All Seminars

Title: Relative local-global principles
Seminar: Algebra
Speaker: Danny Krashen of Rutgers University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-12-03 at 4:00PM
Venue: MSC W303
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Abstract:
In various contexts, the Hasse principle can be used to transfer questions of rational points and triviality of Galois cohomology classes from global fields to local fields. Some such results have been extended, for example, in the work of Kato, Bayer-Flukiger-Parimala, Parimala-Preeti, Parimala-Sujatha, to apply to function fields over global fields. In this talk, I will discuss recent joint work with David Harbater and Alena Pirutka in which we examine to what extent local-global principles for one field extend to local-global principles for a function field over this field. We focus particularly on the case where one starts with a semiglobal field (a function fields over discretely valued fields).
Title: Swimming bacteria: Mathematical modelling and applications
Seminar: Computational Math
Speaker: Christian Esparza-Lopez of University of Cambridge
Contact: Irving Martinez, irving.martinez@emory.edu
Date: 2019-11-22 at 1:00PM
Venue: MSC W201
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Abstract:
Miniaturisation of actuators and power sources are two of the biggest technical challenges in the design and fabrication of microscopic robots. As it is often the case, Nature can offer insight into overcoming some of these challenges. Swimming bacteria, such as the well-studied flagellated E. coli, are known to be efficient swimmers with intricate sensing capabilities. They have thus inspired scientists to mimic them to improve the design of artificial micro-robots, often with biomedical purposes such as targeted drug delivery. This talk will consist of two parts. After a brief introduction to the study of swimming bacteria I will review the random walk model for bacterial diffusion and chemotaxis, and will show how to use it to describe the diffusive behaviour of artificial micro-swimmers propelled by swimming bacteria. In the second part of the talk I will address the problem of non-flagellated swimming bacteria. Specifically, we will study a minimal model to describe the dynamics of Smeliferum, a helical bacterium that swims by progressively changing the handedness of its body.
Title: Statistical Data Assimilation for Hurricane Storm Surge Modeling
Seminar: Numerical Analysis and Scientific Computing
Speaker: Talea L. Mayo of University of Central Florida
Contact: James Nagy, jnagy@emory.edu
Date: 2019-11-22 at 2:00PM
Venue: MSC W303
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Abstract:
Coastal ocean models are used for a variety of applications, including the simulation of tides and hurricane storm surges. As is true for many numerical models, coastal ocean models are plagued with uncertainty, due to factors including but not limited to the approximation of meteorological conditions and hydrodynamics, the numerical discretization of continuous processes, uncertainties in specified boundary and initial conditions, and unknown model parameters. Quantifying and reducing these uncertainties is essential for developing reliable and robust storm surge models. Statistical data assimilation methods are often used to estimate uncertain model states (e.g. storm surge heights) by combining model output with uncertain observations. We have used these methods in storm surge modeling applications to reduce uncertainties resulting from coarse spatial resolution. While state estimation is beneficial for accurately simulating the surge resulting from a single, observed storm, larger contributions can be made with the estimation of uncertain model parameters. In this talk, I will discuss applications of statistical data assimilation methods for both state and parameter estimation in coastal ocean modeling.
Title: Connected Fair Detachments of Hypergraphs
Seminar: Combinatorics
Speaker: Amin Bahmanian of Illinois State University
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2019-11-22 at 4:00PM
Venue: MSC W303
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Abstract:
Let $G$ be a hypergraph whose edges are colored. A $(u,n)$-detachment of $G$ is a hypergraph obtained by splitting a vertex $u$ into $n$ vertices, say $u_1,\dots, u_n$, and sharing the incident edges among the subvertices. A detachment is fair if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. In this talk we solve an open problem from 1970s by finding necessary and sufficient conditions under which a $k$-edge-colored hypergraph $G$ has a fair detachment in which each color class is connected. Previously, this was not even know for the case when $G$ is an arbitrary graph. We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions, and completing partial regular combinatorial structures.
Title: (-1)-homogeneous solutions of stationary incompressible Navier-Stokes equations with singular rays
Seminar: Analysis and PDEs
Speaker: Xukai Yan of Georgia Institute of Technology
Contact: Maja Taskovic, maja.taskovic@emory.edu
Date: 2019-11-21 at 3:00PM
Venue: MSC E308A
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Abstract:
In 1944, L.D. Landau first discovered explicit (-1)-homogeneous solutions of 3-d stationary incompressible Navier-Stokes equations (NSE) with precisely one singularity at the origin, which are axisymmetric with no swirl. These solutions are now called Landau solutions. In 1998 G. Tian and Z. Xin proved that all solutions which are (-1) homogeneous, axisymmetric with one singularity are Landau solutions. In 2006 V. Sverak proved that with just the (-1)-homogeneous assumption Landau solutions are the only solutions with one singularity. Our work focuses on the (-1)-homogeneous solutions of 3-d incompressible stationary NSE with finitely many singularities on the unit sphere. In this talk we will first classify all (-1)-homogeneous axisymmetric no-swirl solutions of 3-d stationary incompressible NSE with one singularity at the south pole on the unit sphere as a two dimensional solution surface. We will then present our results on the existence of a one parameter family of (-1)-homogeneous axisymmetric solutions with non-zero swirl and smooth on the unit sphere away from the south pole, emanating from the two dimensional surface of axisymmetric no-swirl solutions. We will also present asymptotic behavior of general (-1)-homogeneous axisymmetric solutions in a cone containing the south pole with a singularity at the south pole on the unit sphere . We also constructed families of solutions smooth on the unit sphere away from the north and south poles, and will have obtained some asymptotic stability result of these solutions. This is a joint work with Professor Yanyan Li and Li Li.
Title: Derived Equivalences from Compactifications
Seminar: Algebra
Speaker: Robert Vandermolen of University of South Carolina
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-11-19 at 4:00PM
Venue: MSC W303
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Abstract:
In this talk we will examine a new generalization of a wonderful construction of Drinfeld, producing a new class of kernels which often induce Fourier-Mukai functors which realize the derived equivalences from wall-crossings in Variations of Geometric Invariant Theory. This new class of functors are parameterized by the rational polyhedral in the group equivariant ample line bundles. This program is inspired by recent work of Ballard, Diemer, Favero (2017) and work of Ballard, Chidambaram, Favero, McFaddin, and myself (2019), these papers provide a new class of kernels for realizing the derived equivalence for many interesting birational transformations.
Title: Turning Cancer Discoveries into Effective Targeted Treatments with the Aid of Mathematical Modeling
Colloquium: Applied Mathematics
Speaker: Dr. Trachette Jackson of University of Michigan
Contact: Jim Nagy, jnagy@emory.edu
Date: 2019-11-13 at 4:15PM
Venue: Oxford Road Building, 3rd Floor, Room 305 and Room 311
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Abstract:
The Department of Mathematics is pleased to announce that Dr. Trachette Jackson, Professor of Mathematics at the University of Michigan, will give a general STEM audience talk titled Turning Cancer Discoveries into Effective Targeted Treatments with the Aid of Mathematical Modeling.
Title: Generalized Brauer dimension of semi-global fields
Seminar: Algebra
Speaker: Saurabh Gosavi of Rutgers University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-11-12 at 4:00PM
Venue: MSC W303
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Abstract:
Given a finite set of Brauer classes $B$ of a fixed period $\ell$, we define $ind(B)$ to be the minimum of degrees of field extensions $L/F$ such that $\alpha \otimes_F L = 0$ for every $\alpha$ in $B$. When $F$ is a semi-global field (i.e transcendence degree one field over a complete discretely valued field), we will provide an upper-bound for $ind(B)$ which depends on invariants of fields of lower arithmetic complexity. As a simple application of our result, we will obtain an upper-bound for the splitting index of quadratic forms and finiteness of symbol length for function fields of curves over higher-local fields.
Title: Finite-Time Performance of Distributed Temporal Difference Learning on Multi-Agent Reinforcement Learning
Seminar: Numerical Analysis and Scientific Computing
Speaker: Thinh T. Doan of Georgia Institute of Technology
Contact: Lars Ruthotto, lruthotto@emory.edu
Date: 2019-11-08 at 2:00PM
Venue: MSC W303
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Abstract:
The rapid development of low-cost sensors, smart devices, communication networks, and learning algorithms has enabled data driven decision making in large-scale multi-agent systems. Prominent examples include mobile robotic networks and autonomous systems. The key challenge in these systems is in handling the vast quantities of information shared between the agents in order to find an optimal policy that maximizes an objective function. Among potential approaches, distributed reinforcement learning, which is not only amenable to low-cost implementation but can also be implemented in real time, has been recognized as an important approach to address this challenge. The focus of this talk is to consider the policy evaluation problem in multi-agent reinforcement learning, one of the most fundamental problems in this area. In this problem, a group of agents operate in an unknown environment, where their goal is to cooperatively evaluate the global discounted accumulative reward composed of local rewards observed by the agents. For solving this problem, I consider a distributed variant of the popular temporal difference learning, often referred to as TD(λ) for some constant λ ∈ [0,1]. My main contribution is to provide a finite-analysis on the performance of this distributed TD(λ) for both constant and time-varying step sizes. The key techniques are to utilize tools from distributed optimization and stochastic approximation in analyzing the underlying algorithm. In particular, I derive an explicit formula for the upper bound on the rates of the proposed method as a function of the constant λ and the network topology characterized the communication between the agents. In addition, my results theoretically address an important question of TD learning from numerical observations, that is, λ=1 gives the best approximation of the function values while λ=0 leads to better performance when there is a large variance in the algorithm. Finally, I conclude my talk with some discussion about my future research in the context of distributed decision making on multi-agent systems.
Title: Integers represented by positive-definite quadratic forms and Petersson inner products
Seminar: Algebra
Speaker: Jeremy Rouse of Wake Forest University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-11-05 at 4:00PM
Venue: MSC W303
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Abstract:
We give a survey of results about the problem of determining which integers are represented by a given quaternary quadratic form $Q$. A necessary condition for $Q(x_1,x_2,x_3,x_4)$ to represent $n$ is for the equation $Q(x_1,x_2,x_3,x_4) = n$ to have a solution with $x_1,x_2,x_3,x_4 \in Z_p$ for all $p$. But even when $n$ is sufficiently large, this is not sufficient for $Q$ to represent $n$. The form $Q$ is anisotropic at the prime $p$ if for $x_1,x_2,x_3,x_4 \in Z_p$, $Q(x_1,x_2,x_3,x_4) = 0$ implies that $x_1=x_2=x_3=x_4=0$. Suppose that $A$ is the Gram matrix for $Q$ and $D(Q) = \det(A)$. We show that if $n >> D(Q)^{6+\epsilon}$, $n$ is locally represented by $Q$, but $Q$ fails to represent $n$, then there is an anisotropic prime $p$ so that $p^2 | n$ and $np^{2k}$ is not represented by $Q$ for any $k \geq 1$. We give sharper results when $D(Q)$ is a fundamental discriminant and discuss applications to universality theorems like the 15 and 290 theorems of Bhargava and Hanke.