Events

Tobias Shin, University of Chicago

Almost complex manifolds are (almost) complex

What is the difference topologically between an almost complex manifold and a complex manifold? Are there examples of almost complex manifolds in higher dimensions (complex dimension 3 and greater) which admit no integrable complex structure? We will discuss these two questions with the aid of a deep theorem of Demailly and Gaussier, where they construct a universal space that induces almost complex structures for a given dimension. A careful analysis of this space shows the question of integrability of complex structures can be phrased in the framework of Gromov's h-principle. If time permits, we will conclude with some examples of almost complex manifolds that admit a family of Nijenhuis tensors whose sup norms tend to 0, despite having no integrable complex structure (joint with L. Fernandez and S. Wilson).

MC 5417

Rizwanur Khan, University of Texas at Dallas

Eisenstein series and the Random Wave Conjecture

What do automorphic forms "look" like when plotted on the modular surface? Quantum chaos predictions say that they should tend to look more and more like random waves. We'll discuss the relevant conjecture and report on progress for a fundamental type of automorphic form - the Eisenstein series. This is joint work with Goran Djanković.

Jesse Huang, University of Waterloo

Cohen-Macaulay Modules

Cohen-Macaulay modules are central objects of study in commutative algebra, with deep connections to algebraic geometry, singularity theory, and homological algebra. In this talk, we give a brief overview of the connection between Cohen-Macaulay modules and geometric objects, particularly how these modules can be used to study the local behavior of varieties at singular points. Several classical examples, including modules over regular local rings and isolated singularities, will illustrate the practical utility of Cohen-Macaulay theory in understanding algebraic structures. We will also touch on Cohen-Macaulay modules over toric Gorenstein rings and the role of mirror symmetry in the study of these modules.

MC 5403

Chris Schulz, University of Waterloo

Toward a characterization of k-automatic structures

We consider structures over Presburger arithmetic that include k-automatic sets, that is to say, sets recognized by a base-k finite automaton. The question of how many such structures exist up to interdefinability is a complex one, with a deceptively simple conjectured answer. We give a proof of this conjecture in the restricted case of expansion by a single unary set, and we discuss potential strategies for handling the multivariate case. This talk is based on joint work with Jason Bell and Alexi Block Gorman.

MC 5403

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