Events

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

Yash Singh, University of Waterloo

Buildings of reductive groups.

We study an algebraic construction of the spherical building of the reductive group due to Halpern-Leistner and a connection between this construction and the classification of toric vector bundles by Kiaveh-Manon.

MC 5403

Caleb Suan, Chinese University of Hong Kong

Hull-Strominger Systems and Geometric Flows

The Hull-Strominger system is a system of partial differential equations stemming from heterotic string theory in physics. Mathematically, these equations lead us to consider special structures with torsion and have been proposed as a natural generalization of the Ricci-flat condition on non-Kahler Calabi-Yau threefolds. In this talk, we discuss a geometric flow approach to the system, known as the anomaly flow. We shall also look at 7-dimensional analogues of the system and flow.

MC 5417

Catherine St-Pierre, University of Waterloo

Why does the Spec functor not extend to non-commutative rings?

The functor Spec, which assigns to a commutative ring its prime spectrum, plays a central role in algebraicgeometry. A natural question is whether this construction can be extended in a meaningful way tononcommutative rings. In this talk, we discuss the obstruction to the extension of the functor Spec to non-commutative rings presented by Manuel L. Reyes, showing that any functor extending Spec and satisfyingreasonable compatibility conditions must collapse on certain noncommutative rings, such as matrix algebras$M_3(\mathbb C)$.

MC 5417

William Dan, University of Waterloo

A Characterization of Random, Left C.E. Reals

An immediate property of the halting probability of a prefix-free machine is that it is a left c.e. real. An easycorollary of the Kraft-Chaitin theorem is that the converse is true: any left c.e. real is the halting probability ofsome prefix-free machine. The most common example of a random real is Chaitin's omega, the haltingprobability of a universal prefix-free machine. In fact, it is a random left c.e. real. It is natural then to ask if theconverse holds in this case as well: that any random left c.e. real is the halting probability of some universalprefix-free machine. As it turns out, this is the case, and in this talk I will explain the concept used to solve thisquestion, Solovay reducibility, then prove the theorems demonstrating the converse. This talk follows sections9.1 and 9.2 of the Downey and Hirschfeldt book.

MC 5403