All Seminars

Title: The Problem of Polytope Reconstruction
Job Talk: Combinatorics
Speaker: Hailun Zheng, Assistant Professor of University of Houston-Downtown
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-01-24 at 10:00AM
Venue: MSC W201
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Abstract:
What partial information about a convex d-polytope is enough to uniquely determine its combinatorial type? This problem, known as the problem of polytope reconstruction, has been extensively studied since the sixties. For instance, a famous result of Perles asserts that simplicial d-polytopes are determined by their \lfloor d/2 \floor-skeletons. In this talk, I will survey recent advances in this field, from mainly two perspectives. 1) realizability: can a certain simplicial complex be realized as the (\lfloor d/2 \rfloor-1)-skeleton of a simplicial d-polytope or a simplicial (d-1)-sphere? 2) sufficiency: can the i-skeleton (where i < \lfloor d/2 \rfloor), together with some additional information such as affine (i+1)-stresses, determine the combinatorial or even affine type of the polytope? This is joint work with Satoshi Murai and Isabella Novik.
Title: Randomness in Ramsey Theory and Coding Theory
Job Talk: Combinatorics
Speaker: Xiaoyu He, NSF Postdoctoral Research Fellow of Princeton University
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-01-19 at 10:00AM
Venue: MSC W201
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Abstract:
Two of the most influential theorems in discrete mathematics state, respectively, that diagonal Ramsey numbers grow exponentially and that error-correcting codes for noisy channels exist up to the information limit. The former, proved by Erd?s in 1947 using random graphs, led to the development of the probabilistic method in combinatorics. The latter, proved by Shannon in 1948 using random codes, is one of the founding results of coding theory. Since then, the probabilistic method has been a cornerstone in the development of both Ramsey theory and coding theory. In this talk, we highlight a few important applications of the probabilistic method in these two parallel but interconnected worlds. We then present new results on Ramsey numbers of graphs and hypergraphs and codes correcting deletion errors, all based on probabilistic ideas.
Title: Measure growth in groups and the Kemperman inverse problem
Job Talk: Math
Speaker: Yifan Jing of University of Oxford
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-01-17 at 10:00AM
Venue: MSC W201
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Abstract:
Two perennially studied questions in arithmetic combinatorics are: (i) Given two sets $A,B$ of a given size, how small can $AB$ be? (ii) What structure must $A$ and $B$ have when $AB$ is as small as possible, or nearly as small as possible? Theorems addressing (i) are called direct theorems, and those addressing (ii) are called inverse theorems. The direct theorem for locally compact groups was obtained by Kemperman (well-known special cases include Kneser's inequality and the Cauchy-Davenport inequality). The Kemperman inverse problem (proposed by Kemperman in 1964, also by Griesmer and Tao) corresponds to question (ii) when the ambient group is connected. In this talk, I will discuss the recent solution to this problem, highlighting the new-developed measure growth phenomenon: if $G$ is connected compact equipped with a normalized measure $\mu$, and $G$ is ``sufficiently non-abelian'', $A\subseteq G$ has a sufficiently small measure, then there is a constant gap between $\mu(AA)$ and $2\mu(A)$. We then discuss a few other applications of this phenomenon, including a Brunn-Minkowski inequality in non-abelian groups. This is based on joint work with Chieu-Minh Tran and Ruixiang Zhang.
Title: Bounding the Chromatic Number and Average Degree of Graphs
Job Talk: Combinatorics
Speaker: Rose McCarty, Instructor and NSF Postdoctoral of Princeton University
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2023-01-12 at 10:00AM
Venue: MSC W201
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Abstract:
When can we partition the vertex set of a graph into a few parts so that, within each part, there are no adjacent vertices? One obvious obstruction is the existence of many pairwise adjacent vertices. A class of graphs is called $\chi$-bounded if this is the only obstruction. We introduce this topic by considering classes of graphs with geometric representations. Then we move on to the general case. While it was recently shown that $\chi$-bounding functions can be arbitrarily bad, we prove that "average degree bounding functions" are actually well-behaved. This proof suggests a new approach to the 1983 conjecture of Thomassen about average degree and girth.
Title: Recent Progresses in Kinetic Equations
Colloquium: Analysis and Differential Geometry
Speaker: Maria Pia Gualdani of University of Texas at Austin
Contact: Maja Taskovic, maja.taskovic@emory.edu
Date: 2022-12-08 at 4:00PM
Venue: WH 111
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Abstract:
We will discuss recent mathematical results for the Landau and Boltzmann equation. Kinetic equations are used to describe evolution of interacting particles. The most famous kinetic equation is the Boltzmann equation: formulated by Ludwig Boltzmann in 1872, this equation describes motion of a large class of gases. Later, in 1936, Lev Landau derived a new mathematical model for motion of plasma. This latter equation was named the Landau equation. While many important questions are still partially unanswered due to their mathematical complexity, many others have been solved thanks to novel combinations of analytical techniques, in particular the ones developed by Hoermander, J. Nash, E. De Giorgi and Moser.
Title: Three case studies in greenhouse gas emissions – new insights provided by an expanded atmospheric observing network
Seminar: Numerical Analysis and Scientific Computing
Speaker: Scot Miller of Johns Hopkins University
Contact: Julianne Chung, jmchung@emory.edu
Date: 2022-12-07 at 10:00AM
Venue: MSC W201
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Abstract:
The global record of greenhouse gas measurements is growing rapidly with the launch of several new satellites and an expansion of ground-based monitoring. This new era of big data is set to transform scientific understanding of greenhouse gas emissions, but the volume of new data creates numerous computational challenges for inverse or emissions models that were originally designed for a small number of ground-based observing stations. The beginning of the talk will focus on new, transformative statistical and mathematical approaches to understand emissions using massive satellite datasets. Then, we will apply these techniques to three different case studies in greenhouse gas emissions. The first case study will focus on carbon dioxide sources and sinks estimated using NASA's OCO-2 satellite. Using this data, we find that most existing biogeochemical models underestimate the seasonal amplitude of the global carbon cycle, and we argue that the impacts of climate change on this aspect of the carbon cycle may be larger than previously believed. The second part of the talk focuses on methane emissions from China, the world's largest emitter of anthropogenic greenhouse gases. We specifically use satellite observations to evaluate the success of China's methane emissions policies. Lastly, we will discuss an often-overlooked greenhouse gas called sulfuryl fluoride, which has surprising and unexpected implications for greenhouse gas emissions targets within the US.
Title: Sumset Estimates in Higher Dimensions
Seminar: Combinatorics
Speaker: David Conlon of Caltech
Contact: Liana Yepremyan, liana.yepremyan@emory.edu
Date: 2022-12-07 at 3:00PM
Venue: MSC W301
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Abstract:
We will describe recent progress, in joint work with Jeck Lim, on the study of sumset estimates in higher dimensions. The basic question we discuss is the following: given a subset A of d-dimensional space and a linear transformation L, how large is the sumset A + LA?
Title: Complex Dynamics of Rational Maps
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?
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
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.