Proving that Galois representations in many situations arise from automorphic forms has been a major theme in number theory for at least two decades. However, most of the existing work concerns the situation when the mod p reduction of the Galois representation (i.e., the residual representation) is irreducible and when the number field is totally real. We will present new modularity results for n-dimensional residually reducible Galois representations over arbitrary number fields. This is joint work with T. Berger.

The Fontaine-Mazur conjecture predicts which p-adic Galois representations
arise geometrically. A few years back, Emerton and Kisin made astounding
progress in the two-dimensional case, by employing the p-adic Langlands
correspondence for GL(2) over $Q_p$, which is very well-understood by now.
A key point was the existence of ``locally algebraic" vectors.
For groups of higher rank, even a conjectural generalization remains
elusive. However, there is a conjecture of Breuil & Schneider,
which gives a weak (but precise) analogue for GL(n). Roughly it says
that a certain filtration exists if and only if a certain lattice exists.
In his thesis, Hu completely proved one direction, and produced the
expected filtration (by translating its existence into the so-called
Emerton condition). We will report on progress in the other direction,
and in many cases prove the existence of GL(n)-stable lattices in
locally algebraic representations constructed from p-adic Hodge theoretical
data. This argument is global in nature; the ultimate integral structure
comes from p-adic modular forms. We hope to also hint at a formalism,
in which an eigenvariety for U(n) parametrizes a correspondence between
semisimple Galois representations and Banach-Hecke modules with a
unitary GL(n)-action, and discuss local-global compatibility "at p"
in this context. In particular, we'll settle the Breuil-Schneider conjecture
for dR representations which ``come from an eigenvariety".

Compactified Jacobian of a singular rational curve is homeomorphic to the direct product of Jacobi factors of singularities of the curve. Therefore, to study the topology of Compactified Jacobians in this case, it suffices to study topology of Jacobi factors, which can be defined very explicitly as subvarieties in Grassmanians. J. Piontkowski showed that in certain cases Jacobi factors can be decomposed into affine cells enumerated by semimodules over the semigroup of the singularity. We proved that for quasihomogeneous plane curve singularities the cells can also be enumerated by Young diagrams inscribed in a right triangle, and dimensions of cells can be computed in a combinatorial way. The resulting combinatorial model is closely related to cell decompositions of Hilbert schemes of points on a complex plane, and to rational generalizations of Garsia-Haiman's q,t-Catalan numbers. In this talk, I will discuss our results on topology of Compactified Jacobians and, time permitting, mention connections to the theory of finite dimensional representations of rational Cherednik algebras and homological knot invariants. The talk is based on a joint work with Eugene Gorsky.

Starting from the notion of BN-pair (which is closely related to the concept of Tits building), we explain, following a classical insight, how one interprets buildings and their automorphism groups over the nonexisting ``field with one element". (As such, the symmetric groups become Chevalley groups over this ``field".) We then introduce scheme theories in ``characteristic one", and show how these are governed by a deep combinatorial (and motivic) nature.

As a vast generalization of quadratic reciprocity, class field theory
describes all abelian extensions of a number field.
Over Q, they are precisely those contained in cyclotomic fields.
However, there are a lot more non-abelian extensions, which arise naturally.
The Langlands program attempts to systematize them, by relating Galois
representations and automorphic forms; mathematical objects of rather
disparate nature. We will illustrate the basic plot for GL(2) through
the example of elliptic curves and modular forms - the context of Wiles' proof
of Fermat's Last Theorem. The main goal of the talk will be to motivate
a ``p-adic" Langlands correspondence, which is at the forefront of
contemporary number theory, but still only well-understood for GL(2) over $Q_p$.
We will discuss, in some depth, the case of semistable elliptic curves,
which provide the first non-trivial example. This leads naturally to
a result we proved recently, which shows the existence of (many) integral
structures in locally algebraic representations of ``Steinberg" type,
for any reductive group G (such as GL(n), symplectic, and orthogonal groups).
As a result, there are a host of ways to p-adically complete the Steinberg
representation (tensored with an algebraic representation). The ensuing
Banach spaces should play a role in a (yet elusive) higher-dimensional
p-adic Langlands correspondence. We hope to at least give some idea of the
proof, which goes via automorphic representations and the trace formula.

The notion of mesh patterns in permutations was introduced recently by Petter Branden and Anders Claesson to provide explicit expansions for certain permutation statistics as possibly infinite linear combinations of (classical) permutation patterns. In my talk, I will discuss eight mesh patterns of small lengths. In particular, I will link avoidance of one of the patterns to the harmonic numbers, while for three other patterns I will show their distributions on 132-avoiding permutations are given by the Catalan triangle. Also, I will show that two specific mesh patterns are Wilf equivalent meaning that for any length, the number of permutations avoiding one of the patterns equals that avoiding the other one. As a byproduct of these studies, one defines a new set of sequences counted by the Catalan numbers and provides a relation on the Catalan triangle that seems to be new. This is joint work with Jeffrey Liese.

The Chern-Simons-Schr\"odinger model in two spatial dimensions is a covariant NLS-type problem and is $L^2$ critical. We prove that

Whenever a mathematical structure is given in characteristic $p$, one can ask whether it is the reduction, in some sense, of an analogous structure in characteristic zero. If so, the structure in characteristic zero is called a ``lift'' of the structure in characteristic $p$. The most famous example is Hensel's Lemma about lifting solutions of polynomials in $\mathbb{Z}/p$ to solutions in the $p$-adic integers $\mathbb{Z}_p$. We will consider a more geometric problem: given a curve $X$ in characteristic $p$ with an action of a finite group $G$, is there a curve in characteristic zero with $G$-action that reduces to $X$? Oort conjectured that this could be done when $G$ is cyclic, and his conjecture was recently proven by the speaker, Stefan Wewers, and Florian Pop. It turns out that this question reduces to a more ``local'' question about automorphisms of power series rings in one variable. This local question will occupy most of the talk. Many examples will be given throughout.

Vesture refers to the recipe for finding nontrivial exact solutions to a nonlinear differential equation by "dressing" a known solution (which could be rather trivial) with one or more prescribed singularities. This is in particular possible if the nonlinear equation is completely integrable. We will explain what that means, and provide examples from the theory of harmonic maps, and General Relativity, to demonstrate how this procedure works. We will in particular show how to obtain the nontrivial and physically interesting hyperextreme Kerr metric, which has a naked ring singularity at its center, from the trivial solution of Einstein Vacuum Equations, namely the Minkowski metric. This is joint work with Shabnam Beheshti.

We study Galois covers of the projective line branched at three points with Galois group $G$. When such a cover is defined over a $p$-adic field, it is known to have potentially good reduction to characteristic $p$ if $p$ does not divide the order of $G$. We give a sufficient criterion for good reduction, even when $p$ does divide the order of $G$, so long as the $p$-Sylow subgroup of $G$ is cyclic and the absolute ramification index of a field of definition of the cover is small enough. This extends work of (and answers a question of) Raynaud. Our proof depends on working very explicitly with Kummer extensions of complete discrete valuation rings with imperfect residue fields.

I shall explain how a new approach via the Hodge-Laplace heat equation works in solving the Poincare-Lelong equation. This method essentially is reduced to a uniqueness theorem and some estimates concluding the preservation of the d-closedeness of the solution of the Hodge-Laplace heat equation, and circumvents the essential difficulties of the elliptic method previously adapted by many people without being able to prove the best possible result. This is a joint work with Luen-Fai Tam.

One of the simplest model problems in the kinetic theory of plasma--physics is the Vlasov-Poisson system with periodic boundary conditions. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. In this talk, we present some Discontinuous Galerkin (DG) methods for the approximation of the Vlasov-Poisson system. The schemes are based on the coupling of DG approximation to the Vlasov equation (transport equation) and several finite element (conforming, non-conforming and mixed) approximations to the Poisson problem. We present the error analysis and discuss further properties of the proposed schemes. We also present numerical experiments in the 1D case that verify the theory and validate the performance of the methods in benchmark problems. If time allows, in the last part of the talk, we shall discuss the possibility of combining the proposed methods with some dimension reduction techniques, such as sparse grids. The talk is based on joint works with Saverio Castelanelli (UAB-CRM), J.A. Carrillo (Imperial College-ICREA), Soheil Hajian (Univ. Geneva) and Chi-Wang Shu (Brown University).

I will discuss the computational methods based on a continuum solvent model to calculate binding affinity for protein-ligand systems. I will also show how such models can be combined with statistical learning methods to predict binding specificity of protein recognition. An application of these methods to understand drug resistance will exemplify its usefulness in drug development.

Motivated by the quest to better understand the structure of the adèle class space of a global field in characteristic zero, Connes and Consani recently initiated a theory of hyperfield extensions of the so-called ``Krasner hyperfield'' which is one of the realizations of the field with one element. Remarkably, there appears a deep correspondence with certain group actions on certain combinatorial geometries. We will elaborate on brand new results in this theory, make wild speculations and pose several important open questions on the way.

Description trees were introduced by Cori, Jacquard and Schaeffer in 1997 to give a general framework for the recursive decompositions of several families of planar maps studied by Tutte in a series of papers in the 1960s. We are interested in two classes of planar maps which can be thought as connected planar graphs embedded in the plane or the sphere with a directed edge distinguished as the root. These classes are rooted non-separable (or, 2-connected) and bicubic planar maps, and the corresponding to them trees are called, respectively, $\beta(1,0)$-trees and $\beta(0,1)$-trees.
Using different ways to generate these trees we define two endofunctions on them that turned out to be involutions. These involutions are not only interesting in their own right, in particular, from counting fixed points point of view, but also they were used to obtain non-trivial equidistribution results on planar maps, certain pattern avoiding permutations, and objects counted by the Catalan numbers. The results to be presented in this talk are obtained in a series of papers in collaboration with several researchers.

Parallel replica dynamics was proposed by A.F. Voter as a tool for accelerating molecular dynamics simulations characterized by a sequence of infrequent, but rapid, transitions from one state to another. An example would be the migration of a defect through a crystal. Parallel replica dynamics accelerates this by simulating many replicas simultaneously, concatenating the simulation times of the realizations as though it were a single long trajectory. This motivates important questions: Is parallel replica dynamics algorithm doing what we hope? For what systems will it be useful? How do we implement it efficiently? In this talk, I will thoroughly describe the algorithm and report on progress towards rigorous justification. Open questions and related problems will also be discussed.

Given a permutation w, Stanley defined a symmetric function $F_w$ which encodes information about the reduced words of w, and showed that $F_w$ is a single Schur function exactly when w avoids the pattern 2143. We generalize this statement, showing that the Schur expansion of $F_w$ respects pattern containment in a certain sense, and that the number of Schur function terms is determined by pattern avoidance conditions on w. Along the way, we compute the cohomology of certain subvarieties of Grassmannians, resolving some cases of a conjecture of Liu. This is joint work with Sara Billey.