Events

Nicole Kitt, University of Waterloo

Characterizing Cofree Representations of SL_n x SL_m

The study, and in particular classification, of cofree representations has been an interest of research for over 70 years. The Chevalley-Shepard Todd Theorem provides a beautiful intrinsic characterization for cofree representations of finite groups. Specifically, this theorem says that a representation V of a finite group G is cofree if and only if G is generated by pseudoreflections. Up until the late 1900s, with the exception of finite groups, all of the existing classifications of cofree representations of a particular group consist of an explicit list, as opposed to an intrinsic group-theoretic characterization. However, in 2019, Edidin, Satriano, and Whitehead formulated a conjecture which intrinsically characterizes stable irreducible cofree representations of connected reductive groups and verified their conjecture for simple Lie groups. The conjecture states that for a stable irreducible representation V of a connected reductive group G, V is cofree if and only if V is pure. In comparison to the classifications comprised of a list of cofree representations, this conjecture can be viewed as an analogue of the Chevalley–Shepard–Todd Theorem for actions of connected reductive groups. The aim of this thesis is to further expand upon the techniques formulated by Edidin, Satriano, and Whitehead as a means to work towards the verification of the conjecture for all connected semisimple Lie groups. The main result of this thesis is the verification of the conjecture for stable irreducible representations V\otimes W of SL_n x SL_m satisfying dim V>=n^2 and dim W>=m^2.

Gian Cordana Sanjaya, University of Waterloo

Density of Special Classes of Polynomials with Squarefree Discriminant

In this talk, we discuss the problem of determining the local densities of monic integer polynomials of fixed degree with squarefree discriminant, with some conditions on the coefficients. Previously, Yamamura solved the case where no extra conditions are imposed, and Bhargava, Shankar, and Wang proved that the global density of general monic integer polynomials (of fixed degree) with squarefree discriminant is equal to the product of the local densities.

Our main case of interest is when the first two non-leading coefficients are fixed. However, we also obtain results on other cases, such as the case where the constant coefficient is fixed. Moreover, the technique used here allows us to compute the local densities exactly.

This talk is based on the recent preprint: https://arxiv.org/abs/2505.06820

MC 5417

Andy Zucker, University of Waterloo

Dynamical Partite Construction

We revisit the partite construction using some of the dynamical ideas we have developed.

MC 5417

Cynthia Dai, University of Waterloo

Resolution of Singularities via Stacky Blow-ups

We follow Dan Abramovich and Ming Hao Quek’s paper on resolution of singularity by multi-weighted blowups. This line of work is first motivated to give a more natural and motivated proof of Hironaka’s result, and that leads to the notion of weighted blowup, where you repeatedly blowing up the worst singular locus via weighted blowups. The problem with this is at the end you do not get an ambient space that’s smooth DM, but log smooth(specifically it’s toroidal DM stack). Using the technique of multi-weighted blowup introduced by Satriano, we can improve this result to get a logarithmic resolution of singularity with a smooth ambient space.

MC 5403

Facundo Camano, University of Waterloo

A Gromov—Hausdorff Convergence Result for the Moduli Space of Singular Monopoles

I will introduce singular monopoles on R^3 and their moduli space. We will then focus on U(2) singular monopoles, which have known explicit expressions, and look at the Gromov—Hausdorff convergence of the moduli space as one singularity is sent off towards infinity.

MC 5403

Yash Singh, University of Waterloo

Vector Bundles on Toric Stacks

We give moduli interpretations of toric vector bundles and generalize this approach to a classification of bundles on arbitrary toric stacks.

MC 5403

Sean Lee, University of Waterloo

Topological dynamics of the Rado graph

We introduce some concepts from topological dynamics, in particular the universal minimal flow, with the goal of showing that the universal minimal flow of the automorphism group of the Rado graph is the space of linear orders of the Rado graph.

MC 5417

Rachael Alvir, University of Waterloo

More Torsion-Free Abelian Groups

We will continue learning about torsion-free Abelian Groups following the Monograph by Downey and Melnikov.

MC 5417

Faisal Romshoo, University of Waterloo

Constructing calibrated submanifolds through evolution equations

I will talk about how we can construct examples of calibrated submanifolds using the techniques of evolution equations. We will begin by defining the ideas involved in coming up with these evolution equations and then look at some of the examples of calibrated submanifolds that are constructed this way, following arXiv:math/0008021, arXiv:math/0008155, arXiv:math/0010036 and arXiv:math/0401123.

MC 5403

Xuemiao Chen, University of Waterloo

On the space of lines

I will make a story about the space of oriented lines in the three dimensional Euclidean space.

MC 5403