Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs.

We introduce the concept of functional equivariance, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic.

Tissue 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 Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form. The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, compensated compactness, uniform $L^4$ estimate on the pressure gradient and emergence of instabilities.

Tissue  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 Hele-Shaw equation. In the stiff pressure limit, one can derive a weak formulation of the corresponding Hele-Shaw free boundary problem and one can make the connection with its geometric form.

The mathematical tools related to these questions include multi-scale analysis, Aronson-Benilan estimate, compensated compactness, uniform $L^4$ estimate on the  pressure gradient and emergence of instabilities.

The original version of the Quillen-Lichtenbaum Conjecture, proved by Voevodsky and Rost,  connects special values of Dedekind zeta functions and algebraic K-groups of number fields. In this talk, I will discuss a generalization of this conjecture to Dirichlet L-functions. The key idea is to twist algebraic K-theory spectra with the equivariant Moore spectra introduced in my thesis. This is joint work in progress with Elden Elmanto.

Donaldson-Thomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, 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 d-critical locus structure in the definition of motivic DT invariant, following the definition by Bussi-Joyce-Meinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by Ricolfi-Savvas, but techniques used here are different. 

Branching Brownian motion (BBM) is a random particle system which incorporates both the tree-like structure and the diffusion process. In this talk, we will consider two variant models of BBM, BBM with absorption and BBM with an inhomogeneous branching rate. In the first model, we will study the transition from the slightly subcritical regime to the critical regime and obtain a Yaglom type asymptotic result of the expected number of particles conditioned on survival as the process gets closer to being critical. In the second model, we will see how it can be used to study the evolution of populations undergoing selection. We will provide a mathematically rigorous justification for the biological observation that the distribution of the fitness levels of individuals in a population evolves over time like a traveling wave with a profile defined by the Airy function. The second part of the talk is based on joint work with Jason Schweinsberg.

In this talk, we will discuss the dynamics on Teichmüller space and moduli space of square-tiled surfaces. For square-tiled surfaces, one can explicitly write down the $SL(2,\mathbb{R})$-orbit on the moduli space. To study the dynamics of Teichmüller flow of the $SL(2,\mathbb{R})$-action, we study its derivative, namely the Kontsevich--Zorich cocycle (KZ cocycle). In this talk, we will define what a KZ cocycle is, and by following explicit examples, we will show how one can compute the KZ monodromy. This is part of an ongoing work with Anthony Sanchez.

This talk is mainly about a new Minkowski formula for  hypersurfaces with free boundary or capillary boundary supported on the unit sphere. With it we have classified all stable free boundary CMC hypersurfaces.  Using it we have introduced a inverse curvature flow, which is used to prove Alexandrov-Fenchel type inequalities for newly introduced quermassintegrals  for free boundary  hypersurfaces. At the end we will talk about various generalizations. The talk is based on the joint work with J. Scheuer and C. Xia and other collaborators.

Modern data analysis involves large sets of structured data, where the structure carries critical information about the nature of the data. These relationships between entities, such as features or data samples, are usually described by a graph structure. While many real-world data are intrinsically graph-structured, e.g. social and traffic networks, there is still a large number of applications, where the graph topology is not readily available. For instance, gene regulations in biological applications or neuronal connections in the brain are not known. In these applications, the graphs need to be learned since they reveal the relational structure and may assist in a variety of learning tasks. Graph learning (GL) deals with the inference of a topological structure among entities from a set of observations on these entities, i.e., graph signals. Most of the existing work on graph learning focuses on learning a single graph structure, assuming that the relations between the observed data samples are homogeneous. However, in many real-world applications, there are different forms of interactions between data samples, such as single-cell RNA sequencing (scRNA-seq) across multiple cell types. This talk will present a new framework for multiview graph learning in two settings: i) multiple views of the same data and ii) heterogeneous data with unknown cluster information. In the first case, a joint learning approach where both individual graphs and a consensus graph are learned will be developed. In the second case, a unified framework that merges classical spectral clustering with graph signal smoothness will be developed for joint clustering and multiview graph learning. 
This is joint work with Abdullah Karaaslanli, Satabdi Saha and Taps Maiti.

In 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.

We recently described the construction and theoretical analysis of a framework (competitive-refractory dynamics model) derived from the canonical neurophysiological principles of spatial and temporal summation. The framework models the competing interactions of signals incident on a target downstream node (e.g. a neuron) along directed edges coming from other upstream nodes that connect into it. The model takes into account how temporal latencies produce offsets in the timing of the summation of incoming discrete events due to the geometry (physical structure) of the network, and how this results in the activation of the target node. It provides a computable representation of how local computations result in global network dynamics. Grounded in this neurophysiological model, we are beginning to explore the use some aspects of category theory and related ideas in order to abstract up and understand how the brain might produce generative (emergent) non-trivial computational properties. In particular, we are interested in understanding the emergence of creativity and imagination.

https://mathweb.ucsd.edu/~bli/research/mathbiosci/MBBseminar/

We will discuss a $p$-adic cohomology theory for rigid analytic varieties with overconvergent structure (dagger spaces) over a local field of characteristic $p$. After explaining the motivation, we will define a site (Robba site) and discuss its basic properties.

The development of vaccines for COVID-19 has enabled us to nearly return to pre-pandemic life. However, while vaccines are becoming globally widespread, high alert levels prevail. Even with vaccines, monitoring for the evolution of mutations or detecting new outbreaks calls for continued vigilance. Therefore, testing is likely to prevail to be a vital mechanism to inform decision making in the near future. In order to conserve scarce testing resources, many nations have endorsed so-called group/pooling test methods. Such methods can be expressed using linear algebra. The basic principle underlying pooling tests is the observation that to efficiently detect positive cases among a population with a very low occurrence prevalence, it can be advantageous to test groups of samples instead of testing all individual samples. We develop matrix designs, which encode all relevant information for doing pooling tests and that enable high compression rates when exactly identifying up to a certain number of positive cases.

Analyzing a linear model is a fundamental topic in statistical inference and has been well-studied. 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 residual-based 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 residual-based bootstrap. This result shows that the residual-based bootstrap prediction interval has about $50\%$ possibility of yielding conditional under-coverage. Moreover, it introduces a new bootstrap prediction interval that has the desired asymptotic conditional coverage probability and the possibility of conditional under-coverage.