Events

Yunqing Tang, Berkeley

The arithmetic of power series and applications to irrationality

We will discuss a new approach to prove irrationality of certain periods, including the value at 2 of the Dirichlet L-function associated to the primitive quadratic character with conductor -3. Our method uses rational approximations from the literature and we develop a new framework to make use of these approximations. The key ingredient is an arithmetic holonomy theorem built upon earlier work by André, Bost, Charles (and others) on arithmetic algebraization theorems via Arakelov theory. This is joint work with Frank Calegari and Vesselin Dimitrov.

MC 5417

Elan Roth, University of Waterloo

A Continuation of Random Binary Sequences

Filling some gaps left by last week's presenter, we'll show the K is a minimal information content measure and finally conclude that 1-Randomness and ML-Randomness are equivalent.

MC 5403

Lionel Nguyen Van Thé, Université d'Aix-Marseille

Revisiting the canonical Ramsey theorem for finite vector spaces

The infinite Ramsey theorem (1931) asserts that for every integer m, if the collection of all m-subsets of natural numbers is finitely colored, then there exists an infinite subset whose m-subsets are all of the same color. This result does not hold anymore if the number of colors is not finite, but Erdös and Rado (1950) showed that there is still an infinite subset where the coloring takes a very particular form, called "canonical". Both of these results admit appropriate finite forms which hold in the context of vector spaces instead of sets by results of Graham-Leeb-Rothschild (1972) and Voigt (1984). The purpose of this talk will be to present a new approach to this latter result.

MC 5403

Facundo Camano, University of Waterloo

Dimensional Reduction of S^1-Invariant Instantons on the Multi-Taub-NUT

In this talk I will discuss the dimensional reduction of S^1-invariant instantons on the multi-Taub-NUT space to singular monopolos on R^3. I will first introduce the multi-Taub-NUT space, followed up by a discussion on S^1-equivariant principal bundles. Next, I will go over the natural decomposition of S^1-invariant connections into horizontal and vertical pieces, and then show how the self-duality equation reduces to the Bogomolny equation under said decomposition. I will then show how the smoothness of the instanton over the NUT points determines the asymptotic conditions for the singular monopole. Finally, I will go over the reverse construction: starting with a singular monopole on R^3 and building up to an S^1-invariant instanton on the multi-Taub-NUT space.

MC 5403

Jennifer Zhu, University of Waterloo

Limits of Quantum Graphs

Quantum graphs were originally introduced as confusability graphs of quantum channels by Duan, Severini, and Winter. Weaver generalized a quantum graph to any weak-* closed operator system $\mathcal V \subseteq B(\mathcal H)$ that is bimodule over the commutant of some von Neumann algebra $\mathcal M \subseteq B(\mathcal H)$. To date, there seem to be two notions of quantum graph morphism. Weaver introduced and Daws extended a notion of CP morphism of quantum graphs. Musto, Reutter, and Verdon have also defined classical morphisms of quantum graphs in finite dimensions which agrees with CP morphisms in finite dimensions. Notably, however, these morphisms are not UCP maps between operator systems of the respective quantum graphs.

      Using a characterization of quantum relations as left ideals in the extended Haagerup tensor product, we will obtain a notion of quantum graph morphism (and hence limit) using the categories of von Neumann algebras and operator spaces. Time permitting, we will show that this limit recovers profinite classical graphs.

QNC 1507

Laindon Burnett, University of Waterloo

A definable criterion for definability

In 2001, A.A. Muchnik proved the surprising result that for any n, there is a formula within Presburger arithmetic which takes in a predicate A and is true if and only if A is definable in Presburger arithmetic; that is, within this setting, the property of being definable is itself definable. We will go over the proof of this result and, time permitting, discuss its applications to automata theory.

MC 5403

Siyuan Yu, Western University

Symplectic embeddings of balls in ℂP² and the generalized configuration space

Let IEmb(B⁴(c),ℂP²) denote the space of unparameterized symplectic embeddings of k balls of capacities (c₁,...,cₖ), where 1 ≤ k ≤ 8. It is known from the work of S. Anjos, J. Li, T.-J. Li, and M. Pinsonnault that the space of capacities decomposes into convex polygons called stability chambers, and that the homotopy type of IEmb(B⁴(c),ℂP²) depends solely on the stability chambers. Based on recent results of M. Entov and M. Verbitsky on Kähler-type embeddings, we show that for 1 ≤ k ≤ 8, IEmb(B⁴(c),ℂP²) is homotopy equivalent to a union of strata F_I of the configuration space of the complex projective plane F(ℂP²,k). The proof relies on constructing an explicit map from the space of Kähler type embeddings to a generalized version of the configuration space that incorporates both configurations of points and compatible complex structures on ℂP².

MC 5417

Jack Jia, University of Waterloo

Langlands Correspondence : Local, Global, and Possibly Geometric.

The classical global Langlands Correspondence is one of the deepest phenomenon in mathematics: In GL_1, it is equivalent to class field theory; in GL_2, it subsumes the modularity conjecture, which leads to the proof of Fermat’s last theorem. (Disclaimer: I will not talk about these). In this talk, I will explain what the local and global L.C. means, and talk about the geometrization if time permits.

Join us for the Grad colloquium to meet your fellow graduate students. Refreshments will start at 5:00, and the talk will start at 5:30.

MC 5417

Mateusz Wasilewski, Polish Academy of Sciences

Quantum graphs and their symmetries

I will present an overview of the theory of quantum graphs, which form a natural generalization of graphs from the point of view of operator algebras/quantum information. I will discuss several approaches to the theory, each coming with its own motivation. Just like in the classical case, studying symmetries is extremely important and this naturally leads to quantum groups. It turns out that to some extent one can go back: for a rich class of quantum groups one can construct quantum graphs, whose symmetries are given by the original quantum group. It is the quantum analogue of Frucht's theorem.

MC 5501

Kaleb Ruscitti, University of Waterloo

Correspondence between logarithmic connections and framed parabolic bundles on the blow up of a nodal Riemann surface

In this seminar I will explain how a Mehta-Seshadri type correspondence between logarithmic connections and parabolic vector bundles works for a specific setting of interest. That is the blow up of the complex curve xy=t at the nodal point.

MC 5403