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4:00pm to 5:00pm - - Logic Set Theory Yeonwook Jung - (UC Irvine) Naive Descriptive Set Theory (Lecture 2) This is the second in a series of lectures going through notes entitled "Naive Descriptive Set Theory" that are available on ArXiV. In the last 20 years the field has had many applications to areas in Analysis and Dynamical Systems The lectures are intended to be an opportunity to learn the subject matter, and will be interspersed with research lectures during the quarter. No background beyond basic elements of the 210 sequence are required. |
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1:00pm to 2:00pm - RH 440R - Dynamical Systems Victor Kleptsyn - (CNRS, University of Rennes 1, France) An example of an atomic physical measure for a non-invertible random dynamical system Stationary measures are one of the key tools in the theory of random dynamical systems, and this naturally motivates the study of their properties. It is known that for an invertible RDS with no finite invariant sets, the corresponding stationary measures cannot have atoms. There are also generalizations of this property; in particular, in a recent joint work with A. Gorodetski and G. Monakov, we have established Hölder regularity of stationary measures for a system of bi-Lipschitz homeomorphisms under very mild assumptions: absence of a common invariant measure and some positive moment condition for the Lipschitz constant. It was also further recently generalized by G. Monakov, obtaining log-Hölder regularity under even milder assumptions for the system. A next natural direction of generalization would be to attempt to remove the invertibility assumption, considering the maps that are only locally invertible. However, it turns out that such a generalization is impossible! Namely, in a recent joint work with V. Goverse, we construct an example of a smooth random dynamical system on the circle, consisting of locally invertible maps, having no common invariant measure, and for which there is an atomic stationary measure. Moreover, this stationary measure is physical: for a Lebesgue-generic initial point, its trajectory is almost surely distributed with respect to this measure. My talk will be devoted to the presentation of this example. |
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3:00pm to 3:50pm - - Analysis Chong-Kyu Han - (Seoul National University) Degenerate Pfaffian systems and evolution of hypersurfaces toward an isolated singularity We briefly survey some history of the exterior differential system, especially the problem raised by J. Pfaff in 1814 that culminates with the Frobenius theorem on integrability and the notion of the first integral and their generalizations. Then we introduce degenerate Pfaffian systems with solutions that admit isolated singularities. As an application we discuss examples of evolution of hypersurfaces including the eikonal equations and the mean curvature flows. |
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4:00pm to 5:00pm - RH 306 - Differential Geometry Henri Guenancia - (Université Paul Sabatier) Bogomolov-Gieseker inequality for log terminal Kahler threefolds In this joint work with Mihai Paun, we show that a so-called Q-sheaf on a log terminal Kahler threefold satisfies a suitable Bogomolov-Gieseker inequality as soon as it is stable with respect to a Kahler class. I will discuss the strategy of the proof as well as some applications if time permits. |
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4:00pm to 5:00pm - RH 306 - Analysis Henri Guenancia - (Université Paul Sabatier) Bogomolov-Gieseker inequality for log terminal Kahler threefolds Joint with Differential Geometry seminar. In this joint work with Mihai Paun, we show that a so-called Q-sheaf on a log terminal Kahler threefold satisfies a suitable Bogomolov-Gieseker inequality as soon as it is stable with respect to a Kahler class. I will discuss the strategy of the proof as well as some applications if time permits. |
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3:00pm to 4:00pm - 510R Rowland Hall - Combinatorics and Probability Marcelo Sales - (UCI) Independence number of hypergraphs under degree conditions A well-known result of Ajtai et al. from 1982 states that every $k$-graph $H$ on $n$ vertices, with girth at least five, and average degree $t^{k-1}$ contains an independent set of size $c n \frac{(\log t)^{1/(k-1)}}{t}$ for some $c>0$. In this talk, we explore a related problem where we relax the girth condition, allowing certain cycles of length 2, 3, and 4. We will also present lower bounds on the size of independent sets in hypergraphs under specific degree conditions. This is joint work with Vojtěch Rödl and Yi Zhao. |
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9:00am to 9:50am - Zoom - Inverse Problems Laure Giovangigli - (ENSTA) Propagation of ultrasounds in random multi-scale media and effective speed of sound estimation |
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1:00pm - 306 Rowland Hall - Harmonic Analysis Petros Valettas - (University of Missouri) On the clustering of Padé zeros and poles of random power series We estimate non-asymptotically the probability of uniform clustering around the unit circle of the zeros of the $[m,n]$-Padé approximant of a random power series $f(z) = \sum_{k=0}^\infty a_k z^k$, for $a_k$ independent, with finite first moment, and anti-concentrated. Under the same assumptions we show that almost surely $f$ has infinitely many zeros in the unit disc, with the unit circle serving as a (strong) natural boundary for $f$. For $R_m$ the radius of the largest disc containing at most $m$ zeros of $f$, a deterministic result of Edrei implies that in our setting the poles of the $[m,n]$-Padé approximant almost surely cluster uniformly at the circle of radius $R_m$ when $n\to \infty$ and $m$ stays fixed, and we provide almost sure rates of convergence of these $R_m$'s to $1$. We also show that our results on the clustering of the zeros hold for log-concave vectors $(a_k)$ with not necessarily independent coordinates. This is joint work with S. Dostoglou (University of Missouri). |