In the late 80’s Henniart used the then recently introduced Laumon l-adic local Fourier transform to prove the numerical local Langlands correspondence for GL(n). More recently, Bloch-Esnault and independently Lopez have developed a complex version of this transform. I will explain the fascinating picture that emerges when Henniart’s strategy of proof is translated to this setting of local geometric Langlands parameters.

The classical central limit theorem (CLT) states that for sums of a large number of i.i.d. random variables with finite variance, the distribution of the rescaled sum is approximately Gaussian. However, the statement of the central limit theorem doesn't give any quantitative error estimates for this approximation. Under slightly stronger moment assumptions, quantitative bounds for the CLT are given by the Berry-Esseen estimates. In this talk we will consider similar questions for CLTs for random walks in random environments (RWRE). That is, for certain models of RWRE it is known that the position of the random walk has a Gaussian limiting distribution, and we obtain quantitative error estimates on the rate of convergence to the Gaussian distribution for such RWRE. This talk is based on joint works with Sungwon Ahn and Xiaoqin Guo.

Consistent reconstruction is a linear programming technique for reconstructing a signal $x\in\RR^d$ from a set of noisy or quantized linear measurements. In the setting of random frames combined with noisy measurements, we prove new mean squared error (MSE) bounds for consistent reconstruction. In particular, we prove that the MSE for consistent reconstruction is of the optimal order $1/N^2$ where $N$ is the number of measurements, and we prove bounds on the associated dimension dependent constant. For comparison, in the important case of unit-norm tight frames with linear reconstruction (instead of consistent reconstruction) the mean squared error only satisfies a weaker bound of order $1/N$. Our results require a mathematical analysis of random polytopes generated by affine hyperplanes and of associated coverage processes on the sphere. This is joint work with Alex Powell.

In joint work with H. Sá Earp we introduced a method to construct G2–instantons over compact G2–manifolds arising as the twisted connected sum of a matching pair of building blocks. I will recall some of the background (including the twisted connected sum construction and a short discussion as to why one should care about G2–instantons), discuss our main result and explain how to interpret it in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a K3 surface. If time permits, I will discuss an idea to construct the input required by our gluing theorem.

In this talk, we discuss the distributions of class groups of orders in number fields. We explain how studying such distributions is related to counting integral orbits having bounded invariants that lie inside the cusps of fundamental domains for coregular representations. We introduce two new methods to solve this counting problem, and as an application, we demonstrate how to determine the average size of the 2-torsion in the class groups of cubic orders. Much of this work is joint with Arul Shankar, Artane Siad, and Ila Varma.

Work over the past few years by Mark Gross, Ilya Zharkov, and Wei-Dong Ruan has led to a clear conjectural picture for the structure of generic supersymmetric T^3 fibrations on Calabi-Yau threefolds. We will explain this picture, show how mirror symmetry would be a natural consequence of it, and discuss other possible applications.

We study the minimal genus problem for embeddings of closed, non-orientable surfaces in a homology cobordism between rational homology spheres, using obstructions derived from Heegaard Floer homology and from the Atiyah-Singer index theorem. For instance, we show that if a non-orientable surface embeds essentially in the product of a lens space with an interval, its genus and normal Euler number are the same as those of a stabilization of a non-orientable surface embedded in the lens space itself. This is joint work with Danny Ruberman and Saso Strle.

This talk presents a work in progress with Sylvain Billard, Regis Ferriere and Chi Viet Tran. How the neutral diversity is affected by selection and adaptation is investigated in an eco-evolutionary framework. In our model, we study a finite population in continuous time, where each individual is characterized by a trait under selection and a completely linked neutral marker. The dynamics is ruled by births and deaths, mutations at birth and competition between individuals. The ecological phenomena depend only on the trait values but we expect that these effects influence the generation and maintenance of neutral variation. Considering a large population limit with rare mutations, but where the marker mutates faster than the trait, we prove the convergence of our stochastic individual-based process to a new measure-valued diffusive process with jumps that we call Substitution Fleming-Viot Process. This process restricted to the trait space is the Trait Substitution Sequence introduced by Metz et al. (1996). During the invasion of a favorable mutation, the marker associated with this favorable mutant is hitchhiked, creating a genetical bottleneck. The hitchhiking effect and how the neutral diversity is restored afterwards are studied. We show that the marker distribution is approximated by a Fleming-Viot distribution between two trait substitutions and that time-scale separation phenomena occur. The SFVP has important and relevant implications that are discussed and illustrated by simulations. We especially show that after a selective sweep, the neutral diversity restoration depend on mutations, ecological parameters and trait values.

The frog model is a type of branching random walk model. Active "frogs" move according to random walks, and if they encounter a sleeping frog on their walk, the sleeping frog becomes active and begins an independent random walk. Over the past 20 years, recurrence properties and asymptotic behavior of this system (and many generalizations) have been studied extensively. One way to generalize this system is to consider the continuous version: Brownian motion frogs moving in R^d. In this talk, we will describe a continuous variant of the problem and show a limiting shape theorem analogous to prior discrete results.

Khovanov homology is a special case of a process known as categorification. The idea of categorification is to lift a known polynomial invariant of links to a homology theory whose isomorphism type is an invariant of links and whose ¡°Euler characteristic¡± is the original polynomial. In the case of Khovanov homology, this Euler characteristic is the famous Jones polynomial. After reviewing some basic knot theory and the construction of the Jones polynomial, we discuss the construction of Khovanov homology. Finally, we will discuss some topological applications of Khovanov homology.

After the recent exciting progress in understanding the geometry of algebraic varieties over the complex numbers, it is natural to try to understand the geometry of varieties over an algebraically closed field of characteristic $p>0$. Many technical issues arise in this context. Nevertheless, there has been much recent progress. In particular, the MMP was established for 3-folds in characteristic $p>5$ by work of Birkar, Hacon, Xu and others. In this talk we will discuss some of the challenges and recent progress in this active area.

Linear boundary value problems occur ubiquitously in many areas of science and engineering, and the cost of computing approximate solutions to such equations is often what determines which problems can, and which cannot, be modelled computationally. Due to advances in the last few decades (multigrid, FFT, fast multipole methods, etc), we today have at our disposal numerical methods for most linear boundary value problems that are "fast" in the sense that their computational cost grows almost linearly with problem size. Most existing "fast" schemes are based on iterative techniques in which a sequence of incrementally more accurate solutions is constructed. In contrast, we propose the use of recently developed methods that are capable of directly inverting large systems of linear equations in almost linear time. Such "fast direct methods" have several advantages over existing iterative methods: (1) Dramatic speed-ups in applications involving the repeated solution of similar problems (e.g. optimal design, molecular dynamics). (2) The ability to solve inherently ill-conditioned problems (such as scattering problems) without the use of custom designed preconditioners. (3) The ability to construct spectral decompositions of differential and integral operators. (4) Improved robustness and stability. In the talk, we will also describe how randomized sampling can be used to rapidly and accurately construct low rank approximations to matrices. The cost of constructing a rank k approximation to an m x n matrix A for which an O(m+n) matrix-vector multiplication scheme is available is O((m+n)*k). This cost is the same as that of Lanczos, but the randomized scheme is significantly more robust. For a general matrix A, the cost of the randomized scheme is O(m*n*log(k)), which should be compared to the O(m*n*k) cost of existing deterministic methods.

A ureteric stent is a slender polymer tube that can be placed within the ureter (the muscular tube that conveys urine from the kidney to the bladder) to relieve a blockage due, for example, to a kidney stone in transit, or to external pressure from a tumor. A urinary catheter can be placed similarly within the urethra (the muscular tube conveying urine from the bladder out of the body), either again to relieve a blockage, or to allow control of urination in incontinent patients or those recovering from surgery. Several clinical complications are associated with each of these biomedical devices. Both become encrusted, over time, with salts that precipitate out from the urine. Such encrustation is often associated with infection and the presence of bacterial biofilm on the device and, if severe, can make removal of the device difficult and painful. Ureteric stents are also associated with urinary reflux: retrograde flow of urine back towards the kidney. This arises because the stent prevents proper function of the sphincter between ureter and bladder that normally closes off when bladder pressure rises. Such reflux can expose the kidney to dangerously high pressures, and increase the risk of renal infection, both of which can lead to long-term damage. This talk will highlight aspects of our interdisciplinary work on such problems. We present mathematical models of the reflux and encrustation processes and consider the implications for device design and clinical practice.

TBD

TDA is now about fifteen years old, and is quickly becoming a widely applied tool in data analysis. In this talk, I'll describe how homology groups, a traditional algebraic invariant, can be turned into persistence diagrams, a robust statistical tool for dealing with high-dimensional data or embedded geometric objects. Time permitting, this talk should have some theory, some applications, and some algorithms, and perhaps even a proof.

In a world of polarized opinions on many cultural issues, we propose a model for the evolution of opinions on a large complex network. Our model is akin to the popular Friedkin-Johnsen model, with the added complexity of vertex-dependent media signals and confirmation bias, both of which help explain some of the most important factors leading to polarization. The analysis of the model is done on a directed random graph, capable of replicating highly inhomogeneous real-world networks with various degrees of assortativity and community structure. Our main results give the stationary distribution of opinions on the network, including explicitly computable formulas for the conditional means and variances for the various communities. Our results span the entire range of inhomogeneous random graphs, from the sparse regime, where the expected degrees are bounded, all the way to the dense regime, where a graph having n vertices has order n^2 edges.

I will discuss some action functionals constructed by Hitchin which lead to manifolds of special holonomy in six and seven dimensions, and explain how these actions could be related to a topological version of M-theory which deals with variations of G2 structures.

For a K3 surface X over a number field with potentially good reduction everywhere, we prove that there are infinitely many primes modulo which the reduction of X has larger geometric Picard rank than that of the generic fiber X. A similar statement still holds true for ordinary K3 surfaces over global function fields. In this talk, I will present the proofs via the intersection theory on GSpin Shimura varieties and also discuss various applications. These results are joint work with Ananth Shankar, Arul Shankar, and Salim Tayou and with Davesh Maulik and Ananth Shankar.