**Problem of the Week**

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**Hilton Memorial Lecture**

seminars:colloquium

Unless stated otherwise, colloquia are scheduled for Thursdays 4:15-5:15pm in WH-100E with refreshments served from 4:00-4:25 pm in WH-102.

Organizers: Adrian Vasiu, Anton Schick and Vladislav Kargin

Coming up:

**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.

**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

**April 19, 4:15 pm**

*Speaker*: ** William Volterman ** (Syracuse University)

*Topic*: ** CBA**

*Abstract*: (TBA).

**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.

**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.

Archive:

seminars/colloquium.txt · Last modified: 2018/03/15 17:38 by kargin

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