We consider a problem of optimal investment with intermediate consumption and random endowment in an incomplete semimartingale model of a financial market. We establish the key assertions of the utility maximization theory assuming that both primal and dual value functions are finite in the interiors of their domains as well as that random endowment at maturity can be dominated by the terminal value of a self-financing wealth process. In order to facilitate verification of these conditions, we present alternative, but equivalent conditions, under which the conclusions of the theory hold.

The computer scientists K. Mulmuley and M. Sohoni have proposed a variant of the conjecture that the complexity classes P and NP are distinct that can be approached via geometry and representation theory. It turns out other problems in complexity, including the complexity of matrix multiplication can be also viewed from this "GCT" perspective. I'll explain how questions about complexity classes are transformed into questions about orbit closures, and describe the state of the art for matrix multiplication and the Mulmuley-Sohoni conjecture.

In this talk I shall describe a class of algebras called "symplectic reflection algebras". These algebras are related to several other areas of mathematics. In particular, they can be used to answer a question from symplectic algebraic geometry on the existence of certain symplectic resolutions. I'll say exactly what this question is and give an idea of how one goes about answering it using the representation theory of symplectic reflection algebras.

In this talk we begin from the very basis of ergodic theory and build out of it the simple theorems on the relation of Fourier coefficients to the chaotic behavior of a system as well as types of recurrence for group actions. The talk will have many extremely accessible examples, primarily from groups related to number theory, with a single, fantastic application to Diophantine approximation. The talk is designed to be accessible to all graduate students of all experience levels with the only minor recommendation that they have at some point seen the definition of a measure.

I am going to sketch a proof of the hard Lefschetz theorem without the Hodge theory (i.e. harmonic analysis). The basic idea is a double induction on the existence of polarized (mixed) Hodge structures for projective varieties of dimension n and the hard Lefschetz theorem for smooth projective varieties of dimension n. I will basically focus on one part, how to get the polarized Hodge structure from the hard Lefschetz theorem. There is virtue about this indirect proof. On the one hand, it can be generalized (over other fields, or singular varieties). On the other hand, I can introduce some of the basic tools for studying the topology of complex algebraic varieties.

Results of Chatterjee/Diaconis concerning subgraph counts in large dense graphs.

We will review recent work of Hugonnier and Kaniel on mutual fund portfolio selection in the presence of continuous time reallocation by a representative investor. In their model, a mutual fund manager receives proportional fees on investments in the fund. The investor trades in the fund and a riskless asset, and seeks to maximize expected logarithmic utility from terminal wealth. BSDE techniques are used to determine a fund portfolio which maximizes the initial market value of the fund fees. We will also discuss possible approaches for the case of an investor with power utility function. This is a candidacy talk.

In a recent paper Ambrosio Frid and Silva study the homogenization problem for a class of porous-medium type equations where the flux function is a stationary random process on a compact probability space. We extend their result to the general stochastic framework, removing the compactness assumption on the probability space. Passing from the small to the large scale, we show the convergence in probability of the weak solutions of rescaled problem to the weak solution of an homogenized deterministic porous-medium type equation. This is a joint work with Professor P.E. Souganidis (University of Chicago).

In low-dimensional topology we often can understand a lot about smooth manifold topology by restricting to manifolds with a certain geometric structure. In the case of 4-manifolds, a symplectic structure is one useful geometric structure for this purpose. In this talk we will discuss symplectic 4-manifolds with boundary, which have boundary conditions that are amenable to gluing constructions. These are called convex symplectic fillings of the 3-manifold boundary. Classifications of all convex symplectic fillings of a fixed 3-manifold (with specific conditions for how the symplectic structure restricts to that 3-manifold boundary) have been completed for the 3-sphere and lens spaces. I will talk about some techniques that go into extending these results to classify symplectic fillings of certain Seifert fibered spaces. No knowledge of symplectic 4-manifold topology will be assumed but a little familiarity with the 4-manifold topology from the current course may help for understanding proof details.

We will discuss recent developments towards the study of unipotent flows on homogeneous spaces preserving a geometric but infinite measure. This talk is based on joint works with H. Oh.

We will discuss the section conjecture of anabelian geometry and the sufficiency of the finite descent obstruction to the Hasse principle for the moduli spaces of principally polarized abelian varieties and of curves over number fields. For the former we show that both the section conjecture and the finite descent obstruction fail in a very controlled way. For the latter, we prove some partial results that suggest that the finite descent obstruction suffices. We also show how this sufficiency implies the same for all hyperbolic curves.

The collection of special cubic fourfolds of discriminant d forms a divisor C_d in the Hilbert scheme of cubic hypersurfaces in P^5 . We give an explicit formula for the degrees of C_d using the modularity of their generating series.

Bordered Floer Homology is a variant of Heegaard Floer homology that is defined for 3-manifolds with boundary. Probably the most exciting consequence is a pairing theorem which allows one to decompose a 3-manifold along a surface and compute its Heegaard Floer invariant from that of its pieces. In this talk, we'll introduce the bordered invariant and work toward a conceptual understanding of how to use it to compute the original invariant.

Computers are often used in mathematics to give intuition about geometric properties of analytic objects. With care, they can also be used to compute bounds for proofs. The common usages are to show the existence of solutions to functional or differential equations, or the validity of bifurcation diagrams. This talk will discuss the techniques used in these proofs and the care taken to rigorously obtain bounds using computers. The discussion will center on a proof of the existence of a solution f to the functional equation f(x)=f(f(ax))/a.