Events

Viktor Majewski, University of Waterloo

Dirac Operators on Orbifold Resolutions

In this talk we discuss Dirac operators along degenerating families of Riemannian manifolds that converge, in the Gromov-Hausdorff sense, to a Riemannian orbifold. Such degenerations arise naturally when analysing the boundary of Teichmüller spaces of special Riemannian metrics as well as moduli spaces appearing in gauge theory and calibrated geometry. Here sequences of smooth geometric structures on Riemannian manifolds may converge to an orbifold limit. To understand and control these degenerations, we introduce smooth Gromov-Hausdorff resolutions of orbifolds, that are, smooth families (X_t,g_t), which collapse to the orbifold (X_0,g_0) as t goes to 0.

The central analytic problem addressed in this paper is to understand the behaviour of Dirac operators along such resolutions, in particular in collapsing regimes where classical elliptic estimates fail. We develop a uniform Fredholm theory for the family of Dirac operators on the Gromov-Hausdorff resolution. Using weighted function spaces, adiabatic analysis, and a decomposition of X_t into asymptotically conical fibred (ACF), conically fibred (CF) and conically fibred singular (CFS), we obtain uniform realisations of the model operators and prove a linear gluing exact sequence relating global and local (co)kernels. As a consequence, we construct uniformly bounded right inverses for D_t, and derive an index additivity formula.

The theory developed here provides the analytic foundation for nonlinear gluing problems in gauge theory and special holonomy geometry, including torsion-free G-structures, instantons, and calibrated submanifolds of Riemannian manifolds close to an orbifold limit.

MC 5403

Beining Mu, University of Waterloo

Algorithmic randomness and Turing degrees 3

In this seminar we talk about coding strategies to encode an arbitrary set into a 1-random set in a sense that every set is wtt-reducible to a 1-random set. We will also have a review of the jump operator and lowness of Turing degrees to explore the distribution of 1-random sets in terms of Turing degrees.

MC 5403

Luke Postle, University of Waterloo

A New Proof of the Existence Conjecture and its Applications to Extremal and Probabilistic Design Theory

We discuss the recently developed method of refined absorption and how it is used to provide a new proof of the Existence Conjecture for combinatorial designs. This method can also be applied to resolve open problems in extremal and probabilistic design theory while providing a unified framework for these problems. Crucially, the main absorption theorem can be used as a ``black-box'' in these applications obviating the need to reprove the absorption step for each different setup.

MC 5403

Carlo Pagano, Concordia University

Reconstructing curves from their Galois set of points

Mazur—Rubin asked to what extent one can reconstruct a curve C over a number field K from its set of points over bar(K), viewed as a Galois set. They asked the same question specifically about the set of fields where C acquires new points and gave evidence for a positive answer for curves of genus 0. In this talk we will present upcoming work with Zev Klagsbrun where we provide a positive answer for a generic pair of elliptic curves with full 2-torsion over a number field. The method of proof uses the combination of additive combinatorics and descent introduced in joint work of the speaker and Koymans in 2024. I will overview several other recent results obtained, by a number of authors, with that method.

MC 5479

Fateme Peimany and/or Jules Ribolzi, University of Waterloo

Meromorphic groups

We redefine meromorphic groups as group objects in the category of abstract meromorphic varieties, and check this agrees with the notion introduced by Pillay-Scanlon.

MC 5479

Kostiantyn Drach, Universitat de Barcelona

Reverse inradius inequalities for ball-bodies

A ball-body, also called a $\lambda$-convex body, is an intersection of congruent Euclidean balls of radius $1/\lambda$ in $\mathbb{R}^n$, $n \geq 2$. Such bodies arise naturally in optimization problems in combinatorial and convex geometry, in particular when the number of generating balls is finite. In recent years, ball-bodies have also played a central role in an active research program on reverse isoperimetric-type problems under curvature constraints. The general objective of this program is to understand how prescribed curvature bounds restrict the extremal behavior of geometric functionals (e.g., volume, surface area, or mean width), and to identify sharp inequalities between them that reverse the existing classical isoperimetric-type inequalities. In this talk, we focus on the inradius minimization problem for $\lambda$-convex bodies with prescribed surface area or prescribed mean width. Here, the inradius of a convex body $K$ is the radius of the largest ball contained in $K$. In this setting, we establish sharp lower bounds for the inradius and show that equality is attained only by lenses, that is, intersections of two balls of radius $1/\lambda$. This solves a conjecture of Karoly Bezdek. We will outline the main ideas of the proof and pose several open problems. This is joint work with Kateryna Tatarko.

MC 5417

Duncan McCoy, Université du Québec à Montréal

The unknotting number of positive alternating knots

The unknotting number is simultaneously one of the simplest classical knot invariants to define and one of the most challenging to compute. This intractability stems from the fact that typically one has no idea which diagrams admit a collection of crossing changes realizing the unknotting number for a given knot. For positive alternating knots, one can show that if the unknotting number equals the lower bound coming from the classical knot signature, then the unknotting number can be calculated from an alternating diagram. I will explain this result along with some of the main tools in the proof, which are primarily from smooth 4-dimensional topology. This is joint work with Paolo Aceto and JungHwan Park.

MC 5417

Jules Ribolzi, University of Waterloo

Meromorphic groups

We continue the proof that definable groups in CCM are meromorphic.

MC 5479

Beining Mu, University of Waterloo

Algorithmic randomness and Turing degrees 4

In this seminar we will talk about the Hyperimmune-Free Basis Theorem and its application to understanding the distribution of 1-random Turing degrees. In addition, we will also cover Demuth's Theorem and its applications.

MC 5403

Nikita Lvov

Random Walks arising in Random Matrix Theory

The cokernel of a large p-adic random matrix M is a random abelian p-group. Friedman and Washington showed that its distribution asymptotically tends to the well-known Cohen-Lenstra distribution. We study an irreducible Markov chain on the category of finite abelian p-groups, whose stationary measure is the Cohen-Lenstra distribution. This Markov chain arises when one studies the cokernels of corners of M. We show two surprising facts about this Markov chain. Firstly, it is reversible. Hence, one may regard it as a random walk on finite abelian p-groups. The proof of reversibility also explains the appearance of the Cohen-Lenstra distribution in the context of random matrices. Secondly, we can explicitly determine the spectrum of the infinite transition matrix associated to this Markov chain. Finally, we show how these results generalize to random matrices over general pro-finite local rings.

MC 5403