In these talks, we will describe the dynamics of different geometric flows defined on bundles over moduli spaces of Riemann surfaces. We will discuss some applications and related open problems motivated by the analogy between the moduli space of surfaces and negatively curved manifolds. This is the first of a series of 3 lectures to be delivered by the speaker.

We explain some recent progress in the classification of higher dimensional algebraic varieties. It is a classical result that algebraic curves (equivalently compact Riemann surfaces) fall into three classes, determined by the genus. In this talk I will explain why the classification of curves is so appealing and to what extent this extends to higher dimensions.

In my last talk, I discussed several vector space norms on the space of algebraic numbers modulo torsion which are constructed in order to satisfy an extremal property with respect to the Mahler measure. We then proved that the infimum in the construction is achieved in a certain finite dimensional space. In this talk I will discuss the S-unit projections which allow us to reduce to finite dimensions, and how they are special to the L1 case. This is joint work with Paul Fili.

Groupoids are categories in which all morphisms are invertible, and they arise naturally in many areas of mathematics. Surprisingly, perhaps, they provide a framework to better understand the notion of convergence of spaces under geometric flows. I will give a general overview of basic groupoid theory, and then provide an illustration of their use in studying the Ricci flow.

Some discussion of highlights of last week's conference in College Station.

The aim of this talk is to assess the impact of jumps in the asset returns dynamics on the optimal consumption and portfolio problem. We propose a practical approach for modeling high-frequency data by using jump processes. In order to overcome the non-stationary of financial markets we first employ the time change. Then we calibrate two types of models and compare their fit to the real data. We present equations for the optimal consumption and portfolio choice. We also introduce an alternative equations for the optimum controls in terms of cumulants. Using these new equations we discuss the effect of the skewness and the kurtosis on the optimal investment strategy. Finally, we perform a numerical study. We compute the optimal investment strategy for different skewness and kurtosis levels assuming given jump model and compare the optimal investment proportion with the Merton proportion. This presentation is based on joint work with Jakub Petrasek.

I will talk about recent joint work with Rahul Panhdaripande and Bernd Siebert, giving an example of mirror symmetry in which both sides of the mirror symmetry story can be explained in elementary terms. In its original incarnation as proposed by string theorists in the UT Austin physics department around 1989, lines, conics and so forth lying on certain complex manifolds called Calabi-Yau manifolds can be counted via calculating certain integrals on a mirror manifold. Since then, a great deal of mathematical energy has been devoted to understanding the reasons why this calculation works. In the version I will discuss, we relate much simpler counts of curves to commutator formulas in a certain group of symplectomorphisms. We believe that this version lies at the heart of an explanation for mirror symmetry.

Given a Hamiltonian H on a compact manifold, the Mane conjecture states that, for a generic potential V, the Aubry set associated to H+V is either a fixed point or a periodic orbit. In this talk I'll show how, given a Hamiltonian which possesses a sufficiently smooth viscosity (sub)solution to the Hamilton-Jacobi equation, for any here exists a potential with -norm bounded by such that the Aubry set associated to H+V_\epsilon is either a fixed point of a periodic orbit. This represents a first step towards the solution of the Mane conjecture. This is a joint work with Ludovic Rifford.

Being a tale of adventure on the high seas involving great risk to the tale teller, and how an understanding of the mathematical theory of knots saved his bacon. No nautical or mathematical background assumed.

Abstract: Let X be a Banach space and denote by SS1(X) the set of all S1-strictly singular operators in X. We prove that there is a Banach space X such that SS1(X) is not an ideal. More specifically, we construct a space X and operators T1, T2 in SS1(X) such that T1+T2 is not in SS1(X). We show one example where the space X is reflexive and other where it is c0-saturated.

Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diħe-Hellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. The strength of computing machinery has made these techniques theoretically susceptible to attack and hence recently there has been an active line of research to develop cryptosystems and key exchange protocols using noncommutative cryptographic platforms. This line of investigation has been given the broad title of noncommutative algebraic cryptography. This was initiated by two public key protocols that used the braid groups, one by Ko, Lee et.al. and one by Anshel, Anshel and Goldfeld. The study of these protocols and the group theory surrounding them has had a large effect on research in infinite group theory. In this talk I survey some of these noncommutative group based methods and discuss several ideas in abstract infinite group theory that have arisen from them. I will mention about some new results and then present a set of open problems.

I will give a more in depth survey of a program due to myself and Siebert for understanding mirror symmetry. The underlying objects controlling mirror symmetry are manifolds with affine structure. These objects can be interpreted in two different ways, yielding the two sides of mirror symmetry. In particular, certain types of curves on these affine manifolds called "tropical curves" are expected to yield interpretations in terms of rational curve counts and period integrals.

Knots have appeared everywhere from art, literature and religion to seafaring. We will discuss their past history as well as their future, with emphasis on the role they play in mathematics and the applications of that role to DNA and synthetic chemistry. No background assumed. Reception to follow.

Inverse transport consists of reconstructing the optical parameters in a transport equation from knowledge of a measurement operator. We review several uniqueness and stability results obtained in the context of various boundary measurements as they arise e.g. in optical tomography, a medical imaging modality. Accurate numerical reconstructions obtained by carefully capturing the singularities of the measurement operator are also briefly presented. In many practical settings, the typical boundary measurements available in inverse transport do not allow us to reconstruct the optical parameters with sufficient resolution. In the second part of the talk, I present recent results obtained for the inverse transport problem with internal controls as they arise in the application of photoacoustic tomography, a recent hybrid medical imaging modality that combines the large contrast observed in optical parameters with the high resolution of ultrasounds. These are joint works with Alexandre Jollivet, Francois Monard, and Gunther Uhlmann

I will discuss what is known about the behaviour of Ricci-flat metrics on Calabi-Yau manifolds when the metric degenerates. In particular, I shall focus on results and conjectures involving collapsing metrics, such as situations in which metrics on K3 surfaces converge to two-spheres. (This is joint work with Pelham Wilson).