The classical Hardy-Littlewood maximal operators (averaging over families of Euclidean balls and cubes) are known to satisfy L^p bounds that are independent of dimension. This talk will extend these results to spherical maximal functions acting on Cartesian products of cyclic groups equipped with the Hamming metric.

Blood vessel systems and leaf venations are typical biological transport networks. The energy consumption for such a system to perform its biological functions is determined by the network structure. In the first part of this talk, I will discuss the optimized structure of vessel networks, and show how the blood vessel system adapts itself to an optimized structure. Mathematical models are used to predict pruning vessels in the experiments of zebra fish. In the second part, I will discuss our recent modeling work on the initiation process of transport networks. Simulation results are used to illustrate how a tree-like structure is obtained from a continuum adaptation equation system, and how loops can exist in our model. Possible further application of this model will also be discussed.

Using a level set formulation and elliptic regularization we define a notion of weak solutions of nonlinear mean curvature flow with Neumann boundary condition. We then outline the proof of an existence result for the weak level set flow. Finally, we discuss some geometric applications of this flow.

In high-energy physics, interactions between fundamental particles can be represented by Feynman graphs. Almost all predictions for particle collider experiments are obtained by computing certain integrals associated to such graphs, called Feynman integrals, and a vast effort in the physics community worldwide is devoted to studying these quantities. Feynman integrals turn out to be periods, and surprisingly many are multiple zeta values. I will survey what is known and not known about these quantities.

Recently, quantitative stochastic homogenization of operators in divergence form has witnessed important progress. Our goal is to go beyond the error bound to analyze statistical fluctuations around the homogenized limit. We prove a pointwise two-scale expansion and a large scale central limit theorem for the solution. The approach is probabilistic. The main ingredients include the Kipnis-Varadhan method applied to symmetric diffusion in random environment, a quantitative martingale central limit theorem, the Helffer-Sjostrand covariance representation and Stein's method. This is joint work with Jean-Christophe Mourrat.

In geometry, there are certain structures that are "rigid" (like Riemannian manifolds) and others that are "flexible" (like topological manifolds). Symplectic geometry lies in between these two extremes and incorporates some attractive features of both. One consequence is that symplectic techniques have recently been used, to great effect, to give combinatorial approaches to questions in topology that previously required difficult gauge-theoretic and analytic techniques. I will introduce symplectic structures and describe some recent developments linking them to the study of three-dimensional manifolds and knots. No real background will be assumed.

We will begin with the idea of General Relativity, which Einstein called his "happiest thought," and then proceed with a qualitative and quantitative discussion of the curvature of space-time. We will describe the central role of differential geometry in the subject and the important role that mathematicians have played proving the conjectures of the physicists, as well as making a few conjectures of our own. Finally, we will describe the geometry of black holes and their relationship to soap bubbles.

Computational protein design is a transformative field with exciting prospects for advancing both basic science and translational medical research. New algorithms blend discrete and continuous mathematics to address the challenges of creating designer proteins. I will discuss recent progress in this area and some interesting open problems. I will motivate this talk by discussing how, by using continuous geometric representations within a discrete optimization framework, broadly-neutralizing anti-HIV-1 antibodies were computationally designed that are now being tested in humans - the designed antibodies are currently in eight clinical trials (See https://clinicaltrials.gov/ct2/results?cond=&term=VRC07&cntry=&state=&city=&dist= ), one of which is Phase 2a (NCT03721510). These continuous representations model the flexibility and dynamics of biological macromolecules, which are an important structural determinant of function. However, reconstruction of biomolecular dynamics from experimental observables requires the determination of a conformational probability distribution. These distributions are not fully constrained by the limited information from experiments, making the problem ill-posed in the sense of Hadamard. The ill-posed nature of the problem comes from the fact that it has no unique solution. Multiple or even an infinite number of solutions may exist. To avoid the ill-posed nature, the problem must be regularized by making (hopefully reasonable) assumptions. I will present new ways to both represent and visualize correlated inter-domain protein motions (See Figure). We use Bingham distributions, based on a quaternion fit to circular moments of a physics-based quadratic form. To find the optimal solution for the distribution, we designed an efficient, provable branch-and-bound algorithm that exploits the structure of analytical solutions to the trigonometric moment problem. Hence, continuous conformational PDFs can be determined directly from NMR measurements. The representation works especially well for multi-domain systems with broad conformational distributions. Ultimately, this method has parallels to other branches of applied mathematics that balance discrete and continuous representations, including physical geometric algorithms, robotics, computer vision, and robust optimization. I will advocate for using continuous distributions for protein modeling, and describe future work and open problems.

To a Legendrian knot, one can associate an $A_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

We consider an isoperimetric inequality for area minimizing submanifolds $R$ lying outside a convex body $K$ in $\mathbb{R}^{n+1}$. Here $R$ is an $(m+1)$-dimensional submanifold whose boundary consists of a submanifold $T$ in $\mathbb{R}^{n+1} \setminus K$ and a free boundary (possibly not rectifiable) along $\partial K$. An isoperimetric inequality outside a convex body was previously proven by Choe, Ghomi, and Ritore in the codimension one setting where $m = n$. We extend their result to higher codimension. A key aspect of the proof are estimates on the concentration of mass of $T$ and $R$ near $\partial K$.

Decades of research in cancer have yielded scant results other than highlighting the need for new approaches that could go beyond the tried and tested molecular-based ones. Recent clinical studies show that tumour heterogeneity and selection, the ingredients of Darwinian evolution, can explain cancer progression towards malignancy as well as recurrence after treatment. In this talk I will describe mathematical and computational models that explore cancer evolutionary dynamics and that can explain how the interactions between the tumour with its environment (the tumour ecosystem) can yield a better understanding of cancer biology and lead to better and more efficacious treatments for cancer patients.

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Traditional infectious disease epidemiology is built on the foundation of high quality and high accuracy data on disease and behavior. While these data are usually characterized by smallsize, they benefit from designed sampling schemes that make it possible to make population-level inferences. On the other hand, digital infectious disease epidemiology uses existing digital traces, re-purposing them to identify patterns in health-related processes. In this talk, I will discuss our work using data from small epidemiological studies as well as administrative “big data” to understand influenza transmission dynamics and inform disease surveillance and control.

The longest increasing subsequence (LIS) of a uniformly random permutation is a well studied problem. Vershik-Kerov and Logan-Shepp first showed that asymptotically the typical length of the LIS is 2sqrt(n). This line of research culminated in the work of Baik-Deift-Johansson who related this length to the GUE Tracy-Widom distribution. We study the length of the LIS of random permutations drawn from the Mallows measure, introduced by Mallows in connection with ranking problems in statistics. We prove limit theorems for the LIS for different regimes of the parameter of the distribution. I will also describe some recent results on the longest common subsequence of independent Mallows permutations. Relevant background for the talk will be introduced as needed. Based on work with Ron Peled, Riddhi Basu and Ke Jin.

J. Berge discovered a simple condition on a knot, K, in the three-sphere which ensures that Dehn surgery on K yields a lens space. It is an open conjecture, known as the Berge conjecture, that any knot on which one can perform surgery and obtain a lens space satisfies his condition. I will discuss a strategy, developed jointly with Ken Baker and Eli Grigsby by which the knot Floer homology invariants of Ozsvath, Szabo, and Rasmussen could be used to prove this conjecture.

Mathematical models incorporating random forcing, and the resulting stochastic differential equations (SDEs), are becoming increasingly important. However general principles and techniques for their robust and efficient numerical approximation are a very long way behind the corresponding ODE theory. In both cases the idea of adaptivity, that is using varying timesteps to improve convergence, is a key element. In this talk I will describe an approach based upon (low-order) Milstein-type methods using multiple error-controls. The idea is to monitor various terms in the truncation error, both deterministic and stochastic, and then to construct an algorithm that is robust enough to work efficiently in the presence of deterministic/diffusion-dominated regimes and differing accuracy requirements. Such an approach also has other benefits, such as improved numerical stability properties. No knowledge of stochastic calculus will be assumed.

There are two generally accepted models for the cell biological positive feedback loops that allow yeast cells to break symmetry and establish an axis of polarity. Both have been subjects of published mathematical analyses. Here I will argue that the models used to support a vesicle trafficking model incorporated a simplifying assumption that seemed innocuous but in fact was critical to their success. The assumption is not physically plausible, and its removal means that the model fails. I will show how changing other assumptions can make the model work, but there is no experimental support for those changes. And without them, the vesicle trafficking model perturbs polarity, rather than establishing polarity

Random Polymers are modeled as a one dimensional random walk (S_n), with excursion length distribution P(S_1 = n) = \phi(n)/n^\alpha, \alpha > 1 and \phi(n) a slowly varying function. The polymer gets a random reward whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (\omega_n). Depending on the relation between the mean value of the disorder \omega_n and the temperature, the polymer might prefer to stick to the interface (pinnings) or undergo a long excursion away from it (depinning). In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the `Pinning' model and prove its annealed and quenched critical points are distinct. This is joint work with Ken Alexander.

The Uniformization Theorem implies that any compact Riemann surface has a constant curvature metric. Kahler-Einstein (KE) metrics are a natural generalization of such metrics, and the search for them has a long and rich history, going back to Schouten, Kahler (30's), Calabi (50's), Aubin, Yau (70's) and Tian (90's), among others. Yet, despite much progress, a complete picture is available only in complex dimension 2. In contrast to such smooth KE metrics, in the mid 90's Tian conjectured the existence of KE metrics with conical singularities along a divisor (i.e., for which the manifold is `bent' at some angle along a complex hypersurface), motivated by applications to algebraic geometry and Calabi-Yau manifolds. More recently, Donaldson suggested a program for constructing smooth KE metrics of positive curvature out of such singular ones, and put forward several influential conjectures. In this talk I will try to give an introduction to Kahler-Einstein geometry and briefly describe some recent work mostly joint with R. Mazzeo that resolves some of these conjectures. One key ingredient is a new C^{2,\alpha} a priori estimate and continuity method for the complex Monge-Ampere equation. It follows that many algebraic varieties that may not admit smooth KE metrics (e.g., Fano or minimal varieties) nevertheless admit KE metrics bent along a simple normal crossing divisor.

In his recent paper,\Beyond Endoscopy", Langlands proposed an approach to (ultimately) attack the general functoriality conjectures by means of the trace formula. For a (reductive algebraic) group G over a global field F and a representation of its L-group, the strategy, among other things, aims at detecting those automorphic representations of G for which the L-function, L(s;\pi ;\rho ), has a pole at s = 1. The method suggested using the the trace formula together with an averaging process to capture these poles. In this talk we will start by recalling the functoriality conjectures and brie y describe the method suggested by Langlands. Then, specializing on the group GL(2) we will discuss some recent work on Beyond Endoscopy. More precisely, we will discuss the elliptic part of the trace formula and the analytic problems caused by the volumes of tori, singularities of orbital integrals and the non-tempered terms. We will then describe how one can use an approximate functional equation in the trace formula to rewrite the elliptic part which resolves these issues. Finally, we will talk about applications of the resulting formula.

I will discuss various approaches to the question: When does a vector bundle admit a holomorphic structure? I will explore applications of Yang-Mills theory, geometric quantization, and discrete dynamics to this problem.