Self-adjoint $n$-by-$n$ matrices have a natural partial ordering, namely $ A \leq B $ if the matrix $ B-A$ is positive semi-definite.

In 1934 K. Loewner characterized functions that preserve this ordering; these functions are called $n$-matrix monotone. The condition depends on the dimension $n$, but if a function is $n$-matrix monotone for all $n$, then it must extend analytically to a function that maps the upper half-plane to itself.

I will describe Loewner's results, and then discuss what happens if one wants to characterize functions $f$ of two (or more) variables that are matrix monotone in the following sense: If $ A = (A_1, A_2)$ and $B = (B_1,B_2)$ are pairs of commuting self-adjoint $n$-by-$n$ matrices, with $A_1 \leq B_1 $ and $A_2 \leq B_2$, then $f(A) \leq f (B)$.

This talk is based on joint work with Jim Agler and Nicholas Young.

Algebraic Geometry is one of the oldest and most central areas
in Mathematics. Looking at it from different angles may result in
quite different pictures. In this talk we'll take a tropical viewpoint.

This is a point of view which ignores the phase of a complex
number (or the sign of a real number). There is a correspondence
principle between complex and tropical objects (which is related
to the correspondence principle in Quantum Mechanics). In the
talk we will consider some instances of this correspondence.

Whereas Hamiltonian dynamics (be it described by Newton's or Schr̦dinger's equation) is time-reversible, most phenomena in our macroscopic world are clearly irreversible. On the level of theoretical phyiscs, this dichotomy between micro and macro is well-understood since the work of Boltzmann. However, the mathematical questions it raises, remain largely unsolved. Despite progress in the field of dynamical systems and quantum field theory, there are still only a few systems in which we can prove that irreversible behaviour like return to equilibrium, diffusion, friction, actually occur. I will present an overview and some recent progress.

The fractional Brownian motion is a centered self-similar Gaussian process with stationary increments, which depends on a parameter H in (0,1) called the Hurst index. In this talk we will describe some basic properties of the fractional Brownian motion, and we will analyze different approaches to construct a stochastic calculus with respect to this process using path-wise techniques, Riemann sums and Malliavin calculus. We will also present some recent results on the existence and uniqueness of solutions and numerical approximations for stochastic differential equations driven by a fractional Brownian motion.

The study of division algebras, meaning noncommutative ``fields'' finite over their center, is greatly enhanced by their correspondence with Brauer group elements and thereby with etale or Galois cohomology. But often the interesting properties of division algebras, like their degree or index (really their ``size''), is difficult or impossible to discern from the cohomology. Despite this there are results about indices over surfaces of various sorts, associated applications, and just a hint of a general theory. Of course there are many more interesting conjectures and questions without answers.

If $X$ is a projective algebraic variety defined over an algebraically closed field $F$ of characteristic zero, then there is a unique minimal (algebraically closed) subfield $F_0$ of $F$ such that $X$ is isomorphic to a variety defined over $F_0$. In this talk, I shall discuss whether there are analogous results in characteristic p.

Such questions arise in proofs of the geometric version of the Mordell conjecture. (Recall that the original Mordell conjecture asserts that a curve of genus at least 2 defined over a number field has only finitely many rational points).

I will start by giving an overview of the theory of descent from a number of perspectives, sketch a proof of the characteristic zero result (which is to some extent "well known") and then discuss what happens in characteristic p. Along the way, I shall also mention various notions of Galois theory for purely inseparable extensions.

The original motivation for the theory of buildings was to provide a purely combinatorial approach to semi-simple algebraic groups. Indeed, one of the first non-trivial applications of this theory was a combinatorial proof of an earlier version of the Borel-Tits theorem. This proof was indicated by Tits in 1962, but not written up in detail, because there was a more efficient approach for the final result using completely different tools. However, the original geometric ideas regained interest through the work of Serre on complete reducibility of spherical buildings which lead him to promote a question which became known as the 'center conjecture'.

There are several versions of the center conjecture, and its 'simplicial version' has been proved in joint work with Tits for classical groups in 2006. The remaining exceptional -- and considerably more difficult -- cases have been treated by Leeb and Ramos-Cuevas in 2009.

In the first part of my talk I intend to explain the center conjecture for projective spaces and to say something about its history. In the second part I want to present the main ideas of a new proof of the simplicial version of the center conjecture which I found recently in joint work with R. Weiss. It is considerably shorter than the existing proof and has also the advantage that it works uniformly for all types.

Representation theory emerged about 100 years ago with the pioneering work of Frobenius and Schur, and has become a central area of mathematics because of its connections to combinatorics, algebraic geometry, number theory, and applications to physics. Cohomology theories were developed throughout the 20th century by topologists to construct algebraic invariants for the investigation of manifolds and topological spaces. Cohomology was also defined for algebraic structures like groups and Lie algebras to find the possible ways in which their representations can be glued together.

In my talk I will demonstrate how cohomology theories for algebraic structures can be used to reintroduce ambient geometric structures into the picture. These methods have their origins in finite group cohomology. Later in the talk, I will present new applications of these ideas to "categorifying" the important long standing combinatorial notions of defect and atypicality developed by Kac and Wakimoto for Lie superalgebras. Moreover, as another application, I will present recent results on the realization of support varieties for irreducible representations for the small quantum group through the nilpotent cone (ambient geometry) with connections to the Lusztig Character Formula and the properties of Kazhdan-Lusztig polynomials.

In 2009 Mike Hill, Mike Hopkins and I solved the Kervaire invariant problem in stable homotopy theory. In the first talk I will describe the history of the problem beginning with Pontryagin's work on the homotopy groups of spheres in the 1930s and Kervaire-Milnor's work on exotic spheres in the 1960s.

Many classical families of groups, including arithmetic groups, surface mapping class groups and automorphism groups of free groups, are studied by considering the Geometry, Topology of their actions on nice contractible spaces. In the case of the group Out(F_n) of outer automorphisms of a free group the appropriate space is known as Outer space. I will describe Outer space, show how it is used to obtain information about Out(F_n) and then indicate how these ideas have recently been used to study a wider class of automorphism groups.

In the late 1950s Kolmogorov introduced the concept of entropy into ergodic theory using ideas from information theory. Entropy has since become a pervasive presence in dynamics, and as originally conceived by Kolmogorov it was ultimately seen to apply most generally to actions of amenable groups, for which one can average over partial orbits in a way that produces a meaningful invariant.

Recently Lewis Bowen showed, quite surprisingly, that this theory of entropy can be vastly extended to the realm of actions of countable sofic groups. Soficity is a much weaker kind of finite approximation property than amenability, and in fact it is unknown whether there exists a countable group that is not sofic. The definition of entropy in this case required a new strategy that replaces the information-theoretic perspective with the statistical-mechanical idea of counting discrete models. Hanfeng Li and I have subsequently developed an alternative and more general approach to sofic entropy that uses operator algebras in an unexpectedly essential way. I will describe all of these developments and discuss some examples that also connect to operator algebras in a different way.

Suppose $V$ is a subvariety of the bidisk, $\{ (z,w) \in {\mathcal C}^2 : |z| < 1, |w| < 1 \}$. Does every holomorphic function $f$ on $V$ have an extension to a holomorphic function $F$ on the whole bidisk? The answer is yes; this is a special case of a celebrated theorem of H. Cartan in 1951.

Now suppose in addition that the range of $f$ lies in the open unit disk; can $F$ always be chosen so that it also takes values in the disk? The answer now depends on the geometry of $V$.

I shall discuss how the problem arises, and what the solution is.

The Skolem-Mahler-Lech theorem is a well-known theorem from number theory which says that if $f(n)$ satisfies a linear recurrence over a field of characteristic $0$, then the set of $n$ for which $f(n)$ is equal to zero is a finite union of infinite arithmetic progressions along with a finite set. We show how this theorem can be recast geometrically and is connected to the dynamical Mordell-Lang problem in arithmetic dynamics. In addition, we consider what happens if $f(n)$ satisfies a recurrence over a field of characteristic $p>0$. In this case, Derksen has shown that the conclusion does not hold, but that a beautiful characterization can be given in terms of finite-state automata, and we describe Derksen's work. (Part of this talk involves joint work with Dragos Ghioca and Tom Tucker.)