Events

François Greer, Michigan State University

"Finiteness of monodromy for fibered Calabi-Yau threefolds"

An old question going back to S.T. Yau asks whether there are finitely many diffeomorphism types for smooth projective Calabi-Yau manifolds of a given dimension. The answer is affirmative for dimensions one and two (elliptic curves and K3 surfaces). It has recently been settled for Calabi-Yau threefolds admitting elliptic fibrations. We discuss the case of CY3’s admitting abelian surface or K3 fibrations. 

MC 5417

Konstantin Tikhomirov, Carnegie Mellon University

"On the width of random polyhedra"

We consider the problem of estimating the width of a polyhedron defined as the intersection of m i.i.d random affine subspaces of n-dimensional space. Such polyhedra naturally appear in probabilistic analysis of linear programs, as well as in convex geometric analysis as extremizers of various quantities associated with convex sets. For a wide range of parameters m, n, we obtain sharp estimates of the width of the polyhedron in any given direction.

MC 5501

Micah Milinovich, University of Mississippi

"Fourier optimization, prime gaps, and the least quadratic non-residue"

There are many situations where one imposes certain conditions on a function and its Fourier transform and then wants to optimize a certain quantity. I will describe two such Fourier optimization frameworks that can be used to study classical problems in number theory: bounding the maximum gap between consecutive primes assuming the Riemann hypothesis and bounding for the size of the least quadratic non-residue modulo a prime assuming the generalized Riemann hypothesis (GRH) for Dirichlet L-functions. The resulting extremal problems can be stated in accessible terms, but finding the exact answer appears to be rather subtle. Instead, we experimentally find upper and lower bounds for our desired quantity that are numerically close. If time allows, I will discuss how a similar Fourier optimization framework can be used to bound the size of the least prime in an arithmetic progression on GRH. This is based upon joint works with E. Carneiro (ICTP), E. Quesada-Herrera (TU Graz), A. Ramos (SISSA), and K. Soundararajan (Stanford). 

MC 5417

Joey Lakerdas-Gayle, Department of Pure Mathematics, University of Waterloo

"Computable Structure Theory IX"

We will discuss effective interpretability of graphs, following Antonio Montalbán's monograph.

MC 5479

Faisal Romshoo, Department of Pure Mathematics, University of Waterloo

"A theoretical framework for H-structures"

For an oriented Riemannian manifold $(M^n, g)$, and Lie subgroup $H \subset SO(n)$, a compatible $H$-structure on $(M^n,g)$ is a principal $H$-subbundle of the principal $SO(n)$-bundle of oriented orthonormal coframes.  They can be described in terms of the sections of the homogeneous fibre bundle obtained by $H$-reduction of the oriented frame bundle. Examples of these structures include $U(m)$-structures, $G_2$-structures and $\text{Spin(7)}$-structures. In this talk, we will study a general theory for $H$-structures described in a paper of Daniel Fadel, Eric Loubeau, Andrés J. Moreno and Henrique N. Sá Earp titled "Flows of geometric structures" (https://arxiv.org/abs/2211.05197).

MC 5403

AJ Fong, Department of Pure Mathematics, University of Waterloo

"Flat families of schemes"

We will discuss families of schemes, and how flatness is the correct relevant notion for families.

MC 5417

Rahim Moosa, Department of Pure Mathematics, University of Waterloo

"Binding groups for rational dynamics"

I will report on ongoing work with Moshe Kamensky toward developing a theory of binding groups for quantifier-free types in ACFA, well-suited for applications to rational algebraic dynamics.

MC 5479

Talk #1: Ted Fu, University of Waterloo

"On Waring's problem for large powers"

Let G(k) be the least number s having the property that every sufficiently large natural number is the sum of at most s positive integer k-th powers. In this talk, I will present how Brüdern and Wooley implement smooth numbers technologies in their minor arc analysis and derive G(k) ≤ ⌈k(log k + 4.20032)⌉.

Talk #2: Aidan Boyle, University of Waterloo

"Waring’s problem: Beyond Freiman’s Theorem"

Suppose that we are given a non-decreasing sequence of positive integers (k_i) where each term is at least 2. Given a positive integer j, we seek to understand the circumstances in which there exists a positive integer s := s(j) such that every sufficiently large natural number n can be written as a sum of s positive integers to the respective powers k_j, ..., k_j+s-1. Freĭman asserted that such representation exists if and only if the infinite summation of all 1/k_i diverges. We provide an effective version of this theorem, and in particular, comment on instances in which the exponents form a sequence of consecutive terms of an arithmetic progression.

MC 5417

Alex Waldron, University of Wisconsin-Madison

"Parabolic gap theorems for the Yang-Mills functional"

Given a principal bundle over a compact Riemannian 4-manifold or special-holonomy manifold, it is natural to ask whether a uniform gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. As applications, we recover a theorem of Taubes on path-connectedness of instanton moduli spaces on the 4-sphere, and obtain a method to construct instantons on quaternion-Kähler manifolds with positive scalar curvature.

MC 5417

Katarzyna Wyczesany, Carnegie Mellon University

"Dualities on sets and how they appear in optimal transport"

In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We prove that any order reversing quasi-involution is of a special form, which arose from the consideration of optimal transport problem with respect to costs that attain infinite values. We will discuss how this unified point of view on order reversing quasi-involutions helps to deepen the understanding of the underlying structures and principles. We will provide many examples and ways to construct new order reversing quasi-involutions from given ones. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09