Mathematics

Spaces of geodesic triangulations of surfaces

Speaker: 
Yanwen Luo
SFU
Date: 
Wed, Nov 29, 2023
Location: 
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

In 1962, Tutte proposed a simple method to produce a straight-line embedding of a planar graph in the plane, known as Tutte's spring theorem. It leads to a surprisingly simple proof of a classical theorem proved by Bloch, Connelly, and Henderson in 1984, which states that the space of geodesic triangulations of a convex polygon is contractible. In this talk, I will introduce spaces of geodesic triangulations of surfaces, review Tutte's spring theorem, and present this short proof. It time permits, I will briefly report the recent progress in identifying the homotopy types of spaces of geodesic triangulations of general surfaces.

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Around Artin's primitive root conjecture

Speaker: 
Paul Péringuey
UBC
Date: 
Wed, Nov 22, 2023
Location: 
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

In this talk we will first discuss this soon to be 100 years old conjecture, which states that the set of primes for which an integer \(a\) different from \(-1\) or a perfect square is a primitive root admits an asymptotic density among all primes. In 1967 Hooley proved this conjecture under the Generalized Riemann Hypothesis.

After that, we will look into a generalization of this conjecture, where we don't restrain ourselves to look for primes for which \(a\) is a primitive root but instead elements of an infinite subset of \(\mathbb{N}\) for which \(a\) is a generalized primitive root. In particular, we will take this infinite subset to be either \(\mathbb{N}\) itself or integers with few prime factors.

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On the eigenvalues of the graphs D(5,q)

Speaker: 
Himanshu Gupta
URegina
Date: 
Wed, Oct 25, 2023
Location: 
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

In 1995, Lazebnik and Ustimenko introduced the family of q-regular graphs D(k,q), which is defined for any positive integer k and prime power q. The connected components of the graph D(k, q) have provided the best-known general lower bound on the size of a graph for any given order and girth to this day. Furthermore, Ustimenko conjectured that the second largest eigenvalue of D(k, q) is always less than or equal to 2\sqrt{q}, indicating that the graphs D(k, q) are good expanders. In this talk, we will discuss some recent progress on this conjecture. This includes the result that the second largest eigenvalue of D(5, q) is less than or equal to 2\sqrt{q} when q is an odd prime power. This is joint work with Vladislav Taranchuk.

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Bounds on the Number of Solutions to Thue Equations

Speaker: 
Gregory Knapp
UCalgary
Date: 
Wed, Oct 11, 2023
Location: 
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

In 1909, Thue proved that when $F(x,y) \in \mathbb{Z}[x,y]$ is irreducible, homogeneous, and has degree at least 3, the inequality $|F(x,y)| \leq h$ has finitely many integer-pair solutions for any positive $h$. Because of this result, the inequality $|F(x,y)| \leq h$ is known as Thue’s Inequality and much work has been done to find sharp bounds on the number of integer-pair solutions to Thue’s Inequality. In this talk, I will describe different techniques used by Akhtari and Bengoechea; Baker; Bennett; Mueller and Schmidt; Saradha and Sharma; and Thomas to make progress on this general problem. After that, I will discuss some improvements that can be made to a counting technique used in association with "the gap principle’’ and how those improvements lead to better bounds on the number of solutions to Thue’s Inequality.

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Understanding adversarial robustness via optimal transport perspective.

Speaker: 
Jakwang Kim
UBC
Date: 
Wed, Sep 27, 2023
Location: 
Online
Conference: 
Emergent Research: The PIMS Postdoctoral Fellow Seminar
Abstract: 

In this talk, I will present the recent progress of understanding adversarial multiclass classification problems, motivated by the empirical observation of the sensitivity of neural networks by small adversarial attacks. From the perspective of optimal transport theory, I will give equivalent reformulations of this problem in terms of 'generalized barycenter problems' and a family of multimarginal optimal transport problems. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. Furthermore, based on this optimal transport approach I will give the result of the existence of optimal robust classifiers which not only extends the binary setting to the general one but also provides shorter proof and an interpretation between adversarial training problems and related generalized barycenter problems.

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Subject: 

Sofic groups are surjunctive

Speaker: 
Keivan Mallahi-Karai
Constructor University
Date: 
Mon, Nov 27, 2023
Location: 
Online
University of Utah
Conference: 
University of Utah Seminar in Ergodic Theory
Abstract: 

In this talk, which is based on Benjamin Weiss' Sofic groups and dynamical systems, I will give the definition of sofic groups which can be considered as common generalizations of amenable and residually finite groups, and discuss examples and some of their basic properties. Finally, we will give Weiss' alternative proof Gromov's theorem stating that sofic groups are surjunctive. This theorem settles Gottschalk's conjecture for sofic groups.

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The eighth moment of $\Gamma_1(q)$ L-functions

Speaker: 
Vorrapan Chandee
Date: 
Mon, Nov 6, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

In this talk, I will discuss my on-going joint work with Xiannan Li on an unconditional asymptotic formula for the eighth moment of $\Gamma_1(q)$ L-functions, which are associated with eigenforms for the congruence subgroups $\Gamma_1(q)$. Similar to a large family of Dirichlet L-functions, the family of $\Gamma_1(q)$ L-functions has a size around $q^2$ while the conductor is of size $q$. An asymptotic large sieve of the family is available by the work of Iwaniec and Xiaoqing Li, and they observed that this family of harmonics is not perfectly orthogonal. This introduces certain subtleties in our work.

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Twisted moments of characteristic polynomials of random matrices

Speaker: 
Siegfred Baluyot
Date: 
Mon, Nov 27, 2023
Location: 
PIMS, University of Lethbridge
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
Abstract: 

In the late 90's, Keating and Snaith used random matrix theory to predict the exact leading terms of conjectural asymptotic formulas for all integral moments of the Riemann zeta-function. Prior to their work, no number-theoretic argument or heuristic has led to such exact predictions for all integral moments. In 2015, Conrey and Keating revisited the approach of using divisor sum heuristics to predict asymptotic formulas for moments of zeta. Essentially, their method estimates moments of zeta using lower twisted moments. In this talk, I will discuss a rigorous random matrix theory analogue of the Conrey-Keating heuristic. This is ongoing joint work with Brian Conrey.

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A survey of Büthe's method for estimating prime counting functions

Speaker: 
Sreerupa Bhattacharjee
Date: 
Tue, Nov 21, 2023
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

This talk will begin with a study on explicit bounds for $\psi(x)$ starting with the work of Rosser in 1941. It will also cover various improvements over the years including the works of Rosser and Schoenfeld, Dusart, Faber-Kadiri, Platt-Trudgian, Büthe, and Fiori-Kadiri-Swidinsky. In the second part of this talk, I will provide an overview of my master's thesis which is a survey on 'Estimating $\pi(x)$ and Related Functions under Partial RH Assumptions' by Jan Büthe. This article provides the best known bounds for $\psi(x)$ for small values of $x$ in the interval $[e^{50},e^{3000}]$. A distinctive feature of this paper is the use of Logan's function and its Fourier Transform. I will be presenting the main theorem in Büthe's paper regarding estimates for $\psi(x)$ with other necessary results required to understand the proof.

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Some Pólya Fields of Small Degrees

Speaker: 
Abbas Maarefparvar
Date: 
Tue, Nov 7, 2023
Location: 
PIMS, University of Lethbridge
Online
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

Historically, the notion of Pólya fields dates back to some works of George Pólya and Alexander Ostrowski, in 1919, on entire functions with integer values at integers; a number field $K$ with ring of integers $\mathcal{O}_K$ is called a Pólya field whenever the $\mathcal{O}_K$-module $\{f \in K[X] \, : \, f(\mathcal{O}_K) \subseteq \mathcal{O}_K \}$ admits an $\mathcal{O}_K$-basis with exactly one member from each degree. Pólya fields can be thought of as a generalization of number fields with class number one, and their classification of a specific degree has become recently an active research subject in algebraic number theory. In this talk, I will present some criteria for $K$ to be a Pólya field. Then I will give some results concerning Pólya fields of degrees $2$, $3$, and $6$.

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