The combinatorial complexity of the union of a set of geometric objects is the total number of faces of all dimensions that lie on the boundary of the union. We review some recent results on the complexity of the union of geometric objects in 2d and 3D satisfying various natural conditions and on computing the union. We then discuss the critical roles they play in computing an epsilon net and a hitting set of a set system.

Kick off this year's flu season with a better understanding of within-host influenza dynamics. Influenza A is a rapidly-evolving RNA virus that typically escapes herd immunity through the generation of new antigenic variants every 3 to 8 years. An important part of this antigenic evolution is believed to occur at the intrahost level. I will present two competing models of intrahost dynamics and compare their predictions to empirical observations.

Using Heegaard Floer homology, we construct a numerical invariant for any smooth, oriented 4-manifold X with the homology of S^1 x S^3. Specifically, we show that for any smoothly embedded 3-manifold Y representing a generator of H_3(X), a suitable version of the Heegaard Floer d invariant of Y, defined using twisted coefficients, is a diffeomorphism invariant of X. We show how this invariant can be used to obstruct embeddings of certain types of 3-manifolds, including those obtained as a connected sum of a rational homology 3-sphere and any number of copies of S^1 x S^2. We also give similar obstructions to embeddings in certain open 4-manifolds, including exotic R^4s. This is joint work with Danny Ruberman.

Although “big data” is ubiquitous in data science, one often faces challenges of “small data,” as the amount of data that can be taken or transmitted is limited by technical or economic constraints. To retrieve useful information from the insufficient amount of data, additional assumptions on the signal of interest are required, e.g. sparsity (having only a few non-zero elements). Conventional methods favor incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent. I will present a nonconvex approach that works particularly well in the coherent regime. I will also address computational aspects in the nonconvex optimization. Various numerical experiments have demonstrated advantages of the proposed method over the state-of-the-art. Applications, ranging from super-resolution to low-rank approximation, will be discussed.

It is known that the simple random walk on the unique infinite cluster of supercritical percolation on Z^d diffuses in the same way it does on the original lattice. In critical percolation, however, the behavior of the random walk changes drastically. The infinite incipient cluster (IIC) of percolation on Z^d can be thought of as the critical percolation cluster conditioned on being infinite. Alexander and Orbach (1982) conjectured that the spectral dimension of the IIC is 4/3. This means that the probability of an n-step random walk to return to its starting point scales like n^{-2/3} (in particular, the walk is recurrent). In this work we prove this conjecture when d>18; that is, where the lace-expansion estimates hold. Joint work with Gady Kozma.

In this talk, we will introduce a dynamic mathematical model that describes the interaction of the immune system with the human immunodeficiency virus (HIV). Using optimal control theory, we will illustrate that optimal dynamic multidrug therapies can produce a drug dosing strategy that exhibits structured treatment interruption, a regimen in which patients are cycled on and off therapy. In addition, sensitivity analysis of the model including both classical sensitivity functions and generalized sensitivity functions will be presented. Finally, we will describe how stochastic estimation can be used to filter and estimate states and parameters from noisy data. In the course of this analysis it will be shown that automatic differentiation can be a powerful tool for this type of study.

Class groups of cyclotomic fields have long been of central interest in number theory. We consider elements of these class groups that arise as values of a cup product pairing on cyclotomic units. These pairing values yield information on a wealth of algebraic objects, but any analytic interpretation of them was heretofore unknown. We will describe how, conjecturally, modular representations can be used to relate the pairing values to p-adic L-values of cusp forms.

I will introduce and discuss a canonical notion of Brownian motion in the random geometry of Liouville quantum gravity, called Liouville Brownian motion. I will explain the construction and discuss some of its basic properties, for instance related to its heat kernel and to the time spent in the thick points of the Gaussian Free Field. Time permitting I will also discuss a derivation of the KPZ formula based on the Liouville heat kernel (joint work with C. Garban. R. Rhodes and V. Vargas).

Supersymmetry can severely constrain higher derivative terms in the effective action of both Yang-Mills theories and string theories. I will explain how to obtain these constraints.

My group studies colloidal suspensions, which are solid micron-sized particles in a liquid. We use an optical confocal microscope to view the motion of these colloidal particles in three dimensions. In some experiments, these particles arrange into a crystalline lattice, and thus the sample is analogous to a traditional solid. We study the interface between colloidal crystals and colloidal liquids, and find that this interface is quite sharply defined. In other experiments, the sample is analogous to a glass, with particles randomly packed together. The particles correspond to individual molecules in a traditional glass, and the sample exhibits glassy behavior when the particle concentration is high. This allows us to directly study the microscopic behavior responsible for the macroscopic viscosity divergence of glasses.

Motivated by the dynamic feedback systems in the kidney, we consider time-delayed transport equations with stochastic left-hand boundary conditions. We first prove the existence and uniqueness of the steady-state solution for the deterministic case with sufficiently small delay (e.g., zero left-hand boundary). Likewise, we prove those for the stochastic case using the similar analytic techniques. In this talk we will show the process of model formulation with biological motivation and address the role of time-delay and stochastic boundary on the solution behaviors. The talk should be accessible to all graduate students who are familiar with basic ODE theory.

Many machine learning algorithms are based upon estimating eigenvalues and eigenfunctions of certain integral operators. In practice, we have only finitely many randomly drawn points. How close are the eigenvalues and eigenfunctions of the finite dimensional matrix we construct in comparison to the infinite dimensional integral operator? In what way can we say these two are close if they do not even operate on the same spaces? To answer these questions, I will be showing some results from a paper "On Learning with Integral Operators" by Rosasco, Belkin, and De Vito.

Advances in high-throughput genomics have facilitated the development of pooled population sequencing techniques which involve the en masse sequencing of tens to hundreds of individual genomes in a single sequencing reaction. Pooled population sequencing methods have numerous applications in quantitative, population and evolutionary genetics. I will discuss some of the statistical and computational challenges associated with the analysis of pooled sequence data in the context of quantitative trait locus (QTL) mapping and detecting selection during experimental evolution.

In this talk, I will introduce some new relative trace formulas toward the global Gan-Gross-Prasad conjecture for unitary groups, which generalize the trace formulas of Jacquet-Rallis and Flicker. In particular, I will state the corresponding conjecture of relative fundamental lemmas. A relation between the well-studied Jacquet-Rallis case the equal-rank case will also be discussed.

Richard Borcherds won the Fields medal in 1998 for his proof of the Monstrous Moonshine Conjecture. Loosely speaking, the conjecture asserts that the representation theory of the Monster, the largest sporadic finite simple group, is dictated by the Fourier expansions of a distinguished set of modular functions. This conjecture arose from astonishing coincidences noticed by finite group theorists and arithmetic geometers in the 1970s. Recently, mathematical physicists have revisited moonshine, and they discovered evidence of undiscovered moonshine which some believe will have applications to string theory and 3d quantum gravity. The speaker and his collaborators have been developing the mathematical facets of this theory, and have proved the conjectures which have been formulated. These results include a proof of the Umbral Moonshine Conjecture, and Moonshine for the first sporadic finite simple group which does not occur as a subgroup or subquotient of the Monster. The most recent Moonshine (announced here) yields unexpected applications to the arithmetic elliptic curves thanks to theorems related to the Birch and Swinnerton-Dyer Conjecture and the Main Conjectures of Iwasawa theory for modular forms. This is joint work with John Duncan, Michael Griffin and Michael Mertens.

The locust olfactory system interfaces with the external world through antennal receptor neurons (ORNs), which represent odors in a distributed, combinatorial manner. ORN axons bundle together to form the antennal nerve, which relays sensory information centrally to the antennal lobe (AL). Within the AL, an odor produces a stimulus-specific temporal progression of neuronal spiking, inspiring the hypothesis that the AL encodes odors through dynamically evolving ensembles of active cells. Such a coding strategy, however, requires higher olfactory centers to integrate a prolonged dynamic profile of AL signals prior to stimulus assessment, a process that is likely to be slow and inconsistent with the generation of quick behavioral responses. Our modeling work has led us to propose an alternate hypothesis: the dynamical interplay of fast and slow inhibition within the locust AL induces transient correlations in the spiking activity of an odor-dependent neural subset, giving rise to a temporal binding code and allowing rapid stimulus detection by downstream elements.

Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions which lead to the emergence of global behaviors: aggregation and flocking. I will discuss the models in different scales: from microscopic agent-based dynamics, through kinetic mean-field descriptions, to macroscopic fluid systems. In particular, the macroscopic models can be viewed as compressible Euler equations with nonlocal interactions. I will show some recent results on the global wellposedness theory of the systems, large time behaviors, and interesting connections to some classical equations in fluid mechanics.