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3:00pm - RH 306 - Analysis Bogdan Suceavă - (CSU-Fullerton) Fundamental Inequalities in the Geometry of Submanifolds Abstract: In 1956 John F. Nash, Jr. proved that a Riemannian manifold can be immersed isometrically into an Euclidean ambient space of dimension sufficiently large. In 1968, S.-S. Chern pointed out that a key technical element in applying Nash's Theorem effectively is finding useful relationships between intrinsic and extrinsic quantities characterizing immersions. A turning point in the history of this question was an enlightening paper written by B.-Y. Chen in 1993, which paved the way for a deeper understanding of the meaning of the Riemannian inequalities between intrinsic and extrinsic quantities. One important development in the study of such geometric inequalities took place in 2007, with Zhiqin Lu's proof of the DDVV Conjecture. Pursuing this avenue, we present several new results related to the Riemannian study of the geometry of submanifolds. |
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1:00pm to 1:50pm - RH 510R - Algebra Konrad Aguilar - (Pomona College) Inductive limits of C*-algebras and compact quantum metrics spaces Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a compact quantum metric of Rieffel, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov-Hausdorff propinquity of Latremoliere, which is a metric on the class of compact quantum metric spaces. This allows us to place new quantum metrics on all unital AF algebras that extend our previous work with Latremoliere on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov-Hausdorff propinquity topology. |
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2:00pm to 2:50pm - RH 340N - Inverse Problems Boya Liu - (North Dakota State University) Partial Data Inverse Problems for the First Order Perturbation of the Biharmonic Operator In this talk we address the issue of stability for the first order perturbation of the biharmonic operator from partial data, in a bounded domain of dimension three or higher. Specifically, we shall consider two partial data settings: (1) Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement. (2) Assuming that the perturbations are known in a neighborhood of the boundary, measurements are performed only on arbitrarily small open subsets of the boundary. In both settings we obtain log type stability estimates. Part of this talk is based on a joint work with Salem Selim.
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3:00pm to 4:00pm - RH 306 - Number Theory Alexander Smith - (UCLA) The distribution of conjugates of an algebraic integer For every odd prime p, the number 2 + 2cos(2 pi/p) is an algebraic integer whose conjugates are all positive numbers; such a number is known as a totally positive algebraic integer. For large p, the average of the conjugates of this number is close to 2, which is small for a totally positive algebraic integer. The Schur-Siegel-Smyth trace problem, as posed by Borwein in 2002, is to show that no sequence of totally positive algebraic integers could best this bound. In this talk, we will resolve this problem in an unexpected way by constructing infinitely many totally positive algebraic integers whose conjugates have an average of at most 1.899. To do this, we will apply a new method for constructing algebraic integers to an example first considered by Serre. We also will explain how our method can be used to find simple abelian varieties with extreme point counts. |
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4:00pm to 4:50pm - RH 306 - Colloquium Gunther Uhlmann - (University of Washington ) The Fractional Calderón Problem The famous Calderón problem consists in determining the conductivity of a medium by making voltage and current measurements at |