# Scientific

## A walk on Legendre paths

The Legendre symbol is one of the most basic, mysterious and extensively studied objects in number theory. It is a multiplicative function that encodes information about whether an integer is a square modulo an odd prime p. The Legendre symbol was introduced by Adrien-Marie Legendre in 1798, and has since found countless applications in various areas of mathematics as well as in other fields including cryptography. In this talk, we shall explore what we call "Legendre paths", which encode information about the values of the Legendre symbol. The Legendre path modulo p is defined as the polygonal path in the plane formed by joining the partial sums of the Legendre symbol modulo p. In particular, we will attempt to answer the following questions as we vary over the primes p: how are these paths distributed? how do their maximums behave? and what proportion of the path is above the real axis? Among our results, we prove that these paths converge in law, in the space of continuous functions, to a certain random Fourier series constructed using Rademakher random multiplicative functions. Part of this work is joint with Ayesha Hussain.

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## Applications of optimal transportation in causal inference

Optimal transportation, at its core, is a powerful framework for obtaining structured yet general couplings between general probability measures based on matching underlying characteristics. This framework lends itself naturally to applications in causal inference: from matching estimation, to difference-in-differences and synthetic controls approaches, to instrumental variable estimation, just to name a few. In this talk, I will provide a non-exhaustive overview of potential applications of optimal transport approaches to causal inference. I will focus on providing an overview of general concepts and ideas.

The talk is based on joint work with Rex Hsieh, Myung Jin Lee, Philippe Rigollet, William Torous, and Yuliang Xu.

## Negative moments of the Riemann zeta-function

**Alexandra Florea (University of California Irvine, USA)**

I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta-function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can obtain non-trivial upper bounds for smaller shifts. I will also discuss some applications to the question of obtaining cancellation of averages of the Mobius function. Joint work with H. Bui.

## A construction of Bowen-Margulis measure (Main talk)

In this talk we try to understand the Bowen-Margulis measure for geodesic flow on manifolds of (variable) negative curvature. In the first half-hour (pre-talk), I will discuss the necessary backgrounds from hyperbolic geometry and dynamics, and during the next hour (main talk), I will explain a construction due to Hamenstädt, which relates Bowen-Margulis measure to the Hausdorff measure with respect to a certain metric.

## A construction of Bowen-Margulis measure (Pre-Talk)

In this talk we try to understand the Bowen-Margulis measure for geodesic flow on manifolds of (variable) negative curvature. In the first half-hour (pre-talk), I will discuss the necessary backgrounds from hyperbolic geometry and dynamics, and during the next hour (main talk), I will explain a construction due to Hamenstädt, which relates Bowen-Margulis measure to the Hausdorff measure with respect to a certain metric.

## The value distribution of the Hurwitz zeta function with an irrational shift

The Hurwitz zeta function $\zeta(s, \alpha)$ is a shifted integer analogue of the Riemann zeta function which shares many of its properties, but is not an ”L-function” under any reasonable definition of the word. We will first review the basics of the value distribution of the Riemann zeta function in the critical strip (moments, Bohr–Jessen theory...) and then contrast it with the value distribution of the Hurwitz zeta function.

Our focus will be on shift parameters $\alpha / \in \mathbb{Q}$, i.e., algebraic irrational or transcendental. We will present a new result (joint with Winston Heap) on moments of these objects on the critical line.

## On the Quality of the ABC-Solutions

**Solaleh Bolvardizadeh (University of Lethbridge, Canada)**

The quality of the triplet $(a,b,c)$, where $\gcd(a,b,c) = 1$, satisfying $a + b = c$ is defined as

$$

q(a,b,c) = \frac{\max\{\log |a|, \log |b|, \log |c|\}}{\log \mathrm{rad}(|abc|)},

$$

where $\mathrm{rad}(|abc|)$ is the product of distinct prime factors of $|abc|$. We call such a triplet an $ABC$-solution. The $ABC$-conjecture states that given $\epsilon > 0$ the number of the $ABC$-solutions $(a,b,c)$ with $q(a,b,c) \geq 1 + \epsilon$ is finite.

In the first part of this talk, under the $ABC$-conjecture, we explore the quality of certain families of the $ABC$-solutions formed by terms in Lucas and associated Lucas sequences. We also introduce, unconditionally, a new family of $ABC$-solutions that has quality $> 1$.

In the remaining of the talk, we prove a conjecture of Erd\"os on the solutions of the Brocard-Ramanujan equation

$$

n! + 1 = m^2

$$

by assuming an explicit version of the $ABC$-conjecture proposed by Baker.

## Theta-finite pro-Hermitian vector bundles from loop groups elements

**Mathieu Dutour (University of Alberta, Canada)**

In the finite-dimensional situation, Lie's third theorem provides a correspondence between Lie groups and Lie algebras. Going from the latter to the former is the more complicated construction, requiring a suitable representation, and taking exponentials of the endomorphisms induced by elements of the group.

As shown by Garland, this construction can be adapted for some Kac-Moody algebras, obtained as (central extensions of) loop algebras. The resulting group is called a loop group. One also obtains a relevant infinite-rank Chevalley lattice, endowed with a metric. Recent work by Bost and Charles provide a natural setting, that of pro-Hermitian vector bundles and theta invariants, in which to study these objects related to loop groups. More precisely, we will see in this talk how to define theta-finite pro-Hermitian vector bundles from elements in a loop group. Similar constructions are expected, in the future, to be useful to study loop Eisenstein series for number fields.

This is joint work with Manish M. Patnaik.

## Torsion points and concurrent lines on Del Pezzo surfaces of degree one

**Julie Desjardins (University of Toronto, Canada)**

The blow up of the anticanonical base point on X, a del Pezzo surface of degree 1, gives rise to a rational elliptic surface E with only irreducible fibers. The sections of minimal height of E are in correspondence with the 240 exceptional curves on X.

A natural question arises when studying the configuration of those curves: If a point of X is contained in “many” exceptional curves, is it torsion on its fiber on E?

In 2005, Kuwata proved for del Pezzo surfaces of degree 2 (where there is 56 exceptional curves) that if “many” equals 4 or more, then yes. In a joint paper with Rosa Winter, we prove that for del Pezzo surfaces of degree 1, if “many” equals 9 or more, then yes. Moreover, we find counterexamples where a torsion point lies at the intersection of 7 exceptional curves.

## The question of q, a look at the interplay of number theory and ergodic theory in continued fractions

In the theory of continued fractions, the denominator of the truncated fraction (often denoted q) contains a great deal of information important in applications. However, q is a surprisingly complicated object from the point of view of ergodic theory. We will look at a few problems related to q and see how different techniques have overcome these difficulties, including modular properties (Moeckel, Fisher-Schmidt), renewal-type theorems (Sinai-Ulcigrai, Ustinov), and "nonstandard" arrangements of points (Avdeeva-Bykovskii).