Events

Francisco Villacis, University of Waterloo

Convexity of Toric Moment Maps

Toric moment maps are arguably the nicest family of moment maps in symplectic geometry. A classical theorem from the 80s state that the images of these moment maps are convex polytopes, which was proven independently by Atiyah, and Guillemin and Sternberg. In this talk I will go through Atiyah's slick proof of the convexity theorem using Morse theory, and if time permits I will talk about other results in this area.

MC 5479

Yasuyuki Kawahigashi, University of Tokyo

Subfactors, quantum 6j-symbols and alpha-induction

Tensor categories have found many applications in physics and mathematics, particularly quantum field theory and condensed matter physics in recent years, as a new type of symmetry generalizing a classical notion of a group. Operator algebras give useful and efficient tools to study tensor categories. A fusion category, a tensor category with certain finiteness condition, is characterized by a finite set of complex numbers satisfying certain compatibility condition, called quantum 6j-symbols. Its variant, called bi-unitary connections, has played an important role in the Jones theory of subfactors in operator algebras. We have a tensor functor called alpha-induction for a braided fusion category, as a quantum version of a classical machinery of an induced representation for a subgroup. We describe alpha-induction in the framework of quantum 6j-symbols from a viewpoint of being of a canonical form.

MC 4042 or join on Zoom

Larissa Kroell, University of Waterloo

What’s that called again? An incomplete journey through ridiculous math names

Most of us have heard of some interestingly named mathematical theorems and objects — some justified others not so much. Additionally, all of us have to deal with the overuse of certain adjectives leading to some regular confusion and having to delete normal from our day-to-day vocabulary. (And don’t even get me started on anything quantum.) In this talk we will go over some of these weirdly named objects and particularly bad examples of not-so-helpful mathematical names. No math was harmed in the making of this talk.

MC 5417 - Refreshments start at 16:30pm

Astrid an Huef, Victoria University of Wellington

Nuclear dimension of C*-algebras of groupoids.

Let G be a locally compact, Hausdorff groupoid.  Guentner, Willet and Yu defined a notion of dynamic asymptotic dimension (dad) for etale groupoids, and used it to find  a bound on the nuclear dimension of C*-algebras of principal groupoids with finite dad.  To have finite dad, a groupoid must have locally finite isotropy subgroups which rules out, for example, the graph groupoids and twists of etale groupoids by trivial circle bundles. I will discuss how the techniques of Guentner, Willett and Yu can be adjusted to include some groupoids with large isotropy subgroups, including an applications to C*-algebras of directed graphs that are AF-embeddable. This is joint work with Dana Williams.

MC 5417

Or join on Zoom with the link below

Jon Brundan, University of Oregon

Classical representation theory via categorification

The standard approach to many sorts of representation theory related to reductive algebraic groups and semisimple Lie algebras is based on the combinatorics of the underlying Weyl group (and its Hecke algebra). In Cartan type A, there is another approach exploiting combinatorics of an underlying Kac-Moody algebra (or its quantized enveloping algebra). This was developed in examples over many decades, and fits into a unified general framework which we now call `Heisenberg categorification'. Analogous approaches are slowly emerging for the other families of classical groups (and supergroups). I will explain the general setup and some of its consequences, with examples.

MC 5501

Tanley Xiao, University of Northern British Columbia

On Buchi's problem

In 1970, J. Richard Buchi showed that there is no general algorithm which decides whether a general quadratic equation in arbitrarily many variables has a solution in the integers, subject to a hypothesis which would be named Buchi's Problem. Buchi's result is a strengthening of the negative answer of Hilbert's Tenth Problem.

Buchi's problem is an elegant number theoretic problem in its own right. It asserts that there exists a positive integer M such that whenever a finite sequence x_0^2, x_1^2,..., x_n^2 of increasing square integers has constant second difference equal to 2 (that is, x_{j+2}^2 - 2 x_{j+1}^2 + x_j^2 = 2 for j = 0, ..., n-2), then either n \leq M or x_j^2 = (x_0 + j)^2 for j = 1, ..., n.

In this talk,  we show that Buchi's problem has an affirmative answer with M = 5. In other words, there are no non-trivial quintuple of increasing square integers with constant difference equal to 2.

MC 5479

Rachael Alvir, University of Waterloo

More Fundamentals of Computability Theory

We continue to present results from Soare's book on computability theory.

MC 5403

Alex Pawelko, University of Waterloo

Prequantum Line Bundles and Geometric Quantization

Prequantum line bundles are objects in symplectic geometry that play a somewhat analogous role to holomorphic line bundles in complex geometry. In this talk, we will discuss the existence of prequantum line bundles, examples of them, and their uses in symplectic geometry, most notably in geometric quantization.

MC 5479

Owen Patashnick, King's College London

Motive-ating formal periods via special values of L-functions

In this talk we will use special values of L-functions as a gateway drug to explore the motivic periods that underlie geometric content associated to these L-values. In particular, we will "motivate" an explicit construction of classes built out of algebraic cycles associated with the L-values L(Sym^n(E), n+m) and muse on the consequences. We will try to make the talk as accessible as possible, and hopefully keep discussion of machinery to a minimum.

MC 5479

Brady Ali Medina, University of Waterloo

Co-Higgs bundles and Poisson structures.

There is a correspondence between co-Higgs fields and holomorphic Poisson structures on P(V) established by Polishchuk in the rank 2 case and by Matviichuk in the case where the co-Higgs field is diagonalizable. In this talk, I will extend this correspondence by providing necessary and sufficient conditions for when a co-Higgs field induces an integrable Poisson structure on V and P(V), showing that the co-Higgs field must either be a function multiple of a constant matrix or have only one non-zero column. We will also analyze this correspondence for co-Higgs fields over curves of genus g greater than one. Finally, I will show how stability can be understood geometrically through the zeros of the induced Poisson structure, establishing connections between \Phi-invariant subbundles, Poisson subvarieties, and the spectral curve. As this talk is a preparation for my thesis defense, please ask me many questions!

MC 5403