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
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:
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:
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
Anne Dranowski, University of Southern California
"Some spaces associated to KLR/W algebras"
We discuss two projects related to the KLR algebra which is a diagrammatic algebra categorifying representations of the quantum group U_q(sl_n). In one project, joint with Guo, Lauda and Manion, we *adapt* a spectral refinement of Khovanov homology to a foamy version thereof, and *extend* it to KLR-mod. In another project, joint with Leroux-Lapierre, we study an equivalence relating KLR-mod and quantizations of slices in the affine Grassmannian using canonical bases and characters. We hope to be as self-contained as possible up to a first course in algebra or topology.
M3 3127
Anwesh Ray, Chennai Mathematical Institute
"Diophantine stability for elliptic curves on average"
Let K be a number field and ℓ≥ 5 be a prime number. Mazur and Rubin introduced the notion of diophantine stability for a variety X/K at a prime ℓ. Under the hypothesis that all elliptic curves E/ℚ have finite Tate-Shafarevich group, we show that there is a positive density set of elliptic curves E/ℚ of rank 1, such that E/K is diophantine stable at ℓ. This result has implications to Hilbert's tenth problem for number rings. This is joint work with Tom Weston.
Zoom link: https://uwaterloo.zoom.us/j/2433704471?pwd=aXJoSDh0NDF0aFREbkthSnFBOUI4UT09
Chatchai Noytaptim, Department of Pure Mathematics, University of Waterloo
"The Laplacian on the Berkovich projective line"
We discuss various examples in computing the Laplacian of functions of bounded differential variation on the Berkovich projective line. We follow closely Chapter 5 in “Potential Theory and Dynamics on the Berkovich Projective Line” by Baker and Rumely.
MC 5479
Adam Jelinsky, University of Waterloo
"Properties of the Pseudo-randomness of a³ mod p²"
In this talk, I will be discussing methods and tactics used to quantify and understand the apparent pseudo-random distribution of a³ mod p², the evidence we have to fit the random model proposed, and the required steps in order to formally prove it.
MC 5403
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
Huixi Li, Nankai University
"On Covering Systems of Polynomial Rings Over Finite Fields"
In 1950, Erd\H{o}s posed a question known as the minimum modulus problem on covering systems for $\mathbb{Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough in 2015. In this presentation, we will discuss the analogous minimum modulus problem for $\mathbb{F}_q[x]$. We proof that the smallest degree of the moduli in any covering system for $\mathbb{F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. This is a joint work with Shaoyun Yi, Biao Wang, and Chunlin Wang.
Zoom: https://uwaterloo.zoom.us/j/98950813087?pwd=SEl1NlNqNHl0QzlYNGJzeDVla204QT09