
9:30 am
Sam Spiro  UCSD
Extremal Problems for Random Objects
Final Defense
AP&M 6402
Zoom link available upon request
AbstractThis dissertation lies at the intersection of extremal combinatorics and probabilistic combinatorics. Roughly speaking, extremal combinatorics studies how large a combinatorial object can be. For example, a classical result of Mantel's says that every $n$vertex trianglefree graph has at most $\frac{1}{4} n^2$ edges. The area of probabilistic combinatorics encompasses both the application of probability to combinatorial problems, as well as the study of random combinatorial objects such as random graphs and random permutations. In this dissertation we study problems related to extremal properties of random objects. In particular we study a certain card guessing game, $F$free subgraphs of random hypergraphs, and thresholds of random hypergraphs. Minimal prerequisites will be assumed.

10:00 am
Robin TuckerDrob  University of Florida
Amenable subrelations of treed equivalence relations and the Paddleball lemma
Math 211B  Group Actions Seminar
AP&M 6402
Zoom ID 967 4109 3409
Email an organizer for the passwordAbstractWe give a comprehensive structural analysis of amenable subrelations of a treed quasimeasure preserving equivalence relation. The main philosophy is to understand the behavior of the RadonNikodym cocycle in terms of the geometry of the amenable subrelation within the tree. This allows us to extend structural results that were previously only known in the measurepreserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. The two main ingredients are an extension of Carrière and Ghys's criterion for nonamenability, along with a new PingPongstyle argument we call the "Paddleball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.

11:00 am
Ching Wei Ho  University of Notre Dame
Heat flow conjecture in random matrices
Math 288  Probability and Statistics Seminar
AP&M 6402 with live streaming via Zoom
Email ynemish@ucsd.edu for zoom ID and password

11:00 am
Brett Kotschwar  ASU
Backward propagation of warpedproduct structures under the Ricci flow and asymptotically conical shrinkers
MATH 258  Differential Geometry Seminar
Zoom ID 924 6512 4982
AbstractWe establish sufficient conditions for a locallywarped product structure to propagate backward in time under the Ricci flow. As an application, we show that if a gradient shrinking soliton is asymptotic to a cone whose crosssection is a locally warped product of Einstein manifolds, the soliton must itself be a warped product over the same manifolds.

1:30 pm
Christos Mantoulidis  Rice University
A nonlinear spectrum on closed manifolds
MATH 258  Differential Geometry Seminar (Special Time)
AP&M 5218
AbstractThe pwidths of a closed Riemannian manifold are a nonlinear analog of the spectrum of its LaplaceBeltrami operator, which was defined by Gromov in the 1980s and correspond to areas of a certain minmax sequence of hypersurfaces. By a recent theorem of LiokumovichMarquesNeves, the pwidths obey a Weyl law, just like eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any ≥ 2dimensional manifold for which all the pwidths are known. In recent joint work with Otis Chodosh, we found all pwidths on the round 2sphere and thus the previously unknown LiokumovichMarquesNeves Weyl law constant in dimension 2.

2:00 pm
Michelle Manes  U. Hawaii
Iterating Backwards in Arithmetic Dynamics
Math 209  Number Theory Seminar
Pretalk at 1:20 PM
APM 6402 and Zoom
See https://www.math.ucsd.edu/~nts/ AbstractIn classical real and complex dynamics, one studies topological and analytic properties of orbits of points under iteration of selfmaps of $\mathbb R$ or $\mathbb C$ (or more generally selfmaps of a real or complex manifold). In arithmetic dynamics, a more recent subject, one likewise studies properties of orbits of selfmaps, but with a number theoretic flavor. Many of the motivating problems in arithmetic dynamics come via analogy with classical problems in arithmetic geometry: rational and integral points on varieties correspond to rational and integral points in orbits; torsion points on abelian varieties correspond to periodic and preperiodic points of rational maps; and abelian varieties with complex multiplication correspond to postcritically finite rational maps.
This analogy focuses on forward iteration, but sometimes surprising and interesting results can be found by thinking instead about preimages of rational points under iteration. In this talk, we will give some background and motivation for the field of arithmetic dynamics in order to describe some of these "backwards iteration" results, including uniform boundedness for rational preimages and open image results for Galois representations associated to dynamical systems.

3:15 pm
Shuang Liu  UCSD
Level set simulations of cell polarity and movement
Postdoc Seminar
AP&M B402A
AbstractWe develop an efficient and accurate level set method to study numerically a crawling eukaryotic cell using a minimal model. This model describes the cell polarity and movement using a reactiondiffusion system coupled with a sharpinterface model.
We employ an efficient finite difference method for the reactiondiffusion equations with noflux boundary conditions. This results in a symmetric positive definite system, which can be solved by the conjugate gradient method accelerated by preconditioners. To track the longtime dynamics, we employ techniques of the moving computational window to keep the efficiency. Our levelset simulations capture well the cell crawling, the straight line trajectory, the circular trajectory, and other features.
Our efficient and accurate computational techniques can be extended to a broad class of biochemical descriptions of cell motility, for which problems are posed on moving domains with complex geometry and fast simulations are very important. This is a joint work with LiTien Cheng and Bo Li.

11:15 am

3:00 pm
Harish Kannan  UCSD
Spatiotemporal dynamics of dense bacterial colonies growing on hard agar
Advancement to Candidacy
Urey Hall 6120
Zoom Meeting ID: 951 7114 5365
Please email hkannan@ucsd.edu for password

4:00 pm
Yassine El Maazouz  UC Berkeley
Sampling from padic varieties
Math 208  Algebraic Geometry Seminar
Pretalk at 3:30 PM
Contact Samir Canning at srcannin@ucsd.edu
for zoom accessAbstractWe give a method for sampling points from an affine algebraic variety over a local field with a prescribed probability distribution. In the spirit of the previous work by Breiding and Marigliano on real algebraic manifolds, our method is based on slicing the given variety with random linear spaces of complementary dimension. We also provide an implementation of our sampling method and discuss a few applications, in particular we sample from algebraic padic matrix groups and modular curves.

4:30 pm

11:00 am
Benoit Perthame  Sorbonne University
Porous media based models of living tissues and free boundary problems
Math 248  Analysis Seminar
AbstractTissue growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics.
We will give an overview of the modeling aspects and focus on the links between those mathematical models. Then, we will focus on the `compressible' description describing the cell population density based on systems of porous medium type equations with reaction terms. A more macroscopic 'incompressible' description is based on a free boundary problem close to the classical HeleShaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding HeleShaw free boundary problem and one can make the connection with its geometric form.
The mathematical tools related to these questions include multiscale analysis, AronsonBenilan estimate, compensated compactness, uniform $L^4$ estimate on the pressure gradient and emergence of instabilities.

11:00 am
Benoit Perthame  Sorbonne University
Porous media based models of living tissues and free boundary problems
Math 248  Analysis Seminar
https://ucsd.zoom.us/j/
99515535778
Zoom meeting ID 995 1553 5778AbstractTissue growth, as it occurs during solid tumors, can be described at a number of different scales from the cell to the organ. For a large number of cells, 'fluid mechanical' approaches have been advocated in mathematics, mechanics or biophysics. We will give an overview of the modeling aspects and focuss on the links between those mathematical models. Then, we will focus on the `compressible' description describing the cell population density based on systems of porous medium type equations with reaction terms. A more macroscopic 'incompressible' description is based on a free boundary problem close to the classical HeleShaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding HeleShaw free boundary problem and one can make the connection with its geometric form. The mathematical tools related to these questions include multiscale analysis, AronsonBenilan estimate, compensated compactness, uniform $L^4$ estimate on the pressure gradient and emergence of instabilities.

1:00 pm
Yun Shi  Center of Mathematical Sciences and Applications, Harvard University
Dcritical locus structure for local toric CalabiYau 3folds
Enumerative Geometry Seminar
https://ucsd.zoom.us/j/
96432448457 Meeting ID: 964 3244 8457
AbstractDonaldsonThomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a CalabiYau 3fold. Motivic DonaldsonThomas theory, originally introduced by KontsevichSoibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will explain the role of dcritical locus structure in the definition of motivic DT invariant, following the definition by BussiJoyceMeinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric CalabiYau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by RicolfiSavvas, but techniques used here are different.

1:30 pm
Caroline Moosmueller  UCSD
Optimal transport in machine learning
AWM Colloquium
AP&M 7321
AbstractIn this talk, I will give an introduction to optimal transport, which has evolved as one of the major frameworks to meaningfully compare distributional data. The focus will mostly be on machine learning, and how optimal transport can be used efficiently for clustering and supervised learning tasks. Applications of interest include image classification as well as medical data such as gene expression profiles.

5:00 pm
Yunyi Zhang
Regression with complex data: regularization, prediction and bootstrap
Final Defense
Zoom ID: 657 026 0290
AbstractAnalyzing a linear model is a fundamental topic in statistical inference and has been wellstudied. However, the complex nature of modern data brings new challenges to statisticians, i.e., the existing theories and methods may fail to provide consistent results. Focusing on a high dimensional linear model with i.i.d. errors or heteroskedastic and dependent errors, this talk introduces a new ridge regression method called `the debiased and thresholded ridge regression' that fits the linear model. After that, it introduces new bootstrap algorithms that generate consistent simultaneous confidence intervals/performs hypothesis testing for the linear model. This talk also applies bootstrap algorithm to construct the simultaneous prediction intervals for future observations.
Another topic of this talk is about properties of a residualbased bootstrap prediction interval. It derives the asymptotic distribution of the difference between the conditional coverage probability of a nominal prediction interval and the conditional coverage probability of a prediction interval obtained via a residualbased bootstrap. This result shows that the residualbased bootstrap prediction interval has about $50\%$ possibility of yielding conditional undercoverage. Moreover, it introduces a new bootstrap prediction interval that has the desired asymptotic conditional coverage probability and the possibility of conditional undercoverage.