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Algebra and Number Theory

A wide array of mathematical research consists of topics in the area of “Algebra.” Subjects such as algebraic geometry, number theory, and representation theory are ubiquitous in all of mathematics, and have innumerable applications in the natural sciences, social sciences, computer science, and more. Our research covers number theory, representation theory, and differential algebra. Some specializations within these research areas that we are more focused on are the theory of elliptic curves, automorphic forms, the Langlands program, and the algebraic theory of differential equations.

Moshe Adrian Langlands Program. Number theory and representation theory

Delaram Kahrobaei Combinatorial and Computational Group Theory, Algebraic Post-Quantum Cryptography, Mathematics of Artificial Intelligence

Krysztof Klosin Automorphic Forms, Paramodular Conjecture, Bloch-Kato Conjecture, Galois representations, special L-values

Kenneth Kramer Abelian Varieties, Galois Representations

Alexey Ovchinnikov Differential Galois Theory, Symbolic Computation

Maria Sabitova Algebraic Number Theory, Abelian Varieties

Combinatorics

Combinatorics is the study of discrete objects and counting techniques. Counting questions arise in all areas of mathematics, especially in algebra, representation theory, and number theory. Combinatorics is also intricately entwined with computer science – both because of the discrete nature of data structures and because computer exploration allows combinatorists to generate data that informs proof techniques. We do research in enumerative and algebraic combinatorics, especially the study of flow polytopes, Ehrhart Theory, and affine Coxeter groups.

Christopher Hanusa Algebraic and Enumerative Combinatorics, Mathematical Art

Complex Analysis

The classical theory of functions of a complex variable was a crowning achievement of mathematics in the 19th and early 20th century. The modern theory continues that rich tradition by introducing new powerful tools and establishing deep connections with other fields of mathematics such as PDE’s, differential geometry, low-dimensional topology, and dynamics. A notable example is Teichmüller theory, the study of deformations of complex structures on Riemann surfaces, which has become a fertile area of research largely due the pioneering works of Ahlfors, Bers, Sullivan, and Thurston. Our work covers Riemann surfaces and Kleinian groups, Teichmüller theory, holomorphic motions, and quasiconformal mappings and their generalizations.

Yunping Jiang Smooth Dynamics, Holomorphic Dynamics, Teichmüller Theory

Sudeb Mitra Teichmüller Theory, Riemann surfaces, Quasiconformal Mappings

Dragomir Saric Teichmüller Theory, Hyperbolic Geometry, Quasiconformal Mappings

Saeed Zakeri Holomorphic Dynamics, (Trans-)Quasiconformal Mappings, Geometric Complex Analysis

Data Science and Statistics

Data science is the interdisciplinary study of extracting knowledge from noisy structured or unstructured raw data. It is a “science” in the sense that it is now the backbone of building models that explain and predict phenomena of interest. These models are many times complex and built through the modern methods of artificial intelligence or machine learning. It is “interdisciplinary” in the sense that a data scientist must be well-versed in skills from a vast toolbox drawn from mathematics, statistics, optimization, computing, algorithms, etc. as well as domain-specific and context-specific knowledge. Statistics is arguably the theoretical core of data science. Its main focus is on inference: estimating quantities of interest, assessing the certainty in these estimates and understanding when these estimates apply. Sometimes these quantities represent causal effects measured from randomized experiments. This last setting is the focus of the research in our department.

Adam Kapelner Machine Learning, Experimental Design

Stefan Ralescu Perturbed Empirical and Quantile Processes, Asymptotic Theory, Stein Estimation

Dynamical Systems

The mathematical theory of dynamical systems studies the long term behavior of idealized systems that evolve with time and predicts how their future depends on various parameters involved. It is a branch of pure mathematics with deep connections to many other areas, especially analysis, geometry, topology, probability theory, and combinatorics. It also has historical links and applications to fields outside mathematics such as celestial mechanics, meteorology, and population dynamics. An area of particular interest is holomorphic dynamics, the study of (semi-)groups generated by analytic transformations, where the introduction of powerful tools from complex analysis and conformal geometry has inspired remarkable progress in the past couple of decades. Our research covers analytic, topological, measure-theoretic, and combinatorial aspects of dynamical systems and their moduli spaces. Our main areas of work include the dynamics of rational and transcendental maps, structure of Julia sets and connectedness loci of polynomials, and ergodic theory of circle endomorphisms.

Yunping Jiang Smooth Dynamics, Holomorphic Dynamics, Teichmüller Theory

Saeed Zakeri Holomorphic Dynamics, (Trans-)Quasiconformal Mappings, Geometric Complex Analysis

Geometry and Topology

Geometry and topology are the fundamental mathematical disciplines that study shapes and spaces. Geometry concerns quantitative aspects, such as length, area, and volume, as well as distance, angles, and curvature. Topology is a more qualitative study of space, in which geometric details, such as size, are de-emphasized so that other properties like dimension can be focused on. Both fields are organized around studying transformations and mappings that preserve certain structures and some of the deepest and most interesting results in geometry and topology reveal intricate algebraic objects working behind the scenes. The subjects are further enriched by connections with physics and other mathematical areas like real and complex analysis, number theory, and applied math.

Dan A. Lee Geometric Analysis

John Terilla Algebraic Topology, Quantum Physics, Machine Learning

Yongwu Rong Low Dimensional Topology, Knot Theory, Quantum Invariants

Dragomir Saric Teichmüller Theory, Hyperbolic Geometry, Quasiconformal Mappings

Nicholas Vlamis Low Dimensional Topology, Hyperbolic Geometry, Teichmüller Theory

Scott Wilson Topology and Geometry of Manifolds

 

Logic and Computability

Logic is the field that studies the axiomatic foundations of mathematics and the way mathematicians use the basic axioms to prove mathematical theorems. It divides into four main branches. Proof theory asks what we mean by “a proof of a theorem.” Set theory tries to determine the best collection of axioms to use for mathematics in general, making the common assumption that all of mathematics can be expressed in the language of sets. Model theory examines the relationship between syntax (the proofs that we write down) and semantics (actual mathematical structures such as groups or graphs or vector spaces). Computability theory examines the extent to which all of this can be done effectively, actually computing presentations of the objects in question rather than merely proving that they exist. Computability often intersects with theoretical computer science, while set theory and proof theory are of real interest to philosophers: thus logic often crosses over into computer science and philosophy, as well as into linguistics, probability, and many other academic disciplines.

Russell Miller Computability, Computable Structure Theory, Decidability in Number Theory

Alexey Ovchinnikov Differential Galois Theory, Symbolic Computation

Mathematical and Computational Biology

Mathematical biology aims at analyzing mathematical representation and modeling of biological processes, using techniques and tools of pure and applied mathematics. This involves algebra, graph theory, numerical analysis, and probability and statistics, among many other areas of mathematics. Computational biology focuses on building and using computational models for problems in biology. Describing biological systems in a quantitative manner with precise models results in their behavior being better simulated, and hence properties can be predicted that might not be evident to the experimenter.

Maria Sabitova Algebraic Number Theory, Abelian Varieties, Computational Biology

Alexey Ovchinnikov Differential Galois Theory, Symbolic Computation, Mathematical Biology