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4:00pm to 5:00pm - RH 440 R - Logic Set Theory Yeonwook Jung - (UC Irvine) Naive Descriptive Set Theory (Lecture 3) This is the third 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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4:00pm to 4:50pm - 340P - Inverse Problems Yuzhou Joey Zou - (Northwestern University) Weighted X-ray mapping properties on the Euclidean and Hyperbolic Disks We discuss recent works studying sharp mapping properties of weighted X-ray transforms on the Euclidean disk and hyperbolic disk. These include a C^\infty isomorphism result (joint with R. Mishra and F. Monard) for certain weighted normal operators on the Euclidean disk, whose proof involves studying the spectrum of a distinguished Keldysh-type degenerate elliptic differential operator. We then discuss how to transfer these results to the hyperbolic disk (joint with N. Eptaminitakis and F. Monard), by using a projective equivalence between the Euclidean and hyperbolic disks via the Beltrami-Klein model. |
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11:00am - Rowland Hall 340P - Harmonic Analysis Amir Vig - (University of Michigan ) Marked length spectral invariants of Birkhoff billiard tables and compactness of isospectral sets For planar billiard tables, the marked length spectrum encodes the lengths |
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4:00pm to 5:00pm - ISEB 1010 - Distinguished Lectures Mikhail Lyubich - (Institute for Math Sciences at Stony Brook) Excursion around the Mandelbrot set The Mandelbrot set M is a fascinating fractal that encodes in one image the dynamical complexity of the quadratic family z^2 + c. We will wander around M, trying to make sense of its bubbles and their bifurcations, explain how baby Mandelbrot sets are born and where the herds of elephants march, along with various other observable features of M. Despite its enormous complexity, there is a good chance of obtaining a precise topological and geometric description of M. It depends, though, on confirming a long-standing “MLC Conjecture” (on the local connectivity of M) and building up several “Renormalization Theories” that control the small-scale structure of this set. |
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4:00pm - RH306 - Differential Geometry Mingyang Li - (UC Berkeley) On 4d Ricci-flat metrics with conformally Kähler geometry Ricci-flat metrics are fundamental in differential geometry, and they are easier to study when they have additional structures. I will describe my works on 4d non-trivially conformally Kähler Ricci-flat metrics, which actually is a very natural class of 4d Ricci-flat metrics. This leads to a classification of asymptotic geometries of such metrics at infinity and a classification of such gravitational instantons |
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3:00pm to 4:00pm - 510R Rowland Hall - Combinatorics and Probability Jeck Lim - (Caltech) Sums of dilates For any subset $A$ of a commutative ring $R$ (or, more generally, an $R$-module $M$) and any elements $\lambda_1, \dots, \lambda_k$ of $R$, let \[\lambda_1 \cdot A + \cdots + \lambda_k \cdot A = \{\lambda_1 a_1 + \cdots + \lambda_k a_k : a_1, \dots, a_k \in A\}.\] Such sums of dilates have attracted considerable attention in recent years, with the basic problem asking for an estimate on the minimum size of $|\lambda_1 \cdot A + \cdots + \lambda_k \cdot A|$ given $|A|$. In this talk, I will discuss various generalizations and settings of this problem, and share recent progress. This is based on joint work with David Conlon. |
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4:00pm to 5:00pm - RH 306 - Colloquium Mikhail Lyubich - (Stony Brook University) Story of Holomorphic Dynamics Holomorphic Dynamics (in a narrow sense) is the theory of the iteration of rational maps on the Riemann sphere. It was founded in the classical work by Fatou and Julia around 1918. After about 60 years of stagnation, it was revived in the 1980s, bringing together deep ideas from Conformal and Hyperbolic Geometry, Teichmüller Theory, the Theory of Kleinian Groups, Hyperbolic Dynamics and Ergodic Theory, and Renormalization Theory from physics, illustrated with beautiful computer-generated pictures of fractal sets (such as various Julia sets and the Mandelbrot set). We will highlight some landmarks of this story. |
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4:00pm to 4:50pm - MSTB 124 - Graduate Seminar Isaac Goldbring - (UC Irvine ) Iterated nonstandard extensions in combinatorics Nonstandard analysis is a set of techniques whose central idea is to enlarge a given mathematical structure by adding certain “ideal” elements in such a way that the enlarged structure maintains the same “logical” properties as the original structure. Nonstandard analysis has found applications in nearly every area of mathematics. In this talk, we will explain how this technique works by giving some simple proofs of important theorems from Ramsey theory, which is a branch of combinatorics. These applications involve a relatively recent idea, namely the notion of an iterated nonstandard extension. |