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12:00pm - zoom - Probability and Analysis Webinar Taryn Flock - (Macalester College) TBA |
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4:00pm to 5:00pm - RH306 - Applied and Computational Mathematics Long Chen - (University of California at Irvine) Finite Element Complexes A Hilbert complex is a sequence of Hilbert spaces connected by a sequence of closed densely defined linear operators satisfying the property: the composition of two consecutive maps is zero. The most well-known example is the de Rham complex involving grad, curl, and div operators. A finite element complex is a discretization of a Hilbert complex by replacing infinite dimensional Hilbert spaces by finite dimensional subspaces based on a mesh of the domain. Usually inside each element of the mesh, polynomial spaces are used and suitable degree of freedoms are proposed to glue them to form a conforming subspace. The finite element de Rham complexes are well understood and can be derived from the framework Finite Element Exterior Calculus (FEEC). In this talk, we will survey the construction of finite element complexes. We present finite element de Rham complex by a geometric decomposition approach. We then generalize the construction to smooth FE de Rham complexes and derive more complexes including the Hessian complex, the elasticity complex, and the divdiv complex by the Bernstein-Gelfand-Gelfand (BGG) construction. This is a joint work with Xuehai Huang from Shanghai University of Finance and Economics. |
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1:00pm to 2:00pm - RH 440R - Dynamical Systems William Wood - (UC Irvine) Uniform Hyperbolicity and the Periodic Anderson-Bernoulli Model In this talk we will focus on the notion of uniform hyperbolicity of sets of matrices, and apply it to transfer matrices related to a discrete Schrodinger operator to study its spectrum. Specifically, we will show how to apply Johnson’s Theorem, which claims that a Schrodinger cocycle is uniformly hyperbolic if and only if the corresponding energy value is not in the almost sure spectrum, to the periodic Anderson-Bernoulli Model. As a result, we will prove that the spectrum of period two Anderson-Bernoulli Model consists of at most four intervals. A period 3 model, given specific conditions, can have infinitely many intervals in the spectrum, however.
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4:00pm - RH 306 - Differential Geometry Nicos Kapouleas - (Brown University) Recent and ongoing work on minimal doublings and related topics I will discuss the current status of understanding for the geometry and constructions of minimal surface doublings. In particular I will discuss in more detail results related to the Linearized Doubling (LD) approach (Kapouleas: JDG 2017, Kapouleas-McGrath: CPAM 2019, and Kapouleas-McGrath: Camb. J. Math. (to appear); arXiv:2001.04240v3) and some ongoing work.
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2:00pm - 510R Rowland Hall - Combinatorics and Probability Yizhe Zhu - (UCI) Non-backtracking spectra of random hypergraphs and community detection The stochastic block model has been one of the most fruitful research topics in community detection and clustering. Recently, community detection on hypergraphs has become an important topic in higher-order network analysis. We consider the detection problem in a sparse random tensor model called the hypergraph stochastic block model (HSBM). We prove that a spectral method based on the non-backtracking operator for hypergraphs works with high probability down to the generalized Kesten-Stigum detection threshold conjectured by Angelini et al (2015). We characterize the spectrum of the non-backtracking operator for sparse random hypergraphs and provide an efficient dimension reduction procedure using the Ihara-Bass formula for hypergraphs. As a result, the community detection problem can be reduced to an eigenvector problem of a non-normal matrix constructed from the adjacency matrix and the degree matrix of the hypergraph. Based on joint work with Ludovic Stephan (EPFL). https://arxiv.org/abs/2203.07346 |
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9:00am to 9:50am - Zoom - Inverse Problems Mikko Salo - (University of Jyväskylä) Free boundary methods in inverse scattering |
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3:00pm to 4:00pm - RH 306 - Number Theory Shahed Sharif - (Cal State University, San Marcos) Quantum money from quaternion algebras Public key quantum money is a replacement for paper money which has cryptographic guarantees against counterfeiting. We propose a new idea for public key quantum money. In the abstract sense, our bills are encoded as a joint eigenstate of a fixed system of commuting unitary operators. We show that the proposal is secure against black box attacks. In order to instantiate this protocol, one needs to find a cryptographically complicated system of computable, commuting, unitary operators. To fill this need, we propose using Brandt operators acting on the Brandt modules associated to certain quaternion algebras. This is joint work with Daniel Kane and Alice Silverberg. |