Events

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

Diego Bejarano, York University

Definability and Scott rank in separable metric structures

In [2], Ben Yaacov et. al. extended the basic ideas of Scott analysis to metric structures in infinitary continuous logic. These include back-and-forth relations, Scott sentences, and the Lopez-Escobar theorem to name a few. In this talk, I will talk on my work connecting the ideas of Scott analysis to the definability of automorphism orbits and a notion of isolation for types within separable metric structures. Our results are a continuous analogue of the more robust Scott rank developed by Montalbán in [3] for countable structures in discrete infinitary logic. However, there are some differences arising from the subtleties behind the notion of definability in continuous logic.

[1] Diego Bejarano, Definability and Scott rank in separable metric structures, https://arxiv.org/abs/2411.01017,

[2] Itaï Ben Yaacov, Michal Doucha, Andre Nies, and Todor Tsankov, Metric Scott analysis, Advances in Mathematics, vol. 318 (2017), pp.46–87.

[3] Antonio Montalbán, A robuster Scott rank, Proceedings of the American Mathematical Society, vol.143 (2015), no.12, pp.5427–5436.

MC 5417

Leigh Foster, University of Waterloo

Proving the count of boxed plane partitions (box stackings) via the RSK algorithm

The study of lozenge tilings and of the dimer model is a well-established area of research, going back to the 1960's and still subject to active research at present. We will start the learning seminar on this topic with a series of three meetings giving an introduction to the dimer model in its single-dimer version, and considered on a finite hexagonal grid.

This week, we will present a proof of Percy MacMahon's generating functions plane partitions. We will use (a modified version of) the RSK algorithm, also known as the Robinson–Schensted–Knuth correspondence. This gives a count of dimer covers on the hexagonal grid, lozenge tilings of the triangular lattice, and plane partitions, as well as other combinatorial objects.

No prior knowledge of RSK, plane partitions, or much combinatorics is required, and participation is encouraged! Come and learn and ask your questions.

MC 5403

Aareyan manzoor, University of Waterloo

1 bounded entropy, strong convergence and peterson thom conjecture

I will introduce 1 bounded entropy and show connections to strong convergence. We will discuss how this was used to resolve the peterson thom conjecture, which says that every amenable and diffuse subalgebra of free group factors are contained in a unique maximal amenable subalgebra.

MC 5479

Paul Cusson , University of Waterloo

Spectral curves of Euclidean SU(N)-monopoles

Monopoles over Euclidean R^3 with gauge group SU(N), originally analytic objects, can be studied using the algebro-geometric properties of their spectral curves. We will discuss known results about these curves and how they depend on the asymptotics of the monopole's Higgs field. We will then go over some elementary results that restrict the possible degrees of the spectral curves when we impose symmetries on these monopoles from finite subgroups of SO(3)
MC 5403