Scientific

The influence of the Galois group structure on the Chebyshev bias in number fields

Speaker: 
Mounir Hayani
Date: 
Tue, Jun 18, 2024
Location: 
PIMS, University of British Columbia
Conference: 
Comparative Prime Number Theory
Abstract: 

In 2020, Fiorilli and Jouve proved an unconditional Chebyshev bias result for a Galois extension of number fields under a group theoretic condition on its Galois group. We extend their result to a larger family of groups. This leads us to characterize abelian groups enabling extreme biases. In the case of prime power degree extensions, we give a simple criterion implying extreme biases and we also investigate the corresponding Linnik-type question.

Class: 

The distribution of analytic ranks of elliptic curve over prime cyclic number fields

Speaker: 
Gyeongwon Oh
Date: 
Mon, Jun 17, 2024
Location: 
PIMS, University of British Columbia
Conference: 
Comparative Prime Number Theory
Abstract: 

Let $E$ be an elliptic curve over $\mathbb{Q}$ and $C_l$ be the family of prime cyclic extensions of degree $l$ over $\mathbb{Q}$. Under GRH for elliptic L-functions, we give a lower bound for the probability for $K \in C_l$ such that the difference $r_K(E) − r_\mathbb{Q}(E)$ between analytic rank is less than a for $a \asymp l$. This result gives conjectural evidence that the Diophantine Stability problem suggested by Mazur and Rubin holds for most of $K \in C_l$.

Class: 

Joint distribution of central values and orders of Sha groups of quadratic twists of an elliptic curve

Speaker: 
Peng-Jie Wong
Date: 
Mon, Jun 17, 2024
Location: 
PIMS, University of British Columbia
Conference: 
Comparative Prime Number Theory
Abstract: 

As a refinement of Goldfeld’s conjecture, there is a conjecture of Keating–Snaith asserting that $\log L(1/2,E_d)$ for certain quadratic twists $E_d$ of an elliptic curve $E$ behaves like a normal random variable. In light of this, Radziwill and Soundararajan conjectured that the distribution of $\log(|Sha(E_d)|/\sqrt{|d|}$ is approximately Gaussian for these $E_d$, and proved that the conjectures of Keating–Snaith and theirs are both valid “from above”. More recently, under GRH, they further established a lower bound for the involving distribution towards Keating–Snaith’s conjecture. In this talk, we shall discuss the joint distribution of central values and orders of Sha groups of $E_d$ and how to adapt Radziwill–Soundararajan’s methods to study upper bound and lower bounds for such a joint distribution if time allows.

Class: 

Prime Number Error Terms

Speaker: 
Nathan Ng
Date: 
Mon, Jun 17, 2024
Location: 
PIMS, University of British Columbia
Conference: 
Comparative Prime Number Theory
Abstract: 

In 1980 Montgomery made a conjecture about the true order of the error term in the prime number theorem. In the early 1990s Gonek made an analogous conjecture for the sum of the Mobius function. In 2012 I further revised Gonek’s conjecture by providing a precise limiting constant. This was based on work on large deviations of sums of independent random variables. Similar ideas can be applied to any prime number error term. In this talk I will speculate about the true order of prime number error terms.

Class: 
Subject: 

The Shanks–Rényi prime number race problem

Speaker: 
Youness Lamzouri
Date: 
Mon, Jun 17, 2024
Location: 
PIMS, University of British Columbia
Conference: 
Comparative Prime Number Theory
Abstract: 

Let $\pi(x; q, a)$ be the number of primes $p\leq x$ such that $p \equiv a (\mod q)$. The classical Shanks–Rényi prime number race problem asks, given positive integers $q \geq 3$ and $2 \leq r \leq \phi(q)$ and distinct reduced residue classes $a_1, a_2, . . . , a_r$ modulo $q$, whether there are infinitely many integers $n$ such that $\pi (n; q, a1) > \pi(n; q, a2) > \cdots > \pi(n; q, ar)$. In this talk, I will describe what is known on this problem when the number of competitors $r \geq 3$, and how this compares to the Chebyshev’s bias case which corresponds to $r = 2$.

Class: 

Fake mu's: Make Abstracts Great Again!

Speaker: 
Tim Trudgian
Date: 
Mon, Jun 17, 2024
Location: 
PIMS, University of British Columbia
Conference: 
Comparative Prime Number Theory
Abstract: 

The partial sums of the Liouville function $\lambda(n)$ are "often" negative, and yet the partials sums of the Möbius function $\mu(n)$ are positive or negative "roughly equally". How can this, be, given that $\mu(n)$ and $\lambda(n)$ are so similar? I shall discuss some problems in this area, some joint work with Greg Martin and Mike Mossinghoff, and a possible application to zeta-zeroes.

Class: 

Introduction to unbalanced optimal transport and its efficient computational solutions

Speaker: 
Laetitia Chapel
Date: 
Thu, May 23, 2024
Location: 
PIMS, University of Washington
Zoom
Online
Conference: 
Kantorovich Initiative Seminar
Abstract: 

Optimal transport operates on empirical distributions which may contain acquisition artifacts, such as outliers or noise, thereby hindering a robust calculation of the OT map. Additionally, it necessitates equal mass between the two distributions, which can be overly restrictive in certain machine learning or computer vision applications where distributions may have arbitrary masses, or when only a fraction of the total mass needs to be transported. Unbalanced Optimal Transport addresses the issue of rebalancing or removing some mass from the problem by relaxing the marginal conditions. Consequently, it is often considered to be more robust, to some extent, against these artifacts compared to its standard balanced counterpart. In this presentation, I will review several divergences for relaxing the marginals, ranging from vertical divergences like the Kullback-Leibler or the L2-norm, which allow for the removal of some mass, to horizontal ones, enabling a more robust formulation by redistributing the mass between the source and target distributions. Additionally, I will discuss efficient algorithms that do not necessitate additional regularization on the OT plan.

Class: 
Subject: 

Generic Representations and ABV packets for p-adic Groups

Speaker: 
Sarah Dijols
Date: 
Thu, Apr 18, 2024
Location: 
PIMS, University of British Columbia
Conference: 
UBC Number Theory Seminar
Abstract: 

After a brief introduction on the theory of p-adic groups complex representations, I will explain why tempered and generic Langlands parameters are open. I will further derive a number of consequences, in particular for the enhanced genericity conjecture of Shahidi and its analogue in terms of ABV packets. This is a joint work with Clifton Cunningham, Andrew Fiori, and Qing Zhang.

Class: 

Hypergeometric functions through the arithmetic kaleidoscope

Speaker: 
Ling Long
Date: 
Thu, Apr 11, 2024
Location: 
PIMS, University of British Columbia
Conference: 
UBC Number Theory Seminar
Abstract: 

The classical theory of hypergeometric functions, developed by generations of mathematicians including Gauss, Kummer, and Riemann, has been used substantially in the ensuing years within number theory, geometry, and the intersection thereof. In more recent decades, these classical ideas have been translated from the complex setting into the finite field and p
-adic settings as well.

In this talk, we will give a friendly introduction to hypergeometric functions, especially in the context of number theory.

Class: 

The Distribution of Logarithmic Derivatives of Quadratic L-functions in Positive Characteristic

Speaker: 
Félix Baril Boudreau
Date: 
Thu, Feb 29, 2024
Location: 
PIMS, University of Lethbridge
Zoom
Online
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

To each square-free monic polynomial $D$ in a fixed polynomial ring $\mathbb{F}_q[t]$, we can associate a real quadratic character $\chi_D$, and then a Dirichlet $L$-function $L(s,\chi_D)$. We compute the limiting distribution of the family of values $L'(1,\chi_D)/L(1,\chi_D)$ as $D$ runs through the square-free monic polynomials of $\mathbb{F}_q[t]$ and establish that this distribution has a smooth density function. Time permitting, we discuss connections of this result with Euler-Kronecker constants and ideal class groups of quadratic extensions. This is joint work with Amir Akbary.

Class: 

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