Events

Xinle Dai, Harvard University

Sectorial Decompositions of Symmetric Products and Homological Mirror Symmetry

Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. They are essential ingredients for defining Heegaard Floer homology and serve as important examples of Liouville manifolds when the surfaces are open. In this talk, I will discuss ongoing work on the symplectic topology of these spaces through Liouville sectorial methods, along with examples as applications of this decomposition construction to homological mirror symmetry.

MC 5417

Adrian Dawid, University of Cambridge

A promenade along the A-side

In this talk we will take a closer look at some of the structures that live on the A-side of mirror symmetry. In particular, the Fukaya category and symplectic cohomology. Along the way we will look at concrete examples of homological mirror symmetry. After a reminder about the Fukaya category, we will introduce symplectic cohomology. We will then discuss the relationship between these two given by open-closed and closed-open string maps. We will look at some examples with an emphasis on the mirror symmetry perspective. If time permits, we will also take a look at some structures that do not (yet?) have an obvious analogue under mirror symmetry, such as the action filtration of the Fukaya category and related invariants.

MC 2017 

Adam Logan, CSE & Kevin Hare, University of Waterloo

Research Stream

The Career Talks seminar series invites professionals from various fields to share their personal career journeys and insights on how they achieved success. Each session offers valuable advice and guidance for current graduate students. By hearing firsthand experiences, attendees gain a deeper understanding of the challenges and opportunities that lie ahead in their professional lives.

MC 5501

Sourabhashis Das, University of Waterloo

On the distributions of divisor counting functions: From Hardy-Ramanujan to Erdős-Kac

In 1917, Hardy and Ramanujan established that w(n), the number of distinct prime factors of a natural number n, and Omega(n), the total number of prime factors of n have normal order log log n. In 1940, Erdős and Kac refined this understanding by proving that w(n) follows a Gaussian distribution over the natural numbers.

In this talk, we extend these classical results to the subsets of h-free and h-full numbers. We show that w_1(n), the number of distinct prime factors of n with multiplicity exactly 1, has normal order log log n over h-free numbers. Similarly, w_h(n), the number of distinct prime factors with multiplicity exactly h, has normal order log log n over h-full numbers. However, for 1 < k < h, we prove that w_k(n) does not have a normal order over h-free numbers, and for k > h, w_k(n) does not have a normal order over h-full numbers.

Furthermore, we establish that w_1(n) satisfies the Erdős-Kac theorem over h-free numbers, while w_h(n) does so over h-full numbers. These results provide a deeper insight into the distribution of prime factors within structured subsets of natural numbers, revealing intriguing asymptotic behavior in these settings.

MC 5479

Jack Jia, University of Waterloo

Group Schemes: a Functor of Points Perspective

A group scheme is a group object in a category of schemes. This definition, much like other category theory mantras, is a great way to organize knowledge but falls short when one tries to work with it in a hands-on way. I will introduce a more hands-on classification for group schemes, which is aligned with how people work with them in practice. Time permitting, I will illustrate the advantage of this definition in the case of elliptic curves.

MC 5479

Erik Séguin, University of Waterloo

A Selected Topic on Fourier-Stieltjes Algebras of Locally Compact Hausdorff Groups

We discuss a particular selected topic on Fourier-Stieltjes algebras of locally compact Hausdorff groups. Time permitting, we may complete the proof a lemma.

MC 5403

Kain Dineen, University of Waterloo

Symplectic Capacities and Rigidity

As an application of Gromov's non-squeezing theorem, we'll prove that the symplectomorphisms (and anti-symplectomorphisms) of (ℝ^2m, 𝜔_0) are exactly the diffeomorphisms that additionally preserve the capacity of every compact ellipsoid. If time permits, then we will use this to prove that if a sequence of symplectomorphisms of any symplectic manifold (M, 𝜔) converges in the C^0-sense to a diffeomorphism 𝜓, then 𝜓*𝜔 = ± 𝜔.

MC 5479

Larissa Kroell, University of Waterloo

Analysis Seminar: Injective Envelopes for partial C*-dynamical systems

Given a C*-dynamical system, a fruitful avenue to study its properties has been to study the dynamics on its injective envelope. This approach relies on the result of Kalantar and Kennedy (2017), who show that C*-simplicity can be characterized via the Furstenberg boundary using injective envelope techniques. Inspired by this use case, we generalize the notion of injective envelope to partial C*-dynamical systems. Partial group actions are a generalization of group actions and first introduced for C*-algebras by Ruy Exel (1994) to express certain C*-algebras as crossed products by a single partial automorphism. In this talk, we give a short introduction to partial actions and show the existence of an injective envelope for unital partial C*-dynamical systems. Additionally, we discuss its connection to enveloping actions. This is based on joint work with Matthew Kennedy and Camila Sehnem.

MC 5417 

Kuntal Banerjee, University of Waterloo

Very stable and wobbly loci for elliptic curves

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus 1. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We utilize semistability of trivially twisted very stable bundles to prove that the wobbly locus is always a divisor in the moduli space of semistable bundles on a genus 1 curve. We prove, by extension, a conjecture regarding the closedness and dimension of the wobbly locus in this setting. This conjecture was originally formulated by Drinfeld in higher genus.

MC 5501

Mark Rudelson, University of Michigan

When a system of real quadratic equations has a solution

The existence and the number of solutions of a system of polynomial equations in n variables over an algebraically closed field is a classical topic in algebraic geometry. Much less is known about the existence of solutions of a system of polynomial equations over reals. Any such problem can be reduced to a system of quadratic equations by introducing auxiliary variables. Due to the generality of the problem, a computationally efficient algorithm for determining whether a real solution of a system of quadratic equations exists is believed to be impossible. We will discuss a simple and efficient sufficient condition for the existence of a solution. While the problem and the condition are of algebraic nature, the proof relies on Fourier analysis and concentration of measure.

Joint work with Alexander Barvinok.

MC 5501