Abstract: Mark Kac’s question “Can One Hear the Shape of a Drum?” asks whether one can determine the shape of a planar domain from the eigenvalues of the Laplace operator with the Dirichlet boundary condition. In this expository talk, we shall study the several complex variables analogue of Mark Kac’s question: To what extent can one determine geometric properties of a domain in several complex variables from spectral properties of the complex Neumann Laplacian? We will discuss results and open problems in both real and complex settings. We will also present relevant results on stability of the spectrum of the complex Laplacian.

https://siqifu.camden.rutgers.edu

Positivity of line bundles is a classical topic in projective algebraic geometry. For a vector bundle \(\mathcal E\) of higher rank, positivity is usually imported from the analogous positivity property of \(\mathcal O_{\mathbb P(\mathcal E)}(1)\). Here we explore a stronger generalization via base loci and also investigate their properties. This is in joint work with Nabanita Ray.

The study of elliptic curves is a deep and rich field of research, whose most famous application is to the proof of Fermat’s Last Theorem. In this talk, we will discuss several problems that can be formulated in terms of elliptic curves, including Fermat’s Last Theorem. We will then discuss how the theory of elliptic curves helps us to solve these problems, and many others.

Note: Free refreshments. The talk starts at 5:40.

Abstract: In this talk I will make an overview of some recent advances in applying differential and generalized geometry to exhibiting intrinsic properties of equations describing mechanical systems. Main examples are given by systems with constraints and coupled systems. From the geometric point of view, they correspond to so-called (almost) Dirac structures: those generalize simultaneously symplectic and Poisson manifolds. We will also address the question in the other direction: given a Dirac structure, how to define meaningful dynamics on it. I will conclude by mentioning some consequences of these results for construction of geometric integrators - numerical methods preserving the geometric structure of the equations and qualitative properties of respective mechanical systems.

The talk is based on:

[1] V.Salnikov, A.Hamdouni, D.Loziienko, Generalized and graded geometry for mechanics: a comprehensive introduction, Mathematics and Mechanics of Complex Systems, Vol. 9, No. 1, 2021

[2] O.Cosserat, C.Laurent-Gengoux, A.Kotov, L.Ryvkin, V.Salnikov, On Dirac structures admitting a variational approach, Mathematics and Mechanics of Complex Systems, 2023

[3] V.Salnikov, A.Hamdouni, From modelling of systems with constraints to generalized geometry and back to numerics, Z Angew Math Mech., Vol. 99, Issue 6, 2019

[4] D.Loziienko, V.Salnikov, A. Hamdouni, Construction of Pseudo-Geometric Integrators, Programming and Computing Software, Vol. 48, Issue 2, 2022

https://www.vladimir-salnikov.org

The UDL (Universal Design for Learning) Framework is used to improve and optimize teaching and learning for all of the diverse learners that we encounter in our classes. In this talk, the fundamentals of this framework will be introduced along with an example of how it was implemented to reimagine MATH 3710(Introduction to Mathematical Modeling) to create a modular, inquiry-based course structure here at UConn. The basic process of research in math education will also be highlighted.

Join us for the Annual Logic Lecture, given by the UConn Logic Group 2022/23 Scholar of Consequence:

Maria Aloni (ILLC, Amsterdam)

Details t.b.a.

https://logic.uconn.edu/annual-logic-lecture/

https://nms.kcl.ac.uk/zoe.wyatt/

Speaker: Beatrice de Tiliere
(University Paris Dauphine)