All Seminars
| Title: (-1)-homogeneous solutions of stationary incompressible Navier-Stokes equations with singular rays |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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 |
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| 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. |
| Title: Quasirandomness and hypergraph regularity, II |
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| Seminar: Combinatorics |
| Speaker: Mathias Schacht of The University of Hamburg and Yale University |
| Contact: Dwight Duffus, dwightduffus@emory.edu |
| Date: 2019-11-01 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: This is a continuation of the last talk. We shall focus on the proof of the equivalence of DISC and DEV in 3-uniform hypergraphs. While the simple implication "DEV implies DISC" easily follows by appropriate applications of the Cauchy-Schwarz inequality, the opposite implication is based on the regularity method for hypergraphs. |
| Title: A refined Brill-Noether theory over Hurwitz spaces |
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| Seminar: Algebra |
| Speaker: Hannah Larson of Stanford Univeristy |
| Contact: David Zureick-Brown, dzb@mathcs.emory.edu |
| Date: 2019-10-29 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: The celebrated Brill-Noether theorem says that the space of degree $d$ maps of a general genus $g$ curve to $\mathbb{P}^r$ is irreducible. However, for special curves, this need not be the case. Indeed, for general $k$-gonal curves (degree $k$ covers of $\mathbb{P}^1$), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map $C \rightarrow \mathbb{P}^1$, using the splitting type of push forwards of line bundles to $\mathbb{P}^1$. In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general $k$-gonal curves. |
| Title: Patient-Specific Modeling in Cardiac Electrophysiology: Parameter Estimation and Personalization |
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| Defense: Dissertation |
| Speaker: Alessandro Barone of Emory University |
| Contact: Alessandro Barone, alessandro.barone@emory.edu |
| Date: 2019-10-25 at 3:10PM |
| Venue: MSC W303 |
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| Abstract: Computational modeling in cardiac electrophysiology (EP) has long played a central role in the study of physio-pathological dynamics of electrical propagation. One of the most significant challenges to face is the translation process of numerical (in silico) investigations to clinical practice. In silico simulations can potentially impact the quality of cardiac arrhythmia therapy, reducing the risk of in vivo testing. However, the clinical use of virtual experiments is hindered by the need of customization of mathematical models to patient-specific data. The personalization process involves the fine tuning of many model parameters, that cannot be measured directly, via accurate and efficient data assimilation techniques. This work is particularly focused on the estimation of cardiac conductivities, crucial parameters of the Bidomain and Monodomain models – currently the most used mathematical descriptions of cardiac electrical behavior. This Thesis addresses the challenge described above yielding four main contributions. (1) We perform an extensive and thorough synthetic and experimental validation of the deterministic variational data assimilation method proposed by Yang and Veneziani in 2015 to retrieve conductivities from potential recordings. The results demonstrate that the procedure provides accurate space-dependent conductivity estimates that reproduce most of the observed dynamics. (2) The Proper Generalized Decomposition (PGD) reduced-order model technique is investigated for the first time in EP to improve the efficiency of the variational technique. Relying on the off-line/on-line paradigm and without the need of any preliminary knowledge of the high-fidelity solution, we show in 2D and 3D settings that the strategy enables nearly real-time estimation preserving reasonable accuracy. (3) With the goal of assessing the robustness of the results, we propose a statistical formulation of the estimation problem for Monodomain conductivities. Exploiting the computational convenience of the on-line PGD solution, the methodology allows a reliable quantification of the uncertainty of two-dimensional estimates. (4) Using a virtual personalized heart model efficiently reconstructed from high resolution MRI images and ECG data via a physics-based reduced-order model approach, we perform a preliminary study of the induction of ventricular electrical anomalies with respect to different conduction properties in view of optimizing arrhythmia treatments in silico. |
| Title: Isentropic Approximation |
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| Seminar: Analysis and PDEs |
| Speaker: Ronghua Pan of Georgia Institute of Technology |
| Contact: Maja Taskovic, maja.taskovic@emory.edu |
| Date: 2019-10-24 at 3:00PM |
| Venue: MSC E308A |
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| Abstract: In the study of compressible flows, the isentropic model was often used to replace the more complicated full system when the entropy is near a constant. This is based on the expectation that the corresponding isentropic model is a good approximation to the full system when the entropy is sufficiently close to the constant. We will discuss the mathematical justification of isentropic approximation in Euler flows and in Navier-Stokes-Fourier flows. This is based on the joint work with Y. Chen, J. Jia, and L. Tong. |