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research

The five main areas of concentration are **Algebra**, **Analysis**, **Combinatorics**, **Geometry/Topology**, and **Statistics/Probability**.

**Algebraic and Geometric Topology**

- John Abou-Rached (VAP)
- Ross Geoghegan (emeritus)

**Algebra related to logic and computer science**

- Fernando Guzman (emeritus)

**Applied and Computational Mathematics**

**Arithmetic Algebraic Geometry**

**Combinatorial Geometry**

**Combinatorics**

- Quaid Iqbal (VAP)

**Geometric Analysis**

**Geometric Group Theory**

- John Abou-Rashed (VAP)
- Matthew Brin (emeritus)
- Ross Geoghegan (emeritus)
- Fernando Guzman (emeritus)

**Geometry of Manifolds**

**Group Theory**

- Benjamin Brewster (emeritus)
- Matthew Brin (emeritus)
- Ross Geoghegan (emeritus)
- Fernando Guzman (emeritus)
- Luise-Charlotte Kappe (emeritus)
- Inna Sysoeva (visiting)

**Machine Learning**

**Number Theory**

- Luise-Charlotte Kappe (emeritus)

**Partial Differential Equations**

**Probability**

**Representation Theory**

**John Abou-Rached** - Robert Riley Visiting Assistant Professor

**Areas of Interest:**

**Description:**

**Laura Anderson** - Associate Professor

**Areas of Interest:** Combinatorics, Topology

**Description:** My research focuses on interactions between combinatorics and topology,
particularly those involving oriented matroids, convex polytopes, and other
concepts from discrete geometry. Much of my work involves combinatorial
models for topological structures such as differential manifolds and vector
bundles. The aims of such models include both combinatorial answers to
topological questions (e.g., combinatorial formulas for characteristic
classes), and topological methods for combinatorics (e.g. on topology of
posets).

**Robert Bieri** - Visiting Professor

**Areas of Interest:** Geometric, homological, combinatorial and asymptotic methods in group theory

**Description:** My original interest in homological methods for
infinite groups (cohomological dimension and Poincare type duality) shifted
towards geometric and -- more recently -- asymptotic methods. I find it
interesting to relate geometric properties at infinity of groups and G-spaces
with algebraic properties of these groups, their group rings and their
modules. The focus is on familar groups like metabelian, soluble, free and
linear ones, or fundamental groups of 3-manifolds, but I also met Thompson's
group F and other PL-homeomorphism groups on the way, and had an encounter
with tropical geometry.

**Alexander Borisov** - Associate Professor

**Areas of Interest:**

**Description:** My general research area is algebraic geometry and number theory, broadly interpreted. Particular topics of interest include birational geometry, toric geometry and convex discrete geometry, polynomial morphisms, integer polynomials, Arakelov geometry.

**Walter Carlip** - Visiting Associate Professor

**Areas of Interest:**

**Description:**

**Zeyu Ding** - Assistant Professor (by courtesy)

**Areas of Interest:** Data privacy, statistical disclosure control, formal methods, machine learning

**Description:** My research lies in the intersection of data privacy, statistical disclosure control, formal methods and machine learning. The overarching goal of my work is to protect sensitive personal information from being leaked in unintended ways. My current research focuses on differential privacy and its interactions with formal verification, numerical optimization, privacy-preserving statistical inference and machine learning.

**Michael Dobbins** - Associate Professor

**Areas of Interest:**

**Description:** My research is primarily on discrete geometry, particularly geometric problems arising from computer science. I also work in computational geometry, complexity, convexity, combinatorics, and topology.

**Yuan Fang** - Assistant Professor (by courtesy)

**Areas of Interest:** Statistics, bioinformatics

**Description:** My current research focuses on studying lipid profiles for ceramide pathways in boys with Duchenne Muscular Dystrophy using multi-omics statistical and bioinformatics approaches. I am also interested in extending existing models and statistical approaches for clustering longitudinal data.

**Alex Feingold** - Professor

**Areas of Interest:** Algebra, Lie algebras, conformal field theory

**Description:** Finite dimensional semisimple Lie algebras, tensor product
decomposition of irreducible modules, representation theory of
the infinite dimensional Kac-Moody Lie algebras, bosonic and
fermionic creation and annihilation operators, affine and
hyperbolic Kac-Moody algebras, topics in combinatorics, power
series identities, modular forms and functions, Siegel modular
forms, conformal field theory, string theory, and statistical
mechanical models, vertex operator algebras, their modules and
intertwining operators, the theory of fusion rules.

**Guifang Fu** - Associate Professor

**Areas of Interest:** Statistics, high-dimensional inference, functional data analysis

**Description:** My main focus is to develop advanced statistical models and computational methodologies to unravel the genetic and environmental mechanisms that regulate complex biological traits, including morphology/shape, biomedical problems and disease. I am particularly interested in high-dimensional, "big data" modeling, and functional data analysis. My genetic leaf shape project was awarded a three-year NSF grant. I enjoy collaborating on interdisciplinary projects, working with researchers from the application domains and addressing real-life data analysis questions.

**James Hyde** - Assistant Professor

**Areas of Interest:**

**Description:**

**Quaid Iqbal** - Robert Riley Visiting Assistant Professor

**Areas of Interest:**

**Description:**

**Dikran Karagueuzian** - Associate Professor

**Areas of Interest:** Algebraic topology, representation theory, group cohomology

**Description:** My research for the past few years has been primarily in the
representations and cohomology of finite groups. For the past few
years I have been studying problems in algebra that arise from
techniques of algebraic topology. Sometimes there is a theorem
hidden behind the feasibility of a well-known method. An example
of this phenomenon is my most recent preprint, written in
collaboration with Peter Symonds of the University of Manchester
Institute of Science and Technology. In this case the theorem
was uncovered through exploration with the computer algebra
package Magma, which is well worth checking out. Often such
software lets us investigate mathematical phenomena which would
be very difficult to understand otherwise.

**Vladislav Kargin** - Associate Professor

**Areas of Interest:**

**Description:** I am particularly interested in random matrices and its applications, in particular, statistical analysis of large data, zeroes of zeta functions, statistical mechanics of random media, and free probability.

**Tae Young Lee** - Robert Riley Visiting Assistant Professor

**Areas of Interest:**

**Description:**

**Paul Loya** - Associate Professor

**Areas of Interest:** Global and geometric analysis, Elliptic theory of differential operators on manifolds with singularities, Partial differential equations

**Description:** The underlying theme of my research is the investigation of topological,
geometric, and spectral invariants of (singular) Riemannian manifolds using
techniques from partial differential equations. For example, the Euler
characteristic of a surface is a topological invariant based its usual
definition in terms of a triangulation of the surface. However, it may also
be considered geometric in view of the Gauss-Bonnet theorem or spectral in
view of the Hodge theorem. I am interested in such relationships on general
singular Riemannian manifolds.

**Cary Malkiewich** - Associate Professor

**Areas of Interest:** Algebraic topology, especially stable homotopy theory, algebraic K-theory, applications to manifolds and cell complexes.

**Description:** My primary research area is algebraic topology. I like to apply stable homotopy theory (spectra) to questions about manifolds and cell complexes. My work has taken a recent turn towards scissors congruence: in 2002 I proved that it is described by a Thom spectrum, and I am developing the consequences of this surprising result for the higher scissors congruence groups.

**Marcin Mazur** - Professor

**Areas of Interest:** Algebraic number theory, group theory

**Description:** My research interests concentrate around areas where number theory and group
theory intersect. Topics of particular interest are group rings, group schemes
over rings of algebraic integers, Galois module structures and Galois
representations.

**Ryan McCulloch** - Visiting Associate Professor

**Areas of Interest:** Group theory, combinatorics

**Description:** My research interests are in general group theory, finite group theory, and related structures such as lattices of subgroups of a group. I am also interested in combinatorics, and have recently been looking at relationships between designs and other combinatorial objects.

**Pedro Ontaneda** - Distinguished Professor

**Areas of Interest:** Topology and differential geometry

**Description:** My general interest is the geometry and topology of aspherical spaces.
I have done some work in the study of the relationship between exotic
structures and (negative, non-positive) curvature, and its applications
to the limitations of PDE methods in geometry. Other interests: geometric
group theory, K-theory, mechanics.

**Aleksey Polunchenko** - Associate Professor

**Areas of Interest:** Statistics, sequential analysis.

**Description:** Mathematical
statistics and specifically the problem of sequential (quickest)
change-point detection, currently focusing on the case of composite
hypotheses.

**Xingye Qiao** - Professor and Chair

**Areas of Interest:** Statistics, machine learning, causal inference

**Description:** My research interests encompass statistics, machine learning, and data science. I develop and analyze predictive and inferential tools for complex data problems such as imbalanced classes, high-dimensional data, transfer learning, and observational studies. My focus is on designing theoretically sound and efficient learning algorithms that address sample, time, and space complexity challenges.
I aim to enhance the trustworthiness and reliability of statistics and machine learning methods, particularly in critical domains like healthcare. My work includes developing user-friendly prediction tools with built-in confidence measures and methods for individualized estimation, prediction, and recommendation from observational and interventional data.

**David Renfrew** - Assistant Professor

**Areas of Interest:**

**Description:** My research lies in Probability and Random Matrices. I am particularly interested in non-Hermitian random matrices and the interplay between random matrices and free probability. I am also interested in applications to biologic systems.

**Minghao Rostami** - Associate Professor

**Areas of Interest:**

**Description:**

**Lorenzo Ruffoni** - Assistant Professor

**Areas of Interest:** Algebraic and Geometric Topology, Geometry of Manifolds, Geometric Group Theory

**Description:** I am interested in Geometry and Topology, and in particular: geometric group theory, geometric structures on manifolds and cell complexes.

**Eugenia Sapir** - Assistant Professor

**Areas of Interest:**

**Description:**

**Anton Schick** - Professor

**Areas of Interest:** Statistics, probability

**Description:** Uses of large sample theory in statistics, the characterization
and construction of efficient estimators and tests for
semiparametric and nonparametric models, statistical inference
for Markov chains and stochastic processes, estimation and
comparison of curves, the behavior of plug-in estimators,
optimal inference for bivariate distributions with constraints on
the marginal, modelling with incomplete data, and the theory and
application of finite and infinite order U-statistics.

**Rakhi Singh** - Assistant Professor

**Areas of Interest:** Statistics, machine learning, design of experiments, subdata selection

**Description:** I am interested in gaining a deeper understanding of the high-dimensional stochastic processes with both small and large data sizes. My current and future research focuses on developing this understanding by means of (a) design and analysis of screening experiments, implying the efficient planning and execution of high-dimensional experiments, and (b) developing statistically- and computationally-efficient subdata selection tools, facilitating the extraction of vital insights from large and ever-growing datasets while maintaining quality. In particular, the designs used for the screening experiments frequently exhibit constrained practical applicability due to (a) a disconnect between existing design selection criteria and the methods used for their analysis, (b) their traditional construction and analysis within the confines of the main effects model, despite real experimental dynamics being influenced by interactions, and (c) a scarcity of simulation and empirical studies illustrating the contexts wherein these designs hold significance versus when they do not. I focus on mitigating these challenges by means of developing the efficient design and analysis of screening experiments. I harness high-performance computing to develop analytically rigorous tools for advanced "design of experiments" and "sampling" to better understand these high-dimensional stochastic processes.

**Daniel Studenmund** - Assistant Professor

**Areas of Interest:**

**Description:** My research addresses questions arising at the intersection of geometric group theory and the study of discrete subgroups of Lie groups. I am particularly interested in invariants associated to the collection of finite-index subgroups of a given group G. One example is the abstract commensurator Comm(G), the group of all isomorphisms between finite-index subgroups of G, modulo equivalence. Other examples are growth rates of various functions associated to the collection of finite-index subgroups, which can be thought of as helping to quantify residual finiteness of G. I also study other invariants of groups, such as superrigidity and cohomology of arithmetic groups, using algebraic and geometric methods.

**Hung Tong-Viet** - Professor

**Areas of Interest:** Representation theory and character theory of finite groups, permutation groups and abstract finite groups.

**Description:** My main research interests lie in the representation and character theory of finite groups, permutation groups and applications to number theory and combinatorics, and finite group theory in general. I am interested in studying groups or group structures using several important numerical invariants of the groups such as character degrees (ordinary and modular), p-parts of the degrees or character values such as zeros of characters. In permutation group theory, I study derangements, that is, permutations without fixed points, and their applications in number theory and graph theory, permutation characters and permutation polytopes. Recently, I am also interested in studying the influence of real conjugacy class sizes on the group structures.

**Tan Nhat Tran** - Robert Riley Visiting Assistant Professor

**Areas of Interest:** Algebraic Combinatorics, Hyperplane Arrangement

**Description:** I am dedicated to research in combinatorics, and especially its connections with commutative algebra, algebraic topology and probability theory. My research over the past few years has focused on the theory of arrangements of hyperplanes, especially how the combinatorial properties of hyperplane arrangements interact with the discrete geometric structures (e.g., graph, polytope, root system), topological objects (e.g., Poincare ? polynomial, CW-complex), algebraic concepts (e.g., logarithmic derivation, Hopf algebra) and probabilistic models (e.g., expectation, vine copula).

**Danika Van Niel** - Robert Riley Visiting Assistant Professor

**Areas of Interest:**

**Description:**

**Adrian Vasiu** - Professor

**Areas of Interest:** Arithmetic Algebraic Geometry

**Description:** My area of research is Arithmetic Algebraic Geometry, which is the common
part of Number Theory, Algebra, and Geometry. I am very much interested in
moduli spaces, group schemes, Lie algebras, formal group schemes,
representation theory, cohomology theories, Galois theory, and the
classification of projective, smooth, connected varieties.
My research is focused on:

- Shimura varieties of Hodge type (which are moduli spaces of polarized abelian varieties endowed with Hodge cycles),
- arithmetic properties of abelian schemes,
- classification of $p$-divisible groups,
- representations of Lie algebras and reductive group schemes,
- crystalline cohomology of large classes of polarized varieties, and
- Galois representations associated to abelian varieties.

- Harmonic Analysis on Manifolds: eigenfunction estimates and multiplier problems on Riemannian manifolds, Gibbs' phenomenon and Pinsky's phenomenon for Fourier inversion and eigenfunction expansion.
- Nonlinear differential equations: Well-posedness problems for nonlinear hyperbolic differential equations on manifolds; Boundary stabilization, controllability problems for (linear and nonlinear) parabolic and hyperbolic PDE's on manifolds; Periodic solutions, subharmonics and homoclinic orbits

- Survival analysis. Since 1987, I have been working in this field, in particular on modeling the interval censored data, studying consistency and asymptotic normality of the generalized maximum likelihood estimator (MLE) of survival function or the semi-parametric estimator under linear regression model.
- Statistical decision theory. My thesis was on admissibility and minimaxity of the best invariant estimator of a distribution function.
- Probability model and computing methods for pattern recognition in the Genome project.

- mathematical physics: the long time behavior of Schrodinger-type equations, relations between quantum and PDE models, non-equilibrium statistical mechanics,
- geometric analysis: mean curvature flow and Ricci flows by methods different from the classical ones, formation of singularities in finite time, flow through singularities.

2024/07/17 12:48 · qiao

research.txt · Last modified: 2024/08/08 08:16 by qiao

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