Monday, September 19, 2022
Time  Items 

All day 

3pm 
09/19/2022  3:00pm Abstract: In this talk, we describe the classical magnetostatic approach to the theory of typeI superconductors. The magnetic field and the current in typeI superconductors are related by the London equations and tend to decay exponentially inside the superconducting material with support of the fields contained primarily in O(λL) neighborhood of the superconductor. We present a Debye source based integral representation for the numerical solution of the London equations, and demonstrate the efficacy of our approach for moderate values of λL on complex three dimensional geometries. However, for typical materials λL∼O(1e−7), which makes the PDE and integral equation increasingly difficult to solve in the limit λL→0 due to the presence of two different length scales in the problem. We derive a limiting PDE and a corresponding integral equation, and show that the solutions of this limiting PDE and integral equations are O(λL) accurate as compared to the corresponding solutions of the London equations and the Debye source integral equations respectively. We demonstrate the effectiveness of this asymptotic approach both in terms of speed and accuracy through several numerical examples. Bio: Manas Rachh joined the Simons foundation as part of the Numerical Algorithms group at Flatiron’s Center for Computational Biology in 2018, and is currently a research scientist in the Center for Computational Mathematics. His research interests include partial differential equations (PDEs) arising in mathematical physics, integral equation methods, robust computation of eigenvalues and eigenfunctions of elliptic PDEs, and the development of fast algorithms for applications in electrostatics, acoustics, viscous flow, electromagnetics, biomedical imaging, and data visualization. Before coming to the foundation, he obtained his Ph.D. from the Courant Institute of Mathematical Sciences at New York University with Leslie Greengard followed by a Gibbs Assistant Professorship at Yale University where he worked with Vladimir Rokhlin. Location:
AKW 200

4pm 
09/19/2022  4:00pm For a convex cocompact Kleinian group $\Gamma <\rm{SO}(n,1)$, Sullivan (around 1985) established a fundamental relation among the critical exponent, the bottom of the L^2spectrum of the hyperbolic manifold $\Gamma\backslash {\mathbb H}^n$, the quasiregular representation $L^2(\Gamma\backslash G)$ and the Hausdorff dimension of the limit set. We consider a higher rank analogue of this relation. For selfjoinings of convex cocompact Kleinian groups (or more generally for any Anosov subgroup of a product of rank one simple algebraic groups), we discover a surprising fact that they satisfy a similar relation as convex cocompact groups with “small” critical exponents. This talk is based on joint works with Dongryul Kim and Yair Minsky, and with Sam Edwards in different parts. Location:
LOM 206
09/19/2022  4:30pm The talk is based on the joint paper with Iva Halacheva, Joel Kamnitzer, and Alex Weekes https://arxiv.org/abs/1708.05105 . Solutions of the algebraic Bethe ansatz for quantum magnet chains are, generally, multivalued functions of the parameters of the integrable system. I will explain how to compute some monodromies of solutions of Bethe ansatz for the Gaudin magnet chain assigned to a semisimple finitedimensional Lie algebra g in terms of Kashiwara crystals (which are combinatorial objects modeling finitedimensional representations of g). Namely, the Bethe eigenvectors in the Gaudin model can be regarded as a covering of the DeligneMumford moduli space of stable rational curves, which is unramified over the real locus of the DeligneMumford space. The monodromy action of the fundamental group of the real DeligneMumford space (called cactus group) on the eigenvectors is naturally equivalent to the action of the same group by commutors (i.e. combinatorial analog of a braiding) on a tensor product of Kashiwara crystals. Location:
LOM 214
