### Week of October 24, 2021

 12:00pm - Zoom - Probability and Analysis Webinar Pavel Zorin-Kranich - (University of Bonn) TBA https://sites.google.com/view/paw-seminar/ 4:00pm to 5:00pm - Zoom - https://uci.zoom.us/j/97796361534 - Applied and Computational Mathematics Axel Almet - (UC Irvine) Inferring the flow of cell-cell communication information through single-cell transcriptomics Cell-cell communication governs cell fate and decision in health and disease, primarily in the form of biochemical signaling, Understanding what forms of cell-cell communication are present and which can be perturbed is crucial to fully understanding the functionality of biological systems. The recent explosion of single-cell RNA-sequencing has led to the development of cell-cell communication inference methods from gene expression data, enabling new studies on cell-cell communication at unprecedented depth and breadth. These methods reveal possible, simultaneous networks of relationships between cell types that are mediated by cell signaling. In this talk, we present ongoing work that extends cell-cell communication inference output by inferring possible causal relations between signals. That is, does the presence of one signaling interaction cause a subsequent interaction, leading to a flow of information? We show how cell-cell communication and single-cell RNA-sequencing data can be framed in the language of causality and thus draw from existing tools developed for causal discovery. We present some preliminary results of our method that have been applied to synthetic data generated by mathematical modeling and suitable single-cell datasets.
 1:00pm to 2:00pm - Zoom - Dynamical Systems Alberto Takase - (UC Irvine) Spectral estimates of dynamically-defined and amenable operator families Suppose that at each vertex of the Cayley graph of a finitely generated group G is a person holding a dollar. Everybody is told to pass their dollar bill to a neighbor. This can be done so that each person’s net worth increases if and only if the group G is non-amenable. Thus, one can think of non-amenable groups as those where Ponzi schemes can benefit everyone. The Cayley graph of the free group with two generators is an infinite 4-valent tree. If everyone passes their dollar towards the origin then everyone’s net worth increases! Because we live in a world where Ponzi schemes don't work, we restrict our attention to amenable groups such as the integer lattice. For dynamically-defined operator families, the Hausdorff distance of the spectra is estimated by the distance of the underlying dynamical systems while the group is amenable. We prove that if the group has strict polynomial growth and both the group action and the coefficients are Lipschitz continuous, then the spectral estimate has a square root behavior or, equivalently, the spectrum map is $\frac{1}{2}$-Hölder continuous. 4:00pm to 5:00pm - NS2 1201 - Differential Geometry Chao-Ming Lin - (UC Irvine) The deformed Hermitian-Yang-Mills equation and the C-subsolution The deformed Hermitian-Yang-Mills equation, which will be abbreviated as dHYM equation, was discovered around the same time in the year 2000 by Mariño-Minasian-Moore-Strominger and Leung-Yau-Zaslow using different points of view. In this talk, first, I will skim through Leung-Yau-Zaslow’s approach in a simple way. Then I will introduce the C-subsolution which is introduced by Székelyhidi and Guan, I will go over some known results of the dHYM equation, and I will bring up my previous results. Last, I will show some of my recent works which will appear soon.
 2:00pm to 3:00pm - Rowland Hall 510R - Combinatorics and Probability Paata Ivanishvili - (UCI) Learning low degree functions in logarithmic number of random queries Perhaps a very basic question one asks in learning theory is as follows: we are given a  function f on the hypercube {-1,1}^n, and we are allowed to query samples (X, f(X)) where X is uniformly distributed on {-1,1}^n. After getting these samples (X_1, f(X_1)), ..., (X_N, f(X_N)) we would like to construct a function h which approximates f up to an error epsilon (say in L^2). Of course h is a random function as it involves i.i.d. random variables X_1, ... , X_N in its construction. Therefore, we want to construct such h which can only fail to approximate f with probability at most delta. So given parameters epsilon, delta  in (0,1) the goal is to minimize the number of random queries N. I will show that around log(n) random queries are sufficient to learn bounded "low-complexity" functions. Based on joint work with Alexandros Eskenazis.
 9:00am to 10:00am - Zoom - Inverse Problems Otmar Scherzer - (University of Vienna & RICAM) Projection and Diffraction Tomography of Particles in a Trap https://sites.uci.edu/inverse/ 11:00am - Zoom ID: 949 5980 546, Password: the last four digits of ID in the reverse order - Harmonic Analysis Jesse Gell-Redman - (University of Melbourne ) A Fredholm approach to scattering We will give a friendly introduction to the scattering matrix for Schrodinger operators, and discuss how a new functional analytic approach to analysis of non-elliptic equations, due to Vasy, gives a conceptually attractive method for proving detailed regularity results for nonlinear scattering.  This is joint work with several groups of authors including Andrew Hassell, Sean Gomes, Jacob Shapiro, and Junyong Zhang. 3:00pm to 3:50pm - https://uci.zoom.us/j/99192240652 - Number Theory Allysa Lumley - (CRM) Primes in short intervals - Heuristics and calculations We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length  $y$ around $x$, where $y\ll(\log x)^2$. In particular, we conjecture that the maximum grows surprisingly slowly as $y$ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.