We will start with a 15-minute presentation by Noah Solomon and continue with a free discussion.
We explore the modular properties of generating functions for Hurwitz class numbers endowed with level structure. Our work is based on an inspection of the weight $\frac{1}{2}$ Maass--Eisenstein series of level $4N$ at its spectral point $s=\frac{3}{4}$, extending the work of Duke, Imamo\={g}lu and T\'{o}th in the level $4$ setting. We construct a higher level analogue of Zagier's Eisenstein series and a preimage under the $\xi_{\frac{1}{2}}$-operator. We deduce a linear relation between the mock modular generating functions of the level $1$ and level $N$ Hurwitz class numbers, giving rise to a holomorphic modular form of weight $\frac{3}{2}$ and level $4N$ for $N > 1$ odd and square-free. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla--Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamo\={g}lu. I wil lbe discussing joint work with Andreas Mono and Ngoc Trinh Le.
Dehn surgery (with a fixed slope p/q) associates, to a knot in S^3, a 3-manifold M with first homology isomorphic to the integers mod p. One might wonder if this function is one-to-one or onto; Cameron Gordon (1978) conjectured that it is never injective nor surjective. The surjectivity case was established a decade later, while the injectivity case was only recently proven by Hayden, Piccirillo, and Wakelin. We will survey this latter result and its proof.
I will discuss several classical problems in number theory about representation of numbers and forms by quadratic forms and related counting results. We will describe several applications such as pair correlations and eigenvalue statistics for quantum systems.
Then we will move to related problems about irrational forms, such as the Oppenheim conjecture and explain how homogeneous dynamics can help to tackle such problems.
The talk will be self contained and hopefully accessible.
In this talk we will review past and recent results pertaining to the emerging field of integrable space-time nonlocal nonlinear evolution equations. In particular, we will discuss blow-up in finite time of soliton solutions as well as the physical derivations of many integrable nonlocal systems.
An $S_k$-set is a subset of a group whose $k$-tuples have distinct products. We give explicit constructions of large $S_k$-sets in the groups $\mathrm{Sym}(n)$ and $\mathrm{Alt}(n)$ and of large $S_2$-sets in $\mathrm{Sym}(n)\times\mathrm{Sym}(n)$ and $\mathrm{Alt}(n)\times\mathrm{Alt}(n)$, as well as some probabilistic constructions for 'nice' groups. We show that $k$ is even and $\Gamma$ has a normal abelian subgroup with bounded index then any $S_k$-set has size at most $(1-\varepsilon)|\Gamma|^{1/k}$. We describe some connections between $S_k$-sets and extremal graph theory. We determine up to a constant factor the largest possible minimum outdegree in a digraph with no subgraph in $\{C_{2,2},\ldots,C_{k,k}\}$, where $C_{\ell,\ell}$ is the orientation of $C_{2\ell}$ with two maximal directed $\ell$-paths. Contrasting with undirected cycles, the extremal minimum outdegree for $\{C_{2,2},\ldots,C_{k,k}\}$ is much smaller than that for any $C_{\ell,\ell}$. We count the directed Hamilton cycles in one of our constructions to improve the upper bound for a problem on Hamilton paths introduced by Cohen, Fachini, and Körner. Joint work with Michael Tait.
Inspired by colored Khovanov homology, for any knot K in the 3-sphere, we define n-colored knot Floer homology as the limit of the cobordism maps from the (full) link Floer homology of the (n,mn)-cable of K to the (full) link Floer homology of (n,(m+1)n)-cable as m goes to infinity. Colored knot Floer homology is graded by Alexander multi-grading and Maslov grading and it is finite dimensional at each fixed degree. We discuss the module structure of this invariant and overview some examples. This is a joint work with Eugene Gorsky and Beibei Liu.
This talk presents two recent advances in Neural Network with Local Converging Inputs (NNLCI) —a novel surrogate model for efficiently resolving nonlinear flow dynamics at modest computational cost
First, a powerful and efficient technique is introduced to extend NNLCI to unstructured computational fluid dynamics. The framework is validated on two-dimensional inviscid supersonic flow in channels with varying bump geometries and positions. The NNLCI model accurately captures key flowfield structures and dynamics, including regions with highly nonlinear shock interactions while achieving a speedup of more than two orders of magnitude.
Second, we conduct a comprehensive benchmark study to compare our method with current state-of-the-art AI-based PDE solvers. Across representative hyperbolic conservation law problems, NNLCI consistently deliver superior accuracy, efficiency and robustness in resolving challenging sharp discontinuities and wave interactions. The work provides practical guidance for model selection in scientific machine learning applications
Please Note: There will be a pre-seminar 10:55-11:15 in Skiles 005.
Proving conjectures is an essential part of our job as mathematicians. Another essential part is to come up with plausible conjectures. In the talk, we focus on this part. We present a new twist to an old method from computer algebra for detecting recurrence equations of infinite sequences of which only the first few terms are known. By applying this new version systematically to all the entries of the Online Encyclopedia of Integer Sequences, we detected a number of potential recurrence equations that could not be found by the classical methods. Some of these have meanwhile been proven. This is joint work with Christoph Koutschan.
====(Below is the information on the pre-talk.)====
Titile: Lattice Reduction
Abstract: It is well known how to go from an exact number (e.g. 1/3) into an approximation (e.g. 0.333). But how can we get back? At first glance, this seems impossible, because some information got lost during the approximation. However, there are techniques for doing this and similar seemingly magic tricks. We will discuss some such tricks that rely on an algorithm for finding short vectors in integer lattices.
