Mathematics

I am a faculty member in the Mathematics Department and have been at OSU since 2005.

Research Interests:

My primary areas of research are Number Theory and Representation Theory. In particular, I conduct research in automorphic forms and L-functions, as part of the the Langlangs Program.

Differential Geometry, Geometric Analysis and Several Complex Variables. In particular, conformal and CR geometries and generalizations (such as parabolic geometries), mathematical relativity, mathematical physics.

Dr. Emory is currently an Assistant Professor at Oklahoma State University in the Mathematics Department. Previously she was a NSF Postdoctoral Fellow at the University of Toronto with mentor James Arthur. Dr. Emory earned her PhD from the University of Missouri in 2018 with advisor Shuichiro Takeda. Dr. Emory is currently serving as the faculty advisor for the OSU student chapter of the Association for Women in Mathematics.

Research Interests:

Dr. Emory's research interests include automorphic forms and representations, number theory, and representation theory.

One area of research that Dr. Emory works in deals with a strategy of Langlands known as Beyond Endoscopy which utilizes the trace formula to study the analytic behavior of automorphic L-functions. Its goal is to establish the principle of functoriality which is the fundamental pillar of the Langlands program, with enormous implications for number theory, arithmetic geometry, and representation theory.

Another area of research that Dr. Emory is currently engaged in is on the restriction of representations of a non-classical group, the general Spin group. In general, the question is how a representation of a group decomposes when restricted to a subgroup. The goal of this research is to formulate and prove a local Gan-Gross-Prasad conjecture for general Spin groups. The representation theory of general spin groups is important as it subsumes the representation theory of the special orthogonal groups; and having a non-trivial center (as occurs with general spin groups) is preferred for application purposes.

If instead of algebraic solutions to curves, one is interested in the distributions of solutions mod p, we are interested in the Sato-Tate conjecture. This area of Dr. Emory's research aims to determine the full Sato-Tate group for curves of certain forms. Roughly speaking, we are interested in studying how the coefficients of the (local) normalized L-polynomials of a curve vary as the prime p goes to infinity. This research area is important because the coefficients of the L-polynomial encode arithmetic information about the curve. The generalized Sato-Tate conjecture predicts that as p goes to infinity this distribution converges to the distribution of traces in the Sato-Tate group, a compact subgroup of USp(2g) associated to the Jacobian of the curve.

My research interests are in number theory and analysis. I am primarily interested in topics relating to the distribution of algebraic numbers and points of small height, potential theory, and arithmetic dynamics. I am interested in Bogomolov and Lehmer type problems for heights, unlikely intersections, and equidistribution for dynamical systems. Lately I've become interested in stochastic dynamics.

Dr. Kable is a professor of mathematics.

Research Interests:

Lie Theory, Invariant Theory, and Number Theory.

Lie Theory: I have worked in a number of branches of Lie Theory. Among these, the closest to Lie's original conception of the theory is the study of conformally invariant systems of differential equations. These are systems of one or more partial differential equations whose conformal symmetry group is a large Lie group. I have investigated the construction and classification of such systems, their formal properties, and their solutions in various function spaces. I also study the representation theory of reductive affine algebraic groups and their central extensions. I have worked on problems involving the representations of these groups over finite fields (where they are finite groups of Lie type), non-Archimedean local fields (where they are p-adic analytic groups), Archimedean local fields (where they are Lie groups), and rings of adeles.

Invariant Theory: I have an interest in prehomogeneous vector spaces and their applications. I am also interested in the invariant theory of Lie algebras. The two topics are linked through the occurrence of prehomogeneous vector spaces of parabolic type inside Lie algebras.

Number Theory: I have worked mainly on density and equidistribution problems for families of arithmetical objects, such as number fields, quadratic forms, and division algebras. Often my approach is through parameterizations of these objects by orbits in prehomogeneous vector spaces and the analytic properties of the associated global zeta functions.

Numerical analysis
Mathematical analysis of finite element methods

Several complex variables, complex analysis, real algebraic geometry

I am an assistant professor at the Department of Mathematics at Oklahoma State University. My research interests lie at the interface of math, biology, and data-driven methods, focusing on epidemic forecasting and mathematical modeling of infectious diseases.

In 2021, I held a Research Fellow position at the Machine Intelligence Lab - Boston Children's Hospital /Harvard Medical School under the supervision of Mauricio Santillana. From 2018 to 2020 was a postdoctoral scholar in the Laboratory for Computational Cellular Mechanobiology at the Department of Mechanical and Aerospace Engineering - University of California San Diego, under the supervision of Padmini Rangamani. In 2017, I joined the Scientific Computing Program (PROCC) at Fundação Oswaldo Cruz (FIOCRUZ), as a postdoctoral researcher advised by Cláudia Codeço.



I earned my Ph.D. degree from the Instituto Nacional de Matemática Pura e Aplicada (IMPA) under the supervision of Roberto Imbuzeiro de Oliveira (IMPA) and J. Nathan Kutz (University of Washington). My thesis focused on dimensionality reduction techniques and machine-learning algorithms for Epilepsy and Dengue Epidemiology problems.

Research Interests:

Mathematical Epidemiology, Data-driven Methods, Biological Networks

Commutative algebra

Undergraduate Mathematics Education, in particular, in the setting of mathematics tutoring centers.

Anand Patel is a mathematician with broad interests in algebraic geometry and number theory.

Research Interests:

Algebraic Geometry, Number Theory