Department of Mathematics

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Abstracts

Week of September 18, 2022

September 19, 2022
Applied Mathematics	Static currents in type-I superconductors	3:00pm - AKW 200	Abstract: In this talk, we describe the classical magneto-static approach to the theory of type-I superconductors. The magnetic field and the current in type-I 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.
Group Actions, Geometry and Dynamics	Self-joinings are like convex cocompact groups with "small" critical exponents	4:00pm - LOM 206	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^2-spectrum of the hyperbolic manifold $\Gamma\backslash {\mathbb H}^n$, the quasi-regular representation $L^2(\Gamma\backslash G)$ and the Hausdorff dimension of the limit set. We consider a higher rank analogue of this relation. For self-joinings 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.
Geometry, Symmetry and Physics	Quantum magnet chains and Kashiwara crystals	4:30pm - LOM 214	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 finite-dimensional Lie algebra g in terms of Kashiwara crystals (which are combinatorial objects modeling finite-dimensional representations of g). Namely, the Bethe eigenvectors in the Gaudin model can be regarded as a covering of the Deligne-Mumford moduli space of stable rational curves, which is unramified over the real locus of the Deligne-Mumford space. The monodromy action of the fundamental group of the real Deligne-Mumford 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. If time allows, I will also discuss the generalization of this monodromy theorem to the XXX Heisenberg magnet chain which involves Kirillov-Reshetikhin crystals.

September 20, 2022
Geometry & Topology	Graphically discrete groups and rigidity	4:15pm - LOM 214	Rigidity theorems in geometric group theory prove that a group’s geometric type determines its algebraic type, typically up to virtual isomorphism. We study graphically discrete groups, which impose a discreteness criterion on the automorphism groups of graphs the group acts on and are well suited to studying rigidity problems. Classic examples of graphically discrete groups include virtually nilpotent groups and fundamental groups of closed hyperbolic manifolds; nonabelian free groups are non-examples. We will present new families of graphically discrete groups and demonstrate this property is not a quasi-isometry invariant. We will discuss rigidity phenomena for free products of graphically discrete groups. This is joint work with Alex Margolis, Sam Shepherd, and Daniel Woodhouse.

September 22, 2022
Algebra and Geometry lecture series	Cohomology of Hitchin moduli spaces and the P=W conjecture.	4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214 , 4:00pm - LOM 214

September 23, 2022
Friday Morning Seminar	Friday Morning Seminar	10:30am - LOM 205	A relaxed-pace seminar on impromptu subjects related to the interests of the audience. Everyone is welcome. The subjects are geometry, probability, combinatorics, dynamics, and more!
Graduate Student Seminar	Drunken climber on trees in mapping class groups	12:00pm - LOM	Given a pair of filling multicurves on a closed surface, Thurston ('88) introduced a $\mathrm{PSL}(2, \mathbb{R})$-representation of a subgroup generated by the multitwists along the multicurves by thinking of the subgroup as a stabilizer subgroup (in the Mapping class group) of a hyperbolic disk isometrically embedded in the Teichmüller space. In most cases, such a stabilizer subgroup is free of rank two. We consider a random walk on the stabilizer subgroup and study asymptotic behaviors of the random walk. This talk discusses the following two main theorems, which were known for much more general settings but under finite support conditions by Joseph Maher ('11, '12), Joseph Maher and Giulio Tiozzo ('18), and François Dahmani and Camille Horbez ('18): - Strong law of large numbers of topological entropy along the random walk under finite first-moment conditions. - Almost every sample path consists of all but finitely many pseudo-Anosov mapping classes without any assumption on the moment. This is based on the joint work with Hyungryul Baik and Inhyeok Choi. While the same results for much more general settings were also obtained in the sequel joint work with Hyungryul Baik and Inhyeok Choi, which was reported in Yale Geometry and Topology Seminar in 2021, the argument in the above specific situation is totally different from the general settings and is of independent interest itself.