Events

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

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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

Spiro Karigiannis, University of Waterloo

A tale of two Lie groups

The classical Lie group SO(4) is well-known to possess a very rich structure, relating in several ways to complex Euclidean spaces. This structure can be used to construct the classical twistor space Z over an oriented Riemannian 4-manifold M, which is a 6-dimensional almost Hermitian manifold. Special geometric properties of Z are then related to the curvature of M, an example of which is the celebrated Atiyah-Hitchin-Singer Theorem. The Lie group Spin(7) is a particular subgroup of SO(8) determined by a special 4-form. Intriguingly, Spin(7) has several properties relating to complex Euclidean spaces which are direct analogues of SO(4) properties, but sadly (or interestingly, depending on your point of view) not all of them. I will give a leisurely introduction to both groups in parallel, emphasizing the similarities and differences, and show how we can nevertheless at least partially succeed in constructing a "twistor space" over an 8-dimensional manifold equipped with a torsion-free Spin(7)-structure. (I will define what those are.) This is joint work with Michael Albanese, Lucia Martin-Merchan, and Aleksandar Milivojevic. The talk will be accessible to a broad audience.

MC 5479

Mingyang Li, UC Berkeley

On 4d Ricci-flat metrics with conformally Kahler geometry.

Ricci-flat metrics are fundamental in differential geometry, and they are easier to study when they have additional structures. I will introduce my recent work on 4d conformally Kahler but non-Kahler Ricci-flat metrics, which is a condition analogous to hyperkahler. This leads to a complete classification of asymptotic geometries of such metrics at infinity and a classification of such gravitational instantons.

MC 5417