All Seminars
| Title: Complex Dynamics of Rational Maps |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Ylli Andoni of Emory University |
| Contact: Shanshuang Yang, syang05@emory.edu |
| Date: 2022-12-06 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: We will look at the iteration of rational functions of one complex variable and study the behaviour of points in the complex sphere under such iterations. Such iterations split the plane into two sets, that of well-behaved and that of ill-behaved points known as the Fatou set and the Julia set respectively. The notion of equicontinuity will be used to formally define these two sets and we will relate this to normality as well. Properties of the Fatou and Julia sets will be looked at and topological consequences will be established as well. |
| Title: How much can one learn a PDE from its solution? |
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| Seminar: Analysis and Differential Geometry |
| Speaker: Yimin Zhong of Auburn University |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2022-12-01 at 4:00PM |
| Venue: PAIS 230 |
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| Abstract: In this work we study a few basic questions for PDE learning from observed solution data. Using various types of PDEs, we show 1) how the approximate dimension (richness) of the data space spanned by all snapshots along a solution trajectory depends on the differential operator and initial data, and 2) identifiability of a differential operator from solution data on local patches. Then we propose a consistent and sparse local regression method (CaSLR) for general PDE identification. Our method is data driven and requires minimal amount of local measurements in space and time from a single solution trajectory by enforcing global consistency and sparsity. |
| Title: The Turán Problem for Bipartite Graphs |
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| Seminar: Combinatorics |
| Speaker: Tao Jiang of Miami University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2022-11-18 at 3:30PM |
| Venue: MSC E408 |
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| Abstract: Extremal problems in graph theory, generally speaking, study the interaction between the density of a graph and substructures occurring in it. A natural and central problem of this nature asks for how dense a grap can be when it is missing a particular subgraph. These problems are known as Turán problems. These problems have played a central role in the development of extremal graph theory. While the celebrated he Erd?s–Stone -Simonovits theorem essentially solves the problem when the missing subgraph H is non-bipartite, much less is known when H is bipartite. While there have been steady movements on the problem in the past, there has been an increased amount of progress in recent years due to fresh ideas and angles to approach these problems. In this talk, we will survey some of the recent progresses and techniques/ideas involved in them and suggest further problems to explore. |
| Title: A Simulation Study of the Effects of His Bundle Pacing in Left Bundle Branch Block |
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| Seminar: CODES@Emory |
| Speaker: Leonardo Molinari of Emory University |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2022-11-17 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: His bundle pacing (HBP) has emerged as a feasible alternative to right (RVP) and biventricular pacing (BVP) for Cardiac Resynchronization Therapy (CRT). This study sought to assess, in ex-vivo experimental models, the optimal setup for HBP in terms of electrode placement and pacing protocol to achieve superior electrical synchrony in the case of complete His-Purkinje block and left bundle branch block (LBBB). We developed a 3D model of His bundle and bundle branches, embedded in a patient-specific biventricular heart model reconstructed from CT images. A monodomain reaction-diffusion model was adopted to describe the propagation of cardiac action potential, and a custom procedure was developed to compute pseudo-ECGs. Experimental measurements of tip electrode potential waveforms have been performed on ex-vivo swine myocardium to determine the appropriate boundary condition for delivering the electrical stimulus in the numerical model. An extended parametric analysis, investigating the effect of the electrode orientation and helix length, pacing protocol, and atrioventricular delay, allowed us to determine the optimal setup for HBP therapy. Both selective (S-HBP) and non-selective (NS-HBP) His bundle pacing were tested, as the variable anatomical location of the His bundle can result in the activation of the surrounding myocardium. Our study indicates a perpendicular placement of the electrode as the most advantageous for restoring the physiological function of the His-Purkinje system. We found that higher-energy protocols can compensate for the effects of an angled placement though concurring to potential tip fibrosis. Promisingly, we also revealed that an increased electrode helix length can provide optimal resynchronization even with low-energy pacing protocols. Our results provide informative guidance for implant procedure and therapy optimization, which will hopefully have clinical implications further improving the procedural success rates and patients’ quality of life, due to reduced incidence of lead revision and onset of complications. |
| Title: Class Numbers, Cyclic Simple Groups and Arithmetic |
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| Seminar: Algebra |
| Speaker: John Duncan of Academia Sinica |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-11-15 at 4:00PM |
| Venue: MSC N304 |
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| Abstract: Ogg gave a geometric description of the primes that divide the order of the monster finite simple group in 1973, and thus initiated the theory of monstrous moonshine. In this talk, based on joint work with Cheng and Mertens, we will explain how Ogg’s torsion conjecture (also from 1973) points toward a parallel phenomenon involving all the cyclic groups of prime order. |
| Title: Scalable Bayesian optimal experimental design for efficient data acquisition |
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| Seminar: CODES@Emory |
| Speaker: Peng Chen of Georgia Tech |
| Contact: Matthias Chung, matthias.chung@emory.edu |
| Date: 2022-11-10 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: Bayesian optimal experimental design (OED) is a principled framework for maximizing information gained from limited data in Bayesian inverse problems. Unfortunately, conventional methods for OED are prohibitive when applied to expensive models with high-dimensional parameters. In this talk, I will present fast and scalable computational methods for large-scale Bayesian OED with infinite-dimensional parameters, including data-informed low-rank approximation, efficient offline-online decomposition, projected neural network approximation, and a new swapping greedy algorithm for combinatorial optimization. |
| Title: Athens-Atlanta joint Number Theory Seminar |
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| Seminar: Algebra |
| Speaker: Alina Bucur and Samit Dasgupta of Duke and University of California, San Diego |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-11-07 at 4:00PM |
| Venue: MSC W301 |
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| Abstract: \textbf{Samit Dasgupta} (Duke University), 4:00 \\ Stark’s Conjectures and Hilbert’s 12th Problem \\ In this talk we will discuss two central problems in algebraic number theory and their interconnections: explicit class field theory (also known as Hilbert’s 12th Problem), and the special values of L-functions. The goal of explicit class field theory is to describe the abelian extensions of a ground number field via analytic means intrinsic to the ground field. Meanwhile, there is an abundance of conjectures on the special values of L-functions at certain integer points. Of these, Stark’s Conjecture has special relevance toward explicit class field theory. This conjecture states the existence of certain canonical elements in CM abelian extensions of totally real fields. I will state a conjectural exact formula for Brumer-Stark units that has been developed over the last 15 years. I will conclude with a description of my work with Mahesh Kakde that proves these conjectures away from $p = 2$, thereby giving an explicit class field theory for totally real fields. \\ \textbf{Alina Bucur, USCD} (University of California, San Diego), 5:15 \\ Counting $D_4$ fields \\ A guiding question in number theory, specifically in arithmetic statistics, is that of counting number fields of fixed degree whose normal closure has a given Galois group G as we let their discriminants grow to infinity. In this talk, we will discuss the history of this question and take a closer look at the story in the case that $n=4$, i.e. the counts of quartic fields. |
| Title: Regression with Tropical Rational Functions and Application to Neural Network Initialization |
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| Seminar: CODES@emory |
| Speaker: Alex Dunbar of Emory University |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2022-11-03 at 10:00AM |
| Venue: MSC W301 |
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| Abstract: The tropical semiring and its associated polynomial and rational functions provide an algebraic framework for understanding continuous piecewise linear functions. We propose an alternating minimization heuristic for regression over the space of tropical rational functions. The method alternates between fitting the numerator and denominator via tropical polynomial regression, which is known to admit a closed-form solution. Our work is motivated by applications to ReLU neural network training. ReLU neural networks are a popular class of network architectures in the machine learning community which have recently been connected to tropical rational functions. We present experiments demonstrating the behavior of the alternating minimization method. Additionally, we present preliminary experiments which leverage tropical rational regression to initialize weights in ReLU neural networks and discuss geometric aspects of the network initialization problem. |
| Title: (Lack of) Rank Growth of Elliptic Curves in Quartic Extensions |
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| Seminar: Algebra |
| Speaker: Daniel Keliher of University of Georgia |
| Contact: David Zureick-Brown, david.zureick-brown@emory.edu |
| Date: 2022-11-01 at 4:00PM |
| Venue: MSC N304 |
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| Abstract: Let $E$ be an elliptic curve over the rationals, and let $E(K)$ be the group of K-rational points of E over a number field K. The Mordell-Weil Theorem says that $E(K)$ factors as a finitely generated abelian group; the rank of the free abelian part, $rk(E/K)$, is the rank of E over K. We will consider the related notion of rank growth. That is, for an extension of number fields $F/K$, we will consider the quantity $rk(E/F)-rk(E/K)$. We will survey some results, conjectures, and hopes in this area and outline an approach to produce infinitely many $S_4$ and $A_4$ quartic extensions of the rationals for which an elliptic curve E does not gain rank. The approach adapts methods of Mazur, Rubin, and others to manipulate the 2-Selmer group of a thin family of quadratic twists of our starting E in a prescribed fashion. In doing so, we’ll construct quadratic extensions for which we can understand the local conditions of the corresponding twist and such that the quadratic extensions parameterize the quartic extensions of interest. |
| Title: Algorithmic and combinatorial applications of the cluster expansion |
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| Seminar: Combinatorics |
| Speaker: Matthew Jenssen of University of Birmingham, UK |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2022-10-28 at 3:30PM |
| Venue: MSC E406 |
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| Abstract: The cluster expansion is a classical tool from statistical physics traditionally used to study the phase diagram of lattice spin models. Recently, the cluster expansion has enjoyed a number of applications in two new contexts: i) the design of efficient approximate counting and sampling algorithms for spin models on graphs and ii) classical enumeration problems in combinatorics. In this talk, I’ll give an introduction to the cluster expansion and discuss some of these recent developments. |