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10:00am - RH 306 - Harmonic Analysis Kateryna Tatarko - (University of Waterloo) How often do centroids of sections coincide with centroid of a convex body? In 1961, Grunbaum asked whether the centroid c(K) of a convex body K is the centroid of at least n + 1 different (n − 1)-dimensional sections of K through c(K). A few years later, Lowner asked to find the minimum number of hyperplane section of K passing through c(K) whose centroid is the same as c(K). We give an answer to these questions for n ≥ 5. In particular, we construct a convex body which has only one section whose centroid coincides with the centroid of the body by using Fourier analytic tools and exploiting the existence of non-intersection bodies in these dimensions. Joint work with S. Myroshnychenko and V. Yaskin. |
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3:00pm to 4:00pm - RH 306 - Analysis Xilu Zhu - (USC) Global Behavior of Multispeed Klein-Gordon System Abstract: In this talk, we explore the long-time behavior of multispeed Klein-Gordon systems in space dimension two. In terms of Klein-Gordon systems, the space dimension two is somehow considered as a critical threshold with possible transition form stability to instability. To illustrate this, we first prove a global well-posedness result when Klein-Gordon systems satisfy Ionescu-Pausader nondegeneracy conditions and the nonlinearity is assumed to be semilinear. Second, on the other hand, we construct a specific Klein-Gordon system such that one of the nondegeneracy conditions is violated and its solution has an infinite time blowup, which implies a type of ill-posedness. |
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4:00pm to 5:00pm - Rowland Hall 340P - Differential Geometry Jan Nienhaus - (UCLA) Topology of positive intermediate Ricci curvatures Abstract: Intermediate k^th Ricci curvatures are curvature conditions interpolating between Sectional curvature (k=1) and Ricci curvature(k=n-1). In this talk I will give a broad overview of what is and isn't known or expected about spaces admitting such metrics, on both sides of the apparent behavioral breakpoint of k=n/2. As an example, I will sketch the proof of an upcoming result that spaces with positive Ric_2 and some fixed degree of symmetry (say an action by T^10) must satisfy the Hopf conjecture, i.e. have positive Euler characteristic, and that the possible cohomology of the fixed points are very restricted. This is joint work with Lee Kennard and Lawrence Mouillé |
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3:00pm to 4:00pm - 510R Rowland Hall - Combinatorics and Probability Ishaq Aden-Ali - (UC Berkeley) The injective norm of some random tensors, and a few applications We'll look at the injective norm of random tensors drawn from a fairly general model. Our main result is an upper bound that improves on what was previously known in this setting. As an application, this leads to a very simple proof of Latała's bound on the moments of Gaussian chaoses—an inequality that generalizes the classical Hanson-Wright bound. I'll also explain how I first got interested in this question through a problem in coding theory. |
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3:00pm to 4:00pm - RH 306 - Number Theory Jon Aycock - (UCSD) Jacobians of Graphs via Edges and Iwasawa Theory The Jacobian (or sandpile group) is an algebraic invariant of a graph that plays a similar role to the class group from number theory. There are multiple recent results controlling the sizes of these groups in Galois towers of graphs that mimic the classical results in Iwasawa theory, though the connection to the values of the Ihara zeta function often requires some adjustment. In this talk we will give a new way to view the Jacobian of a graph that more directly centers the edges of the graph, construct a module over the relevant Iwasawa algebra that nearly corresponds to the interpolated zeta function, and discuss where the discrepancy comes from. |
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4:00pm to 5:00pm - Rowland Hall 440R - Differential Geometry Joshua Jordan - (University of Iowa) Canonical Metrics on Complex Surfaces with Split Tangent Bundle Abstract: In joint work with Hao Fang, I introduce and prove the existence of metrics on complex surfaces with split tangent bundle. These metrics are analogous to Calabi-Yau metrics, as they flatten certain holomorphically trivial line bundles adapted to the geometric structure, in this case the splitting. First, we will review the Calabi-Yau theorem in the Kahler setting and some issues with generalizing it to non-Kahler manifolds. Then, I will discuss some machinery — introduced by Streets -- that makes it possible to reduce this problem to the study of a family of non-concave full-nonlinear elliptic PDE. Finally, I will show that these PDE are smoothly solvable and draw some parallels to the twisted Monge-Ampere equation. |
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4:00pm - RH 306 - Number Theory Bryden Cais - (University of Arizona) Iwasawa theory of unramified geometric Z_p-extensions of function fields In this talk, I will describe a novel Iwasawa theory for unramified Z_p-extensions of global function fields over an algebraically closed field of characteristic p. In this context, the p-adic slopes of Frobenius acting on the first crystalline cohomology of the associated Z_p-tower of algebraic curves provide a new kind of Iwasawa-theoretic object to study, and I will present evidence for a recent conjecture about the limiting behavior of these slopes. |