Events

Jiahui Huang, Department of Pure Mathematics, University of Waterloo

"Equivariant invariants for Quot schemes"

Deformation invariants on Quot schemes such as Donaldson-Thomas invariants are useful tools for studying the cohomology of moduli spaces. Equivariant versions of such invariants are obtained by integrating characteristic class of tautological bundles, over Quot schemes of quotients of a rank $N$ bundle on $\mathbb{C}^n$ for $n=2,3,4$. The $n=4$ case has been the subject of recent activity in relation to string theory and the DT/PT conjecture for Calabi-Yau 4-folds. This talk will demonstrate how integrations on Quot schemes are performed via equivariant localization and their connections to the usual invariants for compact manifolds.

MC 5417

Xujia Chen, Harvard University

"Why can Kontsevich's invariants detect exotic phenomena?"

In topology, the difference between the category of smooth manifolds and the category of topological manifolds has always been a delicate and intriguing problem, called the "exotic phenomena". The recent work of Watanabe (2018) uses the tool "Kontsevich's invariants" to show that the group of diffeomorphisms of the 4-dimensional ball, as a topological group, has non-trivial homotopy type. In contrast, the group of homeomorphisms of the 4-dimensional ball is contractible. Kontsevich's invariants, defined by Kontsevich in the early 1990s from perturbative Chern-Simons theory, are invariants of (certain) 3-manifolds / fiber bundles / knots and links (it is the same argument in different settings). Watanabe's work implies that these invariants detect exotic phenomena, and, since then, they have become an important tool in studying the topology of diffeomorphism groups. It is thus natural to ask: how to understand the role smooth structure plays in Kontsevich's invariants? My recent work provides a perspective on this question: the real blow-up operation essentially depends on the smooth structure, therefore, given a manifold / fiber bundle X, the topology of some manifolds / bundles obtained by doing some real blow-ups on X can be different for different smooth structures on X.

Zoom link: https://uwaterloo.zoom.us/j/2433704471?pwd=aXJoSDh0NDF0aFREbkthSnFBOUI4UT09

Rahim Moosa, Department of Pure Mathematics, University of Waterloo

"NIP"

We continue to read through Pierre Simon's "A Guide to NIP Theories".

MC 5403 

Junichiro Matsuda, Kyoto University

"Algebraic connectedness and bipartiteness of quantum graphs"

Quantum graphs are a non-commutative analogue of classical graphs related to operator algebras, quantum information, quantum groups, etc. In this talk, I will give a brief introduction to quantum graphs and talk about spectral characterizations of properties of quantum graphs. We introduce the notion of connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms, and these properties have algebraic characterizations in the same way as classical cases. We also show the equivalence between bipartiteness and two-colorability of quantum graphs defined by two notions of graph homomorphisms: one respects adjacency matrices, and the other respects edge spaces. This talk is based on arXiv:2310.09500.

This seminar will be held both online and in person:

• Room: MC 5479
• Zoom link: https://uwaterloo.zoom.us/j/94186354814?pwd=NGpLM3B4eWNZckd1aTROcmRreW96QT09

Akash Sengupta, Department of Pure Mathematics, University of Waterloo

"Chern classes"

We will talk about the definition and basic properties of Chern classes. We will talk about useful techniques for computing Chern classes and discuss how to count lines on a cubic surface.

This seminar will be held both online and in person:

• MC 5417
• Zoom link: https://uwaterloo.zoom.us/j/93801243240?pwd=blpUMVJ1dExUYVlEMkJVWjlscUNqZz09

Arul Shankar, University of Toronto

"Secondary terms in the first moment of the 2-Selmer groups of elliptic curves"

Ranks of elliptic curves are often studied via their 2-Selmer groups. It is known that the average size of the 2-Selmer group of elliptic curves is 3, when the family of all elliptic curves is ordered by (naive) height. On the computational side, Balakrishnan, Ho, Kaplan, Spicer, Stein, and Weigand collect and analyze data on ranks, 2-Selmer groups, and other arithmetic invariants of elliptic curves, when ordered by height. Interestingly, they find a persistently smaller average size of the 2-Selmer group in the data. Thus it is natural to ask whether there exists a second order main term in the counting function of the 2-Selmer groups of elliptic curves. In this talk, I will discuss joint work with Takashi Taniguchi, in which we prove the existence of such a secondary term.

MC 5501

We will be streaming the last two lectures of Alain Connes’ Lecture Series titled “From rings of operators to noncommutative geometry” given at the Fields Institute. The lecture series details the origin and impact of non-commutative geometry to various areas in mathematics and will end with recent advances in the program of the operator theoretic approach to the Riemann Hypothesis. Everyone is welcome to join us for a joint viewing experience. For more information and the Zoom link for the first part of the Lecture Series see: http://www.fields.utoronto.ca/activities/23-24/Alain-Connes

MC 5479

Kaleb Ruscitti, Department of Pure Mathematics, University of Waterloo

"Homomorphic Encryption (or: my summer as a Fed)"

In this talk, I will describe homomorphic encryption, which are encryption schemes that allow one to evaluate polynomial functions on the encrypted data. I will introduce the basics of encryption and then the general theory of homomorphic encryption, and then discuss some of the applications to online privacy that I looked at during my summer research internship.

MC 5501

Nicolas Chavarria Gomez, Department of Pure Mathematics, University of Waterloo

"NIP"

We continue to read through Pierre Simon's A Guide to NIP Theories.

MC 5403

We will be streaming the last two lectures of Alain Connes’ Lecture Series titled “From rings of operators to noncommutative geometry” given at the Fields Institute. The lecture series details the origin and impact of non-commutative geometry to various areas in mathematics and will end with recent advances in the program of the operator theoretic approach to the Riemann Hypothesis. Everyone is welcome to join us for a joint viewing experience. For more information and the Zoom link for the first part of the Lecture Series see: http://www.fields.utoronto.ca/activities/23-24/Alain-Connes

MC 5479