Seminars and Colloquia by Series

Factorization theorems and canonical representations for generating functions of special sums

Series
Dissertation Defense
Time
Wednesday, July 6, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Hybrid - Skiles 006 and Zoom
Speaker
Maxie Dion SchmidtGeorgia Tech
ABSTRACT: This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. These results are a main focus of the author's publications from 2017-2021. The last primary (non-appendix) section of the thesis explores the topic of how to best rigorously define a so-termed "canonically best" matrix based factorization for a given class of convolution sum sequences. The notion of a canonical factorization for the generating function of such sequences needs to match the qualitative properties we find in the factorization theorems for Lambert series generating functions (LGFs). The expected qualitatively most expressive expansion we find in the LGF case results naturally from algebraic constructions of the underlying LGF series type. We propose a precise quantitative requirement to generalize this notion in terms of optimal cross-correlation statistics for certain sequences that define the matrix based factorizations of the generating function expansions we study. We finally pose a few conjectures on the types of matrix factorizations we expect to find when we are able to attain the maximal (respectively minimal) correlation statistic for a given sum type. COMMITTEE:
  • Dr. Josephine Yu, Georgia Tech
  • Dr. Matthew Baker, Georgia Tech
  • Dr. Rafael de la Llave, Georgia Tech
  • Dr. Jayadev Athreya, University of Washington
  • Dr. Bruce Berndt, University of Illinois at Urbana-Champaign
HYBRID FORMAT LOCATIONS: LINKS:

 

Thesis Defense: James Wenk

Series
Dissertation Defense
Time
Tuesday, July 5, 2022 - 11:00 for 2 hours
Location
Skiles 005
Speaker
James Wenk

Please Note: I will be defending my thesis on the shortest closed curve to inspect a sphere. Time: 11am EST Location: Skiles 005, also on Zoom at https://gatech.zoom.us/j/97708515339 Committee: Dr. Mohammad Ghomi, Advisor School of Mathematics Georgia Institute of Technology Dr. Igor Belegradek School of Mathematics Georgia Institute of Technology Dr. Jason Cantarella Department of Mathematics University of Georgia Dr. Rob Kusner Department of Mathematics University of Massachusetts Dr. Galyna Livshyts School of Mathematics Georgia Institute of Technology Dr. Michael Loss School of Mathematics Georgia Institute of Technology

Erdos-Posa theorems for undirected group-labelled graphs

Series
Dissertation Defense
Time
Friday, June 10, 2022 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 006 (hybrid)
Speaker
Youngho YooGeorgia Tech

Erdos and Posa proved in 1965 that cycles satisfy an approximate packing-covering duality. Finding analogous approximate dualities for other families of graphs has since become a highly active area of research due in part to its algorithmic applications. In this thesis we investigate the Erdos-Posa property of various families of constrained cycles and paths by developing new structural tools for undirected group-labelled graphs.

Our first result is a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs. This structure theorem is then used to prove the Erdos-Posa property of A-paths of length 0 modulo p for a fixed odd prime p, answering a question of Bruhn and Ulmer. Further, we obtain a characterization of the abelian groups G and elements g for which A-paths of weight g satisfy the Erdos-Posa property. These results are from joint work with Robin Thomas.

We extend our structural tools to graphs labelled by multiple abelian groups and consider the Erdos-Posa property of cycles whose weights avoid a fixed finite subset in each group. We find three types of topological obstructions and show that they are the only obstructions to the Erdos-Posa property of such cycles. This is a far-reaching generalization of a theorem of Reed that Escher walls are the only obstructions to the Erdos-Posa property of odd cycles. Consequently, we obtain a characterization of the sets of allowable weights in this setting for which the Erdos-Posa property holds for such cycles, unifying a large number of results in this area into a general framework. As a special case, we characterize the integer pairs (L,M) for which cycles of length L mod M satisfy the Erdos-Posa property. This resolves a question of Dejter and Neumann-Lara from 1987. Further, our description of the obstructions allows us to obtain an analogous characterization of the Erdos-Posa property of cycles in graphs embeddable on a fixed compact orientable surface. This is joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum.

Zoom link: https://gatech.zoom.us/j/96860495360?pwd=cktMRVVqMDRtVnJsb3ZLRll1bFRJQT09

Matching problems in hypergraphs

Series
Dissertation Defense
Time
Thursday, June 9, 2022 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 (hybrid)
Speaker
Xiaofan YuanGeorgia Tech

Kühn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph on n vertices, where n is a multiple of 3 and large, and the minimum vertex degree of H is greater than {(n-1) choose 2} - {2n/3 choose 2}, then H contains a perfect matching.

We show that for sufficiently large n divisible by 3, if F_1, ..., F_{n/3} are 3-uniform hypergraphs with a common vertex set and the minimum vertex degree in each F_i is greater than {(n-1) choose 2} - {2n/3 choose 2} for i = 1, ..., n/3, then the family {F_1, ..., F_{n/3}} admits a rainbow matching, i.e., a matching consisting of one edge from each F_i. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs.

We also prove that, for any integers k, l with k >= 3 and k/2 < l <= k-1, there exists a positive real μ such that, for all sufficiently large integers m, n satisfying n/k - μn <= m <= n/k - 1 - (1 - l/k){ceil of (k - l)/(2l - k)}, if H is a k-uniform hypergraph on n vertices and the minimum l-degree of H is greater than {(n-l) choose (k-l)} - {(n-l-m) choose (k-l)}, then H has a matching of size m+1. This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l <= k-1.  In many cases, our result gives tight bound on the minimum l-degree of H for near perfect matchings. For example, when l >= 2k/3, n ≡ r (mod k), 0 <= r < k, and r + l >= k, we can take m to be the minimum integer at least n/k - 2.

Zoom link: https://gatech.zoom.us/j/91659544858?pwd=SWZtVG15dGFiWEFXSHR1U0JNbVVBZz09

Learning Dynamics from Data Using Optimal Transport Techniques and Applications

Series
Dissertation Defense
Time
Wednesday, June 1, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Shaojun Ma

Zoom link: https://gatech.zoom.us/j/4561289292

Abstract: In recent years we have seen the popularity of optimal transport and deep learning. Optimal transport theory works well in studying differences among distributions, while deep learning is powerful to analyze high dimensional data. In this presentation we will discuss some of our recent work that combine both optimal transport and deep learning on data-driven problems. We will cover four parts in this presentation. The first part is studying stochastic behavior from aggregate data where we recover the drift term in an SDE, via the weak form of Fokker-Planck equation. The second part is applying Wasserstein distance on the optimal density control problem where we parametrize the control strategy by a neural network. In the third part we will show a novel form of computing Wasserstein distance, geometric and map all together in a scalable way. And in the final part, we consider an inverse OT problem where we recover cost function when an observed policy is given.

On embeddings of 3-manifolds in symplectic 4-manifolds

Series
Dissertation Defense
Time
Wednesday, June 1, 2022 - 12:01 for 1.5 hours (actually 80 minutes)
Location
Skiles 006 and ONLINE (zoom)
Speaker
Anubhav MukherjeeGeorgia Institute of Technology

Zoom Link- https://gatech.zoom.us/j/97563537012?pwd=dlBVUVh2ZDNwdDRrajdQcDltMmRaUT09 (Meeting ID: 975 6353 7012 Passcode: 525012)

 

In this talk I will discuss the conjecture that every 3 manifolds can be smoothly embedded in symplectic 4 manifolds. I will give some motivation on why is this an interesting conjecture. As an evidence for the conjecture, I will prove that every 3 manifolds can be embedded in a topological way and such an embedding can be made a smooth one after a single stabilization. As a corollary of the proof, I will prove that integer/rational cobordism group is generated by Stein fillable 3 manifolds. And if time permits, I will give some idea on how one can try to obstruct smooth embeddings of 3 manifolds in symplectic 4 manifolds.

Contact geometric theory of Anosov flows in dimension three

Series
Dissertation Defense
Time
Wednesday, May 25, 2022 - 11:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 005
Speaker
Surena HozooriGeorgia Institute of Technology

Zoom link : https://gatech.zoom.us/j/98171168149

Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in particular in low dimensions, thanks to the use of foliation theory in their study. Although the connection of Anosov flows to contact and symplectic geometry was noted in the mid 1990s by Mitsumatsu and Eliashberg-Thurston, such interplay has been left mostly unexplored. I will present some recent results on the contact and symplectic geometric aspects of Anosov flows in dimension 3, including in the presence of an invariant volume form, which is known to have grave consequences for the dynamics of these flows. Time permitting, the interplay of Anosov flows with Reeb dynamics, Liouville geometry and surgery theory will be briefly discussed as well.

Symmetric Tropical Rank 2 Matrix Completion

Series
Other Talks
Time
Monday, May 23, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
May Cai

An important recent topic is matrix completion, which is trying to recover a matrix from a small set of observed entries, subject to particular requirements. In this talk, we discuss results on symmetric tropical and symmetric Kapranov rank 2 matrices, and establish a technique of examining the phylogenetic tree structure obtained from the tropical convex hulls of their columns to construct the algebraic matroid of symmetric tropical rank 2 $n \times n$ matrices. This matroid directly answers the question of what entries of a symmetric $n \times n$ matrix needs to be specified generically to be completable to a symmetric tropical rank 2 matrix, as well as to a symmetric classical rank 2 matrix.

This is based on joint work with Cvetelina Hill and Kisun Lee.

Concentration of the Chromatic Number of Random Graphs

Series
Graph Theory Seminar
Time
Tuesday, May 17, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lutz WarnkeUCSD
What can we say about the chromatic number \chi(G_{n,p}) of an n-vertex binomial random graph G_{n,p}? From a combinatorial perspective, it is natural to ask about the typical value of \chi(G_{n,p}), i.e., upper and lower bounds that are close to each other. From a probabilistic combinatorics perspective, it is also natural to ask about the concentration of \chi(G_{n,p}), i.e., how much this random variable varies. Among these two fundamental questions, significantly less is known about the concentration question that we shall discuss in this talk. In terms of previous work, in the 1980s Shamir and Spencer proved that the chromatic number of the binomial random graph G_{n,p} is concentrated in an interval of length at most \omega\sqrt{n}, and in the 1990s Alon showed that an interval of length \omega\sqrt{n}/\log n suffices for constant edge-probabilities p\in (0,1). In this talk, we prove a similar logarithmic improvement of the Shamir-Spencer concentration results for the sparse case p=p(n) \to 0, and also discuss several intriguing questions about the chromatic number \chi(G_{n,p}) that remain open. Based on joint work with Erlang Surya; see https://arxiv.org/abs/2201.00906

Approximation of invariant manifolds for Parabolic PDEs over irregular domains

Series
CDSNS Colloquium
Time
Friday, May 13, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Jorge GonzalezGeorgia Tech

Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09

 

The computation of invariant manifolds for parabolic PDE is an important problem due to its many applications. One of the main difficulties is dealing with irregular high dimensional domains when the classical Fourier methods are not applicable, and it is necessary to employ more sophisticated numerical methods. This work combines the parameterization method based on an invariance equation for the invariant manifold, with the finite element method. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two-dimensional polygonal domains (not necessary simply connected), for equilibrium solutions. We implement a-posteriori error indicators which provide numerical evidence of the accuracy of the computations. This is a joint work with J.D Mireles-James, and Necibe Tuncer.  

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