Schedule

Hartogs' Extension Phenomena in Several Complex Variables

 Abstract. Complex analysis deals with extremely well-behaved objects. Compared to real analysis, the statements of theorems are often much easier to state by avoiding extra hypothesis due to pathological cases. The story is different in the world of several complex variables. This distinction makes several complex variables an interesting subject on its own with a wide range of applications to other areas of mathematics.

In this talk, I will explain one of the major differences between functions of one complex variable and several complex variables: the Hartogs' extension phenomena. In particular, Hartogs' theorem tells us that the zero set (or set of singularities) of a holomorphic function of several variables cannot be contained in a compact set. This is in stark contrast with one variable case where zeroes (and singularities) of holomorphic functions are isolated.

This is aimed to be an expository talk and only basic knowledge of analysis is required.

Dr. Luka Mernik, Oklahoma State University

The Einstein Constraint Equations

Abstract. Einstein's gravitational field equations(EFEs) can be considered the most celebrated (arguably) system of PDEs of the last century. In 4 dimensions, combing Gauss and Codazzi equations with Einstein's field equations, one obtains the four Einstein's constraint equations that must satisfy the initial data for the solution of EFEs. In this talk, we show the geometric origin of Einstein's constraint equations and their role to construct solutions to EFEs.

Siddiqur Rahman Milon, Oklahoma State University

The two-dimensional Magneto-Hydro-Dynamic Equations with Magnetic Diffusion

Abstract. Magnetohydrodynamic equations (MHD) are used to study the properties of electrically conducting fluids. MHD equations were introduced by Hannes Alfv´en who won the Nobel prize in 1970. MHD equations play a vital role when understanding the motion of electrically conducting fluids in the presence of a magnetic field. There are two open problems in these equations, namely, the global regularity problem and the stability problem, that require extensive work prior to future applications. Current work has identified the constraints for solving the global regularity problem of 2D MHD resistive equations with initial data. Also presents the work plan for the stability and small data global wellposedness problems for the 2D MHD equations with magnetic diffusion with a background magnetic field. It highlights the importance of studying MHD equations with partial or fractional magnetic diffusion. Specifically, this talk focuses on the 2D resistive MHD equation and the unsolved global regularity and stability problems for this system. 

Lakjayani Hewawasan P., Oklahoma State University

Geometric realization of representations of compact Lie group

 Abstract. Given a compact Lie group G, there is a way to realize the irreducible representations of G as space of holomorphic sections of line bundles. This was independently discovered by Borel and Weil and is called the Borel-Weil theorem. In this talk, I will give the explicit construction to realize the irreducible representations of U(2) as holomorphic sections of line bundles.

Sachin Chandran, Oklahoma State University

Infectious disease forecasting using Machine Learning and Deep Learning

Abstract. In this study, we apply deep learning and statistical time series models to forecast the transmission of Mpox globally. Analyzing the spatial pattern of global Mpox data, we offer short-term (two months) projections by implementing 1D-CNN, LSTM, and hybrid CNN-LSTM on weekly epidemiological data. In addition, we employ three statistical models: ARIMA, SARIMAX, and exponential smoothing of the same time series data for the prediction. We may also implement a deterministic model that provides data-fitting, estimating important epidemiological parameters. By comparing all models with actual Mpox time series data, we demonstrate the potential for a trustworthy prediction model to minimize an epidemic's social and economic impacts on a country and prompt prevention policies and remedial action. Finally, we show a decision tree classifier to identify Monkeypox's major and minor symptoms.

Haridas Kumar Das, Oklahoma State University

On the Correspondence between Hilbert Functions of Lex Ideals and Quotients

Abstract.  Lex ideals and quotients of polynomial rings are important objects of study in commutative algebra and algebraic geometry in that they help determine the behavior of Hilbert functions of graded modules.  Computing a Hilbert function via a lex quotient is far easier and more generalizable than directly from its general counterpart, but the corresponding process for lex ideals is less observed.  We show the similarities between both processes as well as the dual relationship between the Hilbert functions themselves. Given a degree d monomial in n variables, we compute the dimensions of the associated lex segments, viewed as vector spaces, and find that their Macaulay coefficients partition the set {0, 1, ..., n + d - 2}, which is our main result.  

Reid Buchanan, Oklahoma State University

Height Gaps in Abelian Extensions of the Rationals

Abstract. The (absolute logarithmic) Weil height is a measurement of arithmetic complexity for algebraic numbers, which has a codomain in the nonnegative real numbers. The Weil height, and more general height functions, have played essential roles in proving finiteness and equidistribution theorems in number theory and dynamics. A key property of the Weil height is the Northcott property, which states that the set of algebraic numbers with bounded height and bounded degree is finite. This property is one of the main motivations to study algebraic numbers which have relatively small heights. Another property of Weil height is Kronecker's theorem, which states that the points with height 0 are precisely the roots of unity. In this talk, we review a result of Amoroso and Dvornicich, which gives a lower bound for the heights of algebraic numbers that are not roots of unity in abelian extensions of the rational numbers. 

Preston Kelley, Oklahoma State University