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1:00pm to 2:00pm - RH 440R - Dynamical Systems Grigorii Monakov - (UC Irvine) On regularity of stationary measures for random dynamical systems I will talk about some results on regularity of stationary measures of random dynamical systems. Under the assumption of absence of almost sure invariant measures we prove Holder continuity or log-Holder continuity of all stationary measures depending on estimates for the regularity of random maps. The methods I am going to present can also be used in the non-stationary case. This talk is based on a recent joint work with Anton Gorodetski and Victor Kleptsyn. |
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3:00pm to 3:50pm - ISEB 1200 - Analysis Ruobing Zhang - (Princeton University) Collapsing geometry and moduli space of Einstein metrics in dimension four This talk focuses on the recent resolution of the following conjectures in the study of Ricci-flat four manifolds (joint work with Song Sun). (1) Any volume collapsed limit of unit-diameter Einstein metrics on the K3 manifold is isometric to one of the following: the quotient of a flat 3D torus by an involution, a singular special Kaehler metric on the topological 2-sphere, or the unit interval. (2) Any complete non-compact hyperkaehler 4-manifold with quadratically integrable curvature, namely gravitational instanton, must have one of the following asymptotic model geometries: ALE, ALF, ALG, ALH, ALG* and ALH*.
We will also discuss some applications and further directions. |
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4:00pm - ISEB 1200 - Differential Geometry Gaoming Wang - (Cornell University) Second order elliptic operators on triple junction surfaces In this talk, we will consider minimal triple junction surfaces, a special class of singular minimal surfaces whose boundaries are identified in a particular manner. Hence, it is quite natural to extend the classical theory of minimal surfaces to minimal triple junction surfaces. Indeed, we can show that the classical PDE theory holds on triple junction surfaces. As a consequence, we can prove a type of Generalized Bernstein Theorem and talk about the Morse index on minimal triple junction surfaces. |
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2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Yizhe Zhu - (UCI) The characteristic polynomial of sums of random permutations Let $A_n$ be the sum of $d$ permutation matrices of size $n\times n$, each drawn uniformly at random and independently. We prove that the normalized characteristic polynomial $\frac{1}{\sqrt{d}}\det(I_n - z A_n/\sqrt{d})$ converges when $n\to \infty$ towards a random analytic function on the unit disk. As an application, we obtain an elementary proof of the spectral gap of random regular digraphs. Our results are valid both in the regime where $d$ is fixed and for $d$ slowly growing with $n$. Joint work with Simon Coste and Gaultier Lambert. https://arxiv.org/abs/2204.00524 |
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11:00am - RH 306 - Harmonic Analysis Haonan Zhang - (UCI) Noncommutative Bohnenblust--Hille inequalities and applications to quantum learning Bohnenblust--Hille (BH) inequalities are an extension of Littlewood's 4/3 inequality and have found many applications to harmonic analysis. A variant of BH inequalities for Boolean cubes has been proven with constants that are dimension-free and subexponential in degree. Such inequalities have found great applications in learning low-degree Boolean functions. Motivated by learning quantum observables, a quantum analog of BH inequality for Boolean cubes was recently conjectured and resolved unaware of the conjecture. In this talk, we give a simpler proof with better constants. As applications, we study learning problems for quantum observables of low degrees. Joint work with Alexander Volberg. |
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1:00pm - RH 510R - Algebra Vladimir Baranovsky - (UCI) Topological Data Analysis I |
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3:30pm to 4:30pm - Zoom: https://uci.zoom.us/j/95668199292 - Number Theory Christopher Keyes - (Emory University) Local solubility in families of superelliptic curves If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$, what is the probability that the superelliptic curve with equation $C \colon: y^m = f(x, z)$ has a $p$-adic point, or better, points everywhere locally? In joint work with Lea Beneish, we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94\%, with the exact value given by an explicit infinite product of rational function expressions. |
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1:00pm - DBH 1200 - Graduate Seminar Long Chen - (UC Irvine) Transformer Meets Boundary Value Inverse Problems A Transformer-based deep direct sampling method is proposed for solving a class of boundary value inverse problem. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental but critical question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural network? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions in different frequencies as different input channels. Then, by introducing learnable non-local kernel, the approximation of direct sampling is recast to a modified attention mechanism. The proposed method is then applied to electrical impedance tomography, a well-known severely ill-posed nonlinear inverse problem. The new method achieves superior accuracy over its predecessors and contemporary operator learners, as well as shows robustness with respect to noise. This research shall strengthen the insights that the attention mechanism, despite being invented for natural language processing tasks, offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures. This is a joint work with Ruchi Guo (UCI) and Shuhao Cao (University of Missouri-Kansas City). |