A locally compact group $G$ is \emph{Hermitian} if the spectrum $\sigma_{L^1(G)}(f)$ is contained in $\mathbb R$ for every $f=f^*\in L^1(G)$. Examples of Hermitian groups include all abelian locally compact groups. A question from the 1960s asks whether every Hermitian group is amenable. I will speak on the history and recent affirmative solution to this problem.

Recent advances in electron microscopy have, for the first time enabled imaging of single cells in 3D at a nanometer length scale resolution. An uncharted frontier for in silico biology is the ability to simulate cellular processes using these observed geometries. However, this will require a system for going from EM images to 3D volume meshes which can be used in finite element simulations. In this paper, we develop an end-to-end pipeline for this task by adapting and extending computer graphics mesh processing and smoothing algorithms. Our workflow makes use of our recently rewritten mesh processing software, GAMer 2, which implements several mesh conditioning algorithms and serves as a platform to connect different pipeline steps. We apply this pipeline to a series of electron micrographs of dendrite morphology explored at three different length scales and show that the resultant meshes are suitable for finite element simulations. Our pipeline, which consists of free
and open-source community driven tools, is a step towards routine physical simulations of biological processes in realistic geometries. We posit that a new frontier at the intersection of computational technologies and single cell biology is now open. Innovations in algorithms to reconstruct and simulate cellular length scale phenomena based on emerging structural data will enable realistic physical models and advance discovery.

Given a finite set $X$ of size $n$, we can form a metric space on the power set $\mathcal{P}(X)$ by the metric $d(A,B) = |A \triangle B|$ where $A \triangle B := (A \cap B^c) \cup (A^c \cap B)$. An $\alpha$-well separated set system is a subset $\mathcal{F} \subset \mathcal{P}(X)$ so that for all distinct $A, B \in \mathcal{F}$, we have that $d(A,B) \geq \alpha n$. In this talk, we will focus on the case where $\alpha= \frac{1}{2}$ and use linear algebra techniques to explore bounding the size of an $\alpha$-well separated family. We will also discuss the construction of these large $\alpha$-well separated set systems via Hadamard matrices.

The c-projective metrizability equation is an invariant overdetermined linear geometric PDE on an almost c-projective manifold governing the existence of quasi-Kahler metrics compatible with the c-projective structure. I will show that the degeneracy locus of a solution to the c-projective metrizability equation satisfying a generic condition on its prolonged system is a smoothly embedded submanifold of codimension 1 which inherits a partially-integrable nondegenerate almost CR structure. Phrased differently, this result explicitly links the Levi-form of the boundary CR structure of a c-projectively compact quasi-Kahler manifold satisfying a non-vanishing 'generalized scalar curvature' condition to the interior metric.

For a (connected, finite) graph $G$, we define its genus to be the quantity $g := E - V + 1$, where $E$ is the number of edges of $G$ and $V$ is the number of vertices. While it is not the case that graph homomorphisms preserve this invariant, it is the case that contractions between graphs do. In this talk we will consider the category of all genus $g$ graphs and contractions. More specifically, we consider integral representations of the opposite category, i.e. functors from the opposite category to Abelian groups. Using the combinatorics of graph minors, we will show that representations of this kind satisfy a Noetherian property. As applications of this technical result, we show that configuration spaces of graphs as well as Kazhdan-Lusztig polynomials of graphical matroids must satisfy strong finiteness conditions. This is joint work with Nick Proudfoot.

We prove a rigidity result for solutions to the normalized Ricci flow whose rate of convergence is faster than exponential and discuss a connection to the classification problem for noncompact shrinking solitons.

We provide a new modeling framework and adopt modern optimization tools to attack an important open problem in statistics. In particular, we consider the optimal adaptive trial design problem in personalized medicine. Adaptive enrichment designs involve preplanned rules for modifying enrollment criteria based on accruing data in a randomized trial. We focus on designs where the overall population is partitioned into two predefined subpopulations, e.g., based on a biomarker or risk score measured at baseline for personalized medicine. The goal is to learn which populations benefit from an experimental treatment. Two critical components of adaptive enrichment designs are the decision rule for modifying enrollment, and the multiple testing procedure. We provide a general framework for simultaneously optimizing these components for two-stage, adaptive enrichment designs through Bayesian optimization. We minimize the expected sample size under constraints on power and the familywise Type I error rate. It is computationally infeasible to directly solve this optimization problem due to its nonconvexity and infinite dimensionality. The key to our approach is a novel, discrete representation of this optimization problem as a sparse linear program, which is large-scale but computationally feasible to solve using modern optimization techniques. Applications of our approach produce new, approximately optimal designs. In addition, we shall further discuss several extensions to solve other related statistical problems.

The Patlak-Keller-Segel equations (PKS) are widely applied to model
the chemotaxis phenomena in biology. It is well-known that if the
total mass of the initial cell density is large enough, the PKS
equations exhibit finite time blow-up. In this talk, I present some
recent results on applying additional fluid flows to suppress
chemotactic blow-up in the PKS equations. These are joint works with
Jacob Bedrossian and Eitan Tadmor.

I will present two examples of dynamic cell-membrane processes we have been working on, highly inspired by recent experimental results, and described by combining scaling, mathematical modelling and numerical simulations. i) \underline{Immunological synapse}: The cellular basis for the adaptive immune response during antigen recognition relies on a specialized protein interface known as the immunological synapse. We propose a minimal mathematical model for the dynamics of the immunological synapse that encompass membrane mechanics, hydrodynamics and protein kinetics. Simple scaling laws describe the time and length scales of the self-organizing protein clusters as a function of membrane stiffness, rigidity of the adhesive proteins, and the fluid flow in the synaptic cleft. ii) \underline{Formation of Intraluminal Vesicles}: The endosome is a membrane-bound compartment, which encapsulates cargo as it matures into a multi-vescular body that regulate cell activity as well as enabling communication with surrounding cells. The cargo encapsulation process take place as Intraluminal Vesicles form at the endosome membrane, a process in part regulated by the Endosomal Sorting Complex Required for Transport (ESCRTs). We develop a membrane model including membrane elasticity, protein crowding (steric repulsions) and gaussian bending rigidity, which suggests that the vesicles form passively only needing to overcome a small energy barrier.

Escobar proved a sharp Sobolev inequality for the embedding of $W^{1,2}(X^{n+1})$ into $L^{2n/(n-1)}(\partial X)$ by exploiting the conformal properties of the Laplacian in X and the normal derivative along the boundary. More recently, an alternative proof was given by using a Dirichlet-to-Neumann operator along the boundary and its close relationship to the 1/2-power of the Laplacian. In this talk, I describe a new relationship between the conformally covariant fractional powers of the Laplacian due to Graham--Zworski and higher-order Dirichlet-to-Neumann operators in the interior, and use it to prove sharp Sobolev inequalities for embeddings of $W^{k,2}$. Other consequences of this relationship, such as a surprising maximum principle for the conformal 3/2-power of the Laplacian, will also be discussed.

This thesis is devoted to incorporating censoring and truncation to state-of-art Statistical
methodology and theory, to promote the evolution of survival analysis and support Medical
research with up-to-date tools. In Chapter 1, I study the mixture cure-rate model with left
truncation and right-censoring. We propose a Nonparametric Maximum Likelihood Estimation
(NPMLE) approach to effectively handle the truncation issue. We adopt an efficient and stable
EM algorithm. We are able to give a closed form variance estimator giving rise to valid
inference. In Chapter 2, I study the estimation and inference for the Fine-Gray competing
risks model with high-dimensional covariates. We develop confidence intervals based on a
one-step bias-correction to an initial regularized estimator. We lay down a methodological
and theoretical framework for the one-step bias-corrected estimator with the partial
likelihood. In Chapter 3, I study the inference on treatment effect with censored
time-to-event outcome while adjusting for high-dimensional covariates. We propose an
orthogonal score method to construct honest confidence intervals for the treatment effect.
With a slight modification, we obtain a doubly robust estimator extremely tolerant to both
estimation inconsistency and volatility. All the methods in aforementioned chapters are tested
through extensive numerical experiments and applied on real data with authentic medical
interests.