Events

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 monopoloes on \mathbb{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 \mathbb{R}^3 and building up to an S^1-invariant instanton on the multi-Taub-NUT space.

MC 5403

Brent Nelson, Michigan State University

Closable derivations are anticoarse, of course

The anticoarse space of an inclusion $N\subset M$ of tracial von Neumann algebras is an $N$-subbimodule of $L^2(M)$ whose size is sensitive to several structural properties of the inclusion. It has become a staple of so-called microstates techniques in free probability, where it allows one to parlay finite dimensional approximations into algebraic properties. On the other hand, non-microstates techniques, which exploit the regularity of certain derivations on a von Neumann algebra, have not made use of the anticoarse space, until now. In this talk, I will discuss how deformations given by closable derivations provide a natural connection to anticoarse spaces and consequently yield new applications of free probability. This is based on joint work with Yoonkyeong Lee.

QNC 1507 or Join on Zoom

Jérémy Champagne, University of Waterloo

Small fractional parts of polynomials (aka 11J54)

In the eary 1900's, Hardy and Littlewood asked the following question: given a real number α and integer k>1, what is the smallest distance obtained between αn^k and the nearest integer as n runs over the set {1,...,N}? More specifically, does there exist an exponent theta_k>0 such that the smallest distance is at most N^-theta_k for sufficiently large N? This question was answered positively by Vinogradov a couple decades later, but the question of finding the largest possible theta_k with this property is still open.

In this talk, I will discuss some historical results around this problem and present some typical methods used in the literature.

MC 5479

Rahim Moosa, University of Waterloo

The binding group theorem in stable theories, as a bitorsor.

Recently,  Anand Pillay and I developed the theory of binding groups in the setting of CCM — the first order theory of compact complex manifolds. But this talk will not be about that. Rather, I will talk about a somewhat streamlined and simplified presentation of the binding group theorem for stable theories (that appears in the paper with Pillay mentioned above). There is little novelty here, except maybe the bitorsorial presentation that we give (and like).

MC 5403

Spencer Cattalani, Stony Brook University

Ahlfors currents and symplectic non-hyperbolicity

Rational curves are one of the main tools in symplectic geometry and provide a bridge to algebraic geometry. Complex lines are a more general class of curve that has the potential to connect symplectic and complex analytic geometry. These curves are non-compact, which presents a serious difficulty in understanding their symplectic aspects. In this talk, I will explain how Ahlfors currents can be used to resolve this difficulty and produce a theory parallel to that of rational curves. In particular, Ahlfors currents can be constructed via a continuity method, they control bubbling of holomorphic curves, and they form a convex set.

MC 5417

Patrick Naylor, McMaster University

Doubling Gluck twists

The Gluck twist of an embedded 2-sphere in the 4-sphere is a 4-manifold that is homeomorphic but not obviously diffeomorphic to the 4-sphere. Despite considerable study, these strange manifolds have remained a long-standing source of potential counterexamples to the only remaining case of the Poincaré conjecture. In this talk, I will give an overview of this conjecture, a visual introduction to 2-dimensional knot theory, and describe conditions that guarantee that (some) Gluck twists are standard, i.e., diffeomorphic to the 4-sphere. This is based on joint work with Dave Gabai and Hannah Schwartz.

MC 5501

Stanley Xiao, University of Northern British Columbia

Elliptic curves admitting a rational isogeny of prime degree, ordered by conductor

We consider explicit parametrizations of rational points on the modular curves X_0(p) for p in {2,3,5,7}, which corresponds to elliptic curves E/Q$ admitting a rational isogeny of degree p, and consider conductor polynomials of such curves. Conductor polynomials are polynomial divisors of the discriminant which more closely approximate the conductors of elliptic curves. By using results on almost-prime values of polynomials, including recent breakthrough work of Ben Green and Mehtaab Sawhney, we count such curves whose conductors have the least number of distinct prime factors, ordered by conductor. This is joint work with Alia Hamieh and Fatma Cicek. 

MC 5417

Elan Roth, University of Waterloo

A Continuation of Random Binary Sequences

We'll return to ML- and 1-Randomness and prove their equivalence. First, we will define some necessary machinery such as information content measures and the KC theorem.

MC 5403

Paul Cusson, University of Waterloo

Moduli spaces of monopoles part 2

We continue discussing Euclidean SU(n)-monopoles, now in the case n >= 3, and we aim to describe their moduli spaces using spaces of rational maps from the projective line to flag varieties

MC 5403

Thomas Sinclair, Purdue University

Model theory of metric lattices

We propose a general first-order framework for studying geometric lattices within the model theory of metric structures. As an application we develop a novel continuous limiting theory for finite partition lattices and discuss potential implications to their asymptotic combinatorics. This is joint work with Jose Contreras-Mantilla.

QNC 1507 or Join on Zoom