The annual UVa-Vietnam Polymath REU is an opportunity for mathematical interaction and presentation of research between undergraduates in the University of Virginia Department of Mathematics and the Vietnam Polymath REU
Aims and Scope: The seminar serves as an opportunity for students to present mathematical findings, be they from directed readings or from research, aimed at an audience of their peers. Speakers should be undergraduate mathematics majors at the University of Virginia or participants in the Vietnam Polymath REU program, though any interested party is welcome to attend talks.
Format: The seminar is synchronous and online. Sessions last 1 hour and 30 minutes, enough to accommodate two 30-40 minute talks.
2024 Dates: The 2024 installment of the seminar will take place according to the following schedule. All times are given in EST; Vietnam times are +11 hours.
Recordings of morning and evening.
Samuel Goldberg (9 am EST= 20:00 Hanoi time)
Title: Arithmetic Quantum Chaos
Abstract: In this talk, we will introduce quantum chaos and provide an overview of how techniques from number theory (more specifically, automorphic representation theory and L-functions) can be applied to solve problems in this field. The focal point of this talk will be the Quantum Unique Ergodicity conjecture.
Trần Nguyễn Nam Hưng + Lê Xuân Hoàng (9:50 am EST=20:50 Hanoi time)
Title: Surface subgroups of Baumslag doubles along short words
Abstract: Mikhael Gromov asked the following question in Geometric group theory, Volume 2 (1991): “Does every one-ended word-hyperbolic group have a hyperbolic surface subgroup?”. An interesting case is when the group is a Baumslag double of a finite-rank group, which can be reformulated into a graph-theoretic problem via Whitehead graph whose vertices are letters of the alphabet associated with the free group.
Kim and Oum (2010), who worked on two different research areas, solved this problem in two cases: (1) The Whitehead graph has exactly four vertices, (2) Each vertex of the Whitehead graph has the same valence. We developed their arguments and used our computational approaches to discover this problem further, which successfully tackles with two more cases: (1) The number of edges does not exceed the number of vertices plus four, (2) The Whitehead graph has six edges and is simple.
Declan Stacy (8:30 pm EST= 07:30 am Hanoi time May 7 2024)
Title: Stochastic Extinction and Persistence
Abstract: We tackle the problem of determining the stability of an invariant subset M_0 \subset M for an M-valued Markov process X_t. We introduce the technique of average Lyapunov functions to show that (under some mild technical assumptions) both the stability of M_0 and the rate of convergence of X_t to M_0 (in the stable case) can be determined entirely by a type of Lyapunov exponent which only depends on the steady state behavior(s) of X_t on M_0. This is an improvement over previous results which require constructing a Lyapunov function and analyzing its rate of change on the entirety of M. In the talk we will omit the proofs of the main results and instead focus on examples of applying the theory, which may include analysis of stochastic ecological models, a SIS epidemic model with Markovian switching, and a noisy Lorenz system.
Liran Li (9 am EST=20:00 Hanoi time)
Title: Formal Language and Group Theory
Abstract: In this talk, we are going to do a brief survey of formal language and group theory. Finally I will provide a brief introduction to word problem and automatic group.
Nguyễn Đức Thắng (9:50 am=20:50 Hanoi time)
Title: The solvability of systems revisited
Abstract: In this paper, we introduce a new approach to study the solvability of systems of equations generated by bilinear forms. More precisely, let $B (\cdot,\cdot)$ be a non-degenerate bilinear form and $A$ be a set in $\mathbb{F}_q^2$. In this paper, we prove that $\#\{(B(x, y), B(y, z), B(z, x))\colon x, y, z\in A\}\gg q^3$ whenever $|A|\gg q^{5/3}$.
Lê Quang Hưng (9 am EST=20:00 Hanoi time)
Title: Packing sets under the special linear groups
Abstract: Let $E$ be a set in F_q^2 and S be a set in SL_2(F_q). In this talk, we present new and sharp results on the following problems:
- Finding conditions on E and S such that the size of S(E) is at least cq^2. Here S(E)={Mx: M\in S, x\in E}.
- Given two sets E and F in F_q^2, let S(E:F) be the set of matices M in SL_2 such that there exist x in E and y in F such that Mx=y.
Finding conditions of E and F such that the size of S(E:F) is at least cq^3.
As consequences of our results, we are able to show that an energy estimate due to Babai et al. on the growth of sets in SL_2(F_q) is optimal.
Yunsheng Lu (9:50 am EST= 20:50 Hanoi time)
Title: Inverse Problems and Applications in Partial Differential Equations
Abstract: This is a comprehensive introduction of inverse problems of heat equations on both finite and infinite dimensions. In particular, we will explore both deterministic and statistical ways to recover the initial heat distributions after the system evolves for a period of time. In particular, we will first give a general introduction on topics of inverse problems, both the deterministic, including Truncated Singular Value Decomposition (TSVD) and Tikhonov regularization, and then the statistical methods, for example, a Bayesian framework. We then apply the general methodology to specific questions related to heat equations.
Lê Trần Duy Anh (9 am EST=20:00 Hanoi time)
Title: On the Falconer Distance Problem and Projection Theory
Abstract: In this session, we will discuss some results about the Falconer Distance Problem and Projection problems on hyperplanes.
Nguyễn Đắc Quán (9:50 am EST=20:50 Hanoi time)
Title: Entropy methods and Discretized sum-product estimates
Abstract: In this paper, we develop an entropy method to study discretized sum-product problems. As an application, we obtain new results on the sum-product type problem of $\max\{|A^2+A^2|, A+A|\}$. Related results will also be discussed.
Trần Đức Trọng Văn + Nguyễn Đình Đăng Khoa + Võ Kế Hoàng (9 am EST=20:00 Hanoi time)
Title: LOT Groups
Abstract: In this presentation, we will introduce a class of groups called LOT groups, or labelled oriented tree groups. The purpose of studying these groups is to find a counterexample to the Whitehead Conjecture. Some questions we've been considering lately is necessary and sufficient conditions for a LOT group to be abelian, and whether they are torsion-free. We will also introduce our current approach, which is using the Alexander invariant to distinguish between abelian and non-abelian LOT groups.
Mandy Unterhalter (9:50 am EST=20:50 Hanoi time)
Title: A Study of Artifacts in Compton Scattering Tomography with Fixed Source
Abstract: Limited Data Compton Tomography is a relatively new area of limited data tomography that uses circular arcs to determine the scattering density of an object. The Compton Effect allows for the creation of accurate mathematical methods of synthesizing data on MATLAB. We test a particular data collection technique to determine patterns of artifacts given different objects. This technique involves fixing a source along the unit circle and giving the detector complete freedom about the unit circle. Back-scattering and forward-scattering arcs create reconstructions containing predicted end-of-data-set artifacts, as well as currently inexplicable artifacts that seem to occur when back and forward scattering arcs have the same radius. This research will continue by investigating these artifacts and comparing various methods of synthetic data collection to determine the most effective and accurate data collection method for reconstruction.
Zoom link: Meetin ID: 992 5873 1708 Passcode: Contact organizers
Organized by Thomas Koberda (University of Virginia), Thắng Phạm (Vietnam National University), and Quý Phạm (University of Rochester)
Photo credit: Rotunda, University of Virginia, by Georgia O'Keefe. Creative Commons.