Seminars and Colloquia by Series

First passage percolation: exceptional events and asymptotic behavior of invasion restricted geodesics

Series
Dissertation Defense
Time
Thursday, April 13, 2023 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
David Harper – Georgia Tech –

 In first-passage percolation (FPP), we let $\tau_v$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results that show that if the sum of $F^{-1}(1/2+1/2^k)$ diverges, then a.s. there are exceptional times at which the weight grows atypically, but if the sum of $k^{7/8} F^{-1}(1/2+1/2^k)$ converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. Then we will consider what the model looks like when the weight of a long path is unusually small by considering an analogous construction to Kesten's incipient infinite cluster in the FPP setting. This is joint work with M. Damron, J. Hanson, W.-K. Lam.

Finally, we discuss a result related to work of Damron-Lam-Wang ('16) that the growth of the passage time to distance $n$ ($\mathbb{E}T(0,\partial B(n))$, where $B(n) = [-n,n]^2$)  has the same order (up to a constant factor) as the sequence $\mathbb{E}T^{\text{inv}}(0,\partial B(n))$. This second passage time is the minimal total weight of any path from 0 to $\partial B(n)$ that resides in a certain embedded invasion percolation cluster. We discuss a result that claims this constant factor cannot be taken to be 1. This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structures. This was joint work with M. Damron. 

 

 

Quantum trace maps for skein algebras

Series
Dissertation Defense
Time
Friday, April 7, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tao Yu – Georgia Institute of Technology –

We study quantizations of SL_n-character varieties, which appears as moduli spaces for many geometric structures. Our main goal is to establish the existence of several quantum trace maps. In the classical limit, they reduce to the Fock-Goncharov trace maps, which are coordinate charts on moduli spaces of SL_n-local systems used in higher Teichmuller theory. In the quantized theory, the algebras are replaced with non-commutative deformations. The domains of the quantum trace maps are the SL_n-skein algebra and the reduced skein algebra, and the codomains are quantum tori, which are non-commutative analogs of Laurent polynomial algebras. In this talk, I will review the classical theory and sketch the definition of the quantum trace maps.

Algebraic and semi-algebraic invariants on quadrics

Series
Dissertation Defense
Time
Friday, July 22, 2022 - 08:30 for 2 hours
Location
Skiles 006 and Zoom meeting (https://gatech.zoom.us/j/96755126860)
Speaker
Jaewoo Jung – Georgia Institute of Technology –

Dissertation defense information

Date and Time: July 22, 2022, 08:30 am - 10:30 am (EST)

Location:

  • Skiles 006 (In-person)
  • Zoom meeting (Online): https://gatech.zoom.us/j/96755126860

 

Summary

This dissertation consists of two topics concerning algebraic and semi-algebraic invariants on quadrics.

 The ranks of the minimal graded free resolution of square-free quadratic monomial ideals can be investigated combinatorially. We study the bounds on the algebraic invariant, Castelnuovo-Mumford regularity, of the quadratic ideals in terms of properties on the corresponding simple graphs. Our main theorem is the graph decomposition theorem that provides a bound on the regularity of a quadratic monomial ideal. By combining the main theorem with results in structural graph theory, we proved, improved, and generalized many of the known bounds on the regularity of square-free quadratic monomial ideals.

 The Hankel index of a real variety is a semi-algebraic invariant that quantifies the (structural) difference between nonnegative quadrics and sums of squares on the variety. This project is motivated by an intriguing (lower) bound of the Hankel index of a variety by an algebraic invariant, the Green-Lazarsfeld index, of the variety. We study the Hankel index of the image of the projection of rational normal curves away from a point. As a result, we found a new rank of the center of the projection which detects the Hankel index of the rational curves. It turns out that the rational curves are the first class of examples that the lower bound of the Hankel index by the Green-Lazarsfeld index is strict.

 

Advisor: Dr. Grigoriy Blekherman, School of Mathematics, Georgia Institute of Technology

Committee:

  • Dr. Matthew Baker, School of Mathematics, Georgia Institute of Technology
  • Dr. Anton Leykin, School of Mathematics, Georgia Institute of Technology
  • Dr. Rainer Sinn, Institute of Mathematics, Universität Leipzig
  • Dr. Josephine Yu, School of Mathematics, Georgia Institute of Technology

 

Application of Circle Method in Five Number Theory Problems

Series
Dissertation Defense
Time
Friday, July 15, 2022 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hamed Mousavi – Georgia Institute of Technology –

This thesis consists of three applications of the circle method in number theory problems. In the first part, we study the $p-$divisibility of the central binomial coefficients. For a certain set of large prime numbers, we prove that there are infinitely many integers $n$, which $\binom{2n}{n}$ has these primes with unexpectedly small multiplicity in its prime factorization. This result is related to an open problem conjectured by Graham, stating that there are infinitely many integers $n$ which the binomial coefficients $\binom{2n}{n}$ is coprime with $105$. The proof consists of the Fourier analysis method, as well as geometrically bypassing an old conjecture about the primes.

In the second part, we discover an unexpected cancellation on the sums involving the exponential functions. Applying this theorem on the first terms of the Ramanujan-Hardy-Rademacher expansion gives us a natural proof of a ``weak" pentagonal number theorem. We find several similar upper bounds for many different partition functions. Additionally, we prove another set of ``weak" pentagonal number theorems for the primes, which allows us to count the number of primes in certain intervals with small error. Finally, we show an approximate solution to the Prouhet-Tarry-Escott problem using a similar technique. The core of the proofs is an involved circle method argument.

The third part of this thesis is about finding an endpoint $\ell^p-$improving inequality for an ergodic sum involving the primes. As the set of the prime is almost full-dimensional, the question on the endpoint becomes more interesting, because we want to bound $\ell^{\infty}$ to $\ell^{1}$ operator. The weak-type inequality we propose depends on the assumption of the Generalized Riemann Hypothesis. Assuming GRH, we prove the sharpest possible bound up to a constant. Unconditionally, we prove the same inequality up to a $\log $ factor.  The proof is based on a circle method argument and careful use of the Ramanujan sums.

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 Schmidt – Georgia 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 Yoo – Georgia 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 Yuan – Georgia 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 Mukherjee – Georgia 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.

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