1798 Gauss wrote Disquisitiones Arithmeticae, the first rigorous text in number theory. This book laid the groundwork for modern algebraic number theory and arithmetic geometry. Perhaps the most important contribution in the work is Gauss's theory of integral quadratic forms, which appears prominently in modern number theory (sums of squares, Galois theory, rational points on elliptic curves,L-functions, the Riemann Hypothesis, to name a few). Despite the plethora of modern developments in the field, Gauss’s first problem about quadratic forms has not been optimally resolved. Gauss's class number problem asks for the complete list of quadratic form discriminants with class number h. The difficulty is in effective computation, which arises from the fact that the Riemann Hypothesis remains open. To emphasize the subtlety of this problem, we note that the first case, where h=1, remained open until the 1970s. Its solution required deep work of Heegner and Stark, and the Fields medal theory of Baker on linear forms in logarithms. Unfortunately, these methods do not generalize to the cases where h>1. In the 1980s, Goldfeld, and Gross and Zagier famously obtained the first effective class number bounds by making use of deep results on the Birch and Swinnerton-Dyer Conjecture. This lecture will tell the story of Gauss’s class number problem, and will highlight new work by the speaker and Michael Griffin that offers new effective results by different (and also more elementary) means.

Friday March 12, 4:00-5:00pm, THIS IS THE PUBLIC LECTURE
Speaker: Ken Ono (University of Virginia)
Topic: Why does Ramanujan, “The Man Who Knew Infinity”, matter?
Since this talk is open to the general public, we require registration in advance for this meeting: Use this link to preregister. After registering, you will receive a confirmation email containing information about joining the meeting.

Abstract: This lecture is about Srinivasa Ramanujan, “The Man Who Knew Infinity.” Ramanujan was a self-trained two-time college dropout who left behind 3 notebooks filled with equations that mathematicians are still trying to figure out today. He claimed that his ideas came to him as visions from an Indian goddess. This lecture gives many reasons why Ramanujan matters today. The answers extend far beyond his legacy in science and mathematics. The speaker was an Associate Producer of the film “The Man Who Knew Infinity” (starring Dev Patel and Jeremy Irons) about Ramanujan. He will share several clips from the film in the lecture, and will also tell stories about the production and promotion of the film.

Archive:

2014-2015

2015-2016

2016-2017

2017-2018

2018-2019

2019-2020

Colloquium 2021-2022

2022-10-12T12:46:40-04:00

Colloquium 2021-2022

Unless stated otherwise, colloquia are scheduled for Thursdays 4:15-5:15pm in WH-100E with refreshments served from 4:00-4:15 pm in WH-102.

Organizers: Vladislav Kargin, Cary Malkiewich, Anton Schick, and Adrian Vasiu

Spring 2022

Friday May 6, 3:45-4:45pm, WH-100E (NOTE SPECIAL DATE AND TIME)
Speaker: Minghao W. Rostami (Syracuse University)
Topic: Biofluid as a big data challenge

Abstract: The simulation of a fluid around dynamic biological structures, such as bacteria and cilia, entails solving large systems of Partial Differential Equations (PDEs) and Ordinary Differential Equations (ODEs). This boils down to working with large-scale, dense matrices with very few zero entries. We first show that these matrices are “data sparse”, that is, the large amount of data stored in them can be significantly compressed. We then present fast algorithms for matrix-vector multiplication and linear solves involving these matrices. Our methods do not require constructing the large dense matrices and can achieve huge savings in storage and time. We also show how parallel-in-time methods can further speed up the simulation of a biofluid when spatial parallelization “saturates”. In addition, a data-driven, reduced order model discovered by deep learning will be discussed. It allows us to describe the movement of a fluid without using complex PDEs.

Archive:

2014-2015

2015-2016

2016-2017

2017-2018

2018-2019

2019-2020

Colloquium 2022-2023

2024-01-28T09:57:15-04:00

Colloquium 2022-2023

Unless stated otherwise, colloquia are scheduled for Thursdays 4:15-5:15pm in WH-100E with refreshments served from 4:00-4:15 pm in WH-102.

Organizers: Vladislav Kargin, Cary Malkiewich, Hung Tong-Viet, and Adrian Vasiu

Fall 2022

Thursday November 3, 4:15-5:15pm, WH-100E
Speaker: Avy Soffer (Rutgers University)
Topic: The Asymptotic States of Nonlinear Dispersive Equations with Large Initial Data and General Interactions

Abstract: I will describe a new approach to scattering theory, which allows the analysis of interaction terms which are linear and space-time dependent, and nonlinear terms as well. This is based on deriving (exterior) propagation estimates for such equations, which micro-localize the asymptotic states as time goes to infinity. In particular, the free part of the solution concentrates on the propagation set (x=vt), and the localized leftover is characterized in the phase-space as well. The NLS with radial data in three dimensions is considered, and it is shown that besides the free asymptotic wave, in general, the localized part is smooth, and is localized in the region where |x|^2 is less than t. Furthermore, the localized part has a massive core and possibly a halo which may be a self-similar solution. This work is joint with Baoping Liu. This is then followed by new results on the non-radial case and Klein-Gordon equations (Joint works with Xiaoxu Wu).

Thursday November 17, 2:50-3:50pm, WH-100E
Speaker: Emmett Wyman (University of Rochester)
Topic: Improved Weyl law remainders for products of spheres

Abstract: The Laplacian is a fundamental operator in mathematics. It arises, for example, in the heat, wave, and Schrödinger equations. Its eigenfunctions can be viewed as natural vibrational modes and their eigenvalues their respective frequencies.

A fundamental problem in harmonic analysis is to estimate the number $N(\lambda)$ of Laplace eigenvalues less than or equal to $\lambda$, counting multiplicity. The Weyl law gives a main term + remainder term estimate for $N(\lambda)$. The remainder is sharp in general but generically can be improved. It is hard to obtain polynomially improved remainders–let alone optimal remainders–except for a handful of very nice examples.

In this talk, we will discuss the remainder of the Weyl law for two classic cases: the sphere and the torus. We will then compare these results through the lense of the dynamics of the geodesic flow via the Duistermaat-Guillemin theorem. Finally, I will present recent joint work with Iosevich which explores the prospect of obtaining polynomially improved Weyl remainders for products of manifolds, which we illustrate by obtaining polynomially improved remainders for products of spheres.

Tuesday November 22, 4:15–5:15pm, WH-100E
Speaker: Alexander Dunlap (NYU Courant)
Topic: Stochastic partial differential equations in supercritical, subcritical, and critical dimensions

Abstract: A pervading question in the study of stochastic PDE is how small-scale random forcing in an equation combines to create nontrivial statistical behavior on large spatial and temporal scales. I will discuss recent progress on this topic for several related stochastic PDEs - stochastic heat, KPZ, and Burgers equations - and some of their generalizations. These equations are (conjecturally) universal models of physical processes such as a polymer in a random environment, the growth of a random interface, branching Brownian motion, and the voter model. The large-scale behavior of solutions on large scales is complex, and in particular, depends qualitatively on the dimension of the space. I will describe the phenomenology, and then describe several results and challenging problems on invariant measures, growth exponents, and limiting distributions.

Wednesday November 30, 4:30–5:30pm, WH-100E
Speaker: Anibal Medina-Mardones (Universite Sorbonne Paris Nord)
Topic: Effective Algebro-Homotopical constructions and their applications

Abstract: It is necessary in order to incorporate ideas from homotopy theory into concrete contexts – such as topological data analysis and topological lattice field theory – to have effective constructions of concepts defined only indirectly or transcendentally. In groundbreaking work, Mandell showed that the entire homotopy type of a space was encoded in the quasi-isomorphism type of its cochains enhanced with an $E_\infty$-structure. In this talk, we will present a concrete construction of such structure by explicitly restoring up to coherent homotopies the broken symmetry of the diagonal of cellular spaces, and, on the way, we will explore connections of these ideas to data science, theoretical physics, knot theory, higher category theory, and convex and toric geometry.

Friday December 2, 4:30–5:30pm, WH-100E
Speaker: Beibei Liu (MIT)
Topic: The critical exponent: old and new

Abstract: The critical exponent is an important numerical invariant of discrete isometry groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete isometry groups acting on hyperbolic spaces. In particular, I will explain how the numerical invariant is closely related to geometry, dynamics, and representation of the group action on hyperbolic spaces.

Monday December 5, 4:30–5:30pm, WH-100E
Speaker: Jia Zhao (Utah State University)
Topic: Physics-informed Computational Modeling of Multiphase Complex Fluids with Applications in Life Science

Abstract: Complex fluids are ubiquitous in nature and in synthesized materials, such as biofilms, cytoplasm, mucus, synthetic and biological polymeric solutions. Modeling and simulation of complex fluids have been listed as one of the 21st-century mathematical challenges by DARPA, which is therefore of great mathematical and scientific significance. In this talk, I will first explain our research motivations by introducing several complex fluids examples and traditional modeling techniques. We propose physics- informed PDE models for multiphase complex fluid flows by integrating the generalized Onsager Principle and phase-field approaches. Then, I will introduce a numerical analysis platform for developing accurate, efficient, and structure-preserving numerical approximations for solving complex-fluid PDE models. The computational modeling strategy is rather general in that it can be applied to investigate a host of complex-fluid problems. Finally, I will present several applications in life science with our modeling and numerical analysis toolkits.

Wednesday December 7, 4:30–5:30pm, WH-100E
Speaker: Minghao W. Rostami (Syracuse University)
Topic: Learning from data: a beginner’s journey

Abstract: The behavior of a complex physical system is often modeled by Partial Differential Equations (PDEs) derived from fundamental physical laws. We can apply various numerical methods to the PDEs to make predictions about the physical system, uncover the causal factors behind observed behaviors of the system, and optimize the design of the system for a task at hand. When sufficient data and computing resources are available, we can also make discoveries by processing data instead. The speaker, as a classically trained applied mathematician who is “learning the ropes” of this alternative approach, will review and discuss the basics of PDE models, numerical methods, and machine learning models. A new data-driven model for calculating particle trajectories in a fluid flow will also be presented, which, unlike conventional methods, does not entail costly flow computations. This is joint work with Jianchen Wei (PhD student), Lixin Shen from Syracuse University and Melissa Green from University of Minnesota, Twin Cities.

Colloquium 2023-2024

2025-09-30T09:42:49-04:00

Colloquium 2023-2024

Spring 2024

Thursday Feb 29 4:15-5:15pm, WH-100E
Speaker: Alex Iosevich (University of Rochester)
Topic: Signal recovery, uncertainty principles and Fourier restriction theory

Abstract: We are going to consider functions $f: {\mathbb Z}_N \to {\mathbb C}$ and view them as signals. Suppose that we transmit this signal via its Fourier transform $$\widehat{f}(m)=\frac{1}{N} \sum_{x=0}^{N-1} e^{-\frac{2 \pi i xm}{N}} f(x),$$

and that the values of $\widehat{f}(m), m \in S$, are lost. Under what circumstances is it possible to recover the original signal? We shall see how this innocent question quickly leads us into the deep dark forest of Fourier analysis.

Thursday March 14 4:15-5:15pm, WH-100E
Speaker: John Klein (Wayne State University)
Topic: On a rationality problem in quantum information theory

Abstract: In this talk, I shall consider the case of quantum systems consisting of n parties, in which each party is in possession of a qubit, i.e., a two dimensional complex vector space. Each qubit is allowed to evolve independently, and the group G of local symmetries governs the evolution of the n-qubit system.

My goal will be to provide a complete description of the field G-invariant complex valued functions on the space of mixed states of this quantum system. Such functions are to be viewed as detailed measures of entanglement.

PETER HILTON MEMORIAL LECTURE
Thursday April 11th 3:00pm-4:00pm, LH-009
Speaker: Alex Eskin (University of Chicago)
Topic: Polygonal Billiards and Dynamics on Moduli Spaces

Abstract: Billiards in polygons can exhibit bizarre behavior, some of which can be explained by deep connections to several seemingly unrelated branches of mathematics. These include algebraic geometry, Teichmuller theory and ergodic theory on homogeneous spaces. The talk will be an introduction to these ideas, aimed at a general mathematical audience.