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1:00pm to 2:00pm - RH 340N - Dynamical Systems Karl Zieber - (UCI) Localization of the Anderson Model on the Strip Abstract: The Anderson model has been a key tool in the study of disordered alloys and their transport properties. In the one-dimensional discrete model, it is known that any amount of randomness leads to "localization," or a lack of electron transport. Comprehensive results in higher dimensions have been elusive in part due to the loss of one-dimensional tools. As a transitional step to higher dimensions, we consider the quasi-one-dimensional Anderson model on the strip. In this talk, we will discuss existing work to prove spectral localization (with exponentially decaying eigenfunctions) for this class of Anderson model. We will also explore how to extend localization results for IID potentials to potentials that are independent but non-stationary. |
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3:00pm to 4:00pm - RH 440 R - Logic Set Theory Mr. X - (UC Berkeley) Model Theory This is a general talk about a very important corner of Model Theory. |
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4:00pm - 306 Rowland Hall - Differential Geometry Jared Marx-Kuo - (Rice University) Widths, Index, Intersection, and Isospectrality In this talk, I will discuss a series of works on Gromov's p-widths, $\{\omega_p\}$, on surfaces. For ambient dimensions larger than $2$, $\omega_p$ morally realizes the area of an embedded minimal surface of index p. This characterization was historically used to prove the existence of infinitely minimal hypersurfaces in closed Riemannian manifolds. In ambient dimension $2$, $\omega_p$ realizes the length of a union of (potentially immersed) geodesics, and heuristically, $p$ is equal to the sum of the indices of the geodesics plus the number of points of self-intersection. Joint with Lorenzo Sarnataro and Douglas Stryker, we prove upper bounds on the index and vertices, making progress towards this heuristic. Along the way, we prove a generic regularity statement for immersed geodesics. If time allows, we will also discuss the isospectral problem for the p-widths and how surfaces provide a convenient setting to investigate this. |
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4:00pm to 5:00pm - RH 510R - Applied and Computational Mathematics Eun-Jae Park - (Yonsei University) Staggered DG methods: Recent Advances with Raviart-Thomas elements The staggered discontinuous Galerkin (SDG) method is a finite element framework based on staggered primal–dual meshes, where scalar and flux variables are discretized on interlaced grids. This structure naturally ensures local conservation and stability while retaining the flexibility of discontinuous approximations. In this talk, we present a new family of Raviart–Thomas staggered discontinuous Galerkin (RT-SDG) methods on general polygonal meshes. The method is formulated in a mixed setting, with the primal variable approximated in a locally H1-conforming space and the flux in a locally H(div)-conforming Raviart–Thomas space. The staggered mesh structure renders the classical RTk×Pk pair unstable; this is overcome by enriching the primal space with bubble functions on dual elements, leading to inf–sup stability and optimal convergence. Some applications to interface problems are also discussed The RT-SDG framework extends naturally to arbitrary-order discretizations of second-order elliptic eigenvalue problems on polygonal meshes. The resulting schemes are spurious-free, preserve local and global conservation laws, reduce to symmetric positive definite systems involving only the primal variable without hybridization, and require no extrinsic stabilization. Using Fortin-type operators with modified commuting properties, we establish optimal convergence of both eigenvalues and eigenfunctions, together with L2-superconvergence of eigenfunctions. Numerical experiments confirm the theory and demonstrate the robustness of the methods. Short Bio: Eun-Jae Park is Professor of Computational Science and Engineering at Yonsei University, Seoul. He received his Ph.D. from Purdue University in 1993 and held academic positions in the United States and Europe before joining Yonsei. His research focuses on the numerical analysis of partial differential equations, with particular emphasis on structure-preserving finite element methods on general meshes, including staggered discontinuous Galerkin methods, hybrid and mixed formulations, and virtual element methods. He has published extensively in international journals and has delivered plenary and invited talks at several international conferences in numerical analysis. He has also organized a number of international conferences, including ICOSAHOM 2023. |
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1:00pm to 1:50pm - RH 340N - Algebra Andre Kornell - (New Mexico State University) Algebra homomorphisms in quantum information theory Some quantum channels correspond to algebra homomorphisms, but most do not. I will discuss a few properties that characterize these quantum channels from the perspective of quantum information theory and noncommutative geometry. In particular, I will discuss a characterization in terms of a new measure of quantum information that is related to but distinct from von Neumann entropy and Umegaki relative entropy. |