Schedule

Role of Human Mobility in the Disease Dynamics through Network Structure in the Metapopulation Model

Abstract. The metapopulation network model describes the dynamics of an infectious disease in a city where each neighborhood is a node of a network, and the edges of a network represent the daily flux of people between the nodes. Using the metapopulation model, we examined the specific role of human mobility levels and network structure in affecting the dynamics of a disease in a networked metapopulation. We also investigated how human movement between neighborhoods can control or promote the outbreaks of infectious diseases in the networks.

Haridas Kumar Das, Oklahoma State University

Crash Course in Sheaves!

Abstract. What are sheaves? These semi-mysterious objects show up a lot in algebraic and differential geometry. We will go over the basic definitions, examples, and if time permits, go over the structure sheaf of a variety.

John Michael Clark, University of Texas at Austin

A New Symmetric Resolution for the (Pinched) nth Veronese

Abstract. Let S = k[x1, ··· , xn] be a polynomial ring over an arbitrary field k. We construct a new symmetric polytopal minimal resolution of (x1, ··· , xn)^n and a symmetric polytopal minimal resolution of an equigenerated monomial ideal obtained by removing x1x2 ··· xn from the generators of (x1, · · · , xn)^n.

Hoai Dao, Oklahoma State University

Number Theory, Class Numbers, and Geometry

Abstract. The goal of this talk is to describe the basic strategy for showing the finiteness of the class group of an algebraic number field. Interestingly, this strategy involves the geometry of the ring of integers for the given number field, which emerges from embedding the ring into Euclidean space. We will give the background needed for understanding this embedding, and how it informs us on the arithmetic of the ring of integers. No number theory knowledge is assumed for this talk.

Preston Kelley, Oklahoma State University

An Introduction to the theory of Sobolev Spaces

Abstract. The goal of this talk will be to give an expository presentation (at a level appropriate for any graduate student in mathematics) on the key aspects pertaining to the theory of Sobolev spaces. In particular, this talk will aim to demonstrate the motivating utility that Sobolev spaces provide for the analysis that underpins the modern theory within the study of partial differential equations. For example, this talk will cover the notion of weak differentiability, regularity and the approximation of Sobolev functions by smooth functions, Sobolev-type inequalities and associated embedding theorems, and (time-permitting) the characterization of the space $H^s(\mathbb{R}^n)$ via Fourier analysis. Also, some prerequisite material will be briefly reviewed as well.

Tyler Labus, Oklahoma State University

Degrees of Rational Proper Maps between Siegel Upper Half-Spaces

Abstract. Since Alexander proved that every holomorphic proper self map of the unit ball 𝔹n for n > 1 is an automorphism fifty years ago, classification of holomorphic proper maps between balls has stayed an active field of research. It is a classic result that the numerator of a rational proper map between balls achieves the degree of the map. We prove the analog result for rational proper maps between Siegel upper half spaces, another realization of the unit balls.

Abdullah Al Helal, Oklahoma State University

A mini-course on elliptic partial equations

Abstract. Elliptic operators are ubiquitous; they appear frequently in classical mechanics, quantum mechanics, and even in general relativity. This talk aims to explain ellipticity (somewhat generally) from scratch.

I will start with the classic definition of ellipticity for a single linear partial differential equation (PDE) in flat Euclidean space. Then we extend this definition to a system of linear PDEs via the notion of "principal symbols". Then the next step is to define ellipticity for linear operators on vector bundles over manifolds. In this stage, the geometric interpretation of symbols & ellipticity will be addressed. I will extend the notion of ellipticity one more time and explain the Nirenberg-Douglis ellipticity. If time permits, I will give an example from general relativity. 

Siddiqur Rahman Milon, Oklahoma State University

Projections of Richardson Varieties

Abstract. Given an algebraic group G, a Borel subgroup B, and parabolic subgroup P containing B, there is a natural projection map from the flag variety G/B to the partial flag variety G/P. Associated to a flag variety is a Weyl group W, a combinatorial object indexing important subvarieties called Schubert varieties. Interesting in their own right, Schubert varieties also offer insights into Intersection Theory where their characteristic classes form a basis for the Grassmannian's cohomology ring. Richardson varieties are the pairwise intersection of Schubert varieties in general position. One of the most intriguing aspects of these subvarieties is how their geometric properties are often described by the combinatorics of the Weyl group. In this talk, we consider the restriction of the projection map to Richardson varieties and combinatorially describe when this restriction has equidimensional fibers. 

Travis Grigsby, Oklahoma State University