I will discuss recent developments in the theory of elliptic fibrations and how they interact with interesting questions in birational geometry, representation theory, incidence geometry, combinatorics and physics.

# Elliptic Fibrations in String Theory, flop transitions, and Hyperplane Arrangements

# Introduction to *-Autonomous categories, continued

I will mention one example I didn't get to last week: two equivalent categories of locally convex topological vector spaces. The dualizing object is the topological field C (you could just look at real spaces and use R and get the same results). The first category is the weakly topologized spaces--they have the weak topology for their continuous linear functionals. The second is known as the Mackey spaces, but like the other examples have the strongest possible topology for their continuous linear functionals. Then I will describe the "And then a miracle happened" of the Chu and chu constructions that were based on Mackey's "pairs".

Anyone interested in more detail on last week's introduction should look here: http://www.math.mcgill.ca/barr/papers/#dsac

particularly these two papers:

- Topological *-autonomous categories (TAC 2006)
- On duality of topological abelian groups (unpublished note)

Some of the other papers in that list might be interesting, but those are the most relevant to what I said on Feb 2nd.

# Optimal shapes and isoperimetric inequalities for spectral functionals

In this talk I will discuss isoperimetric inequalities involving the spectrum of the Laplace operator (of Faber-Krahn, Saint-Venant or Mahler type) seen from the perspective of "shape optimization". Techniques inspired from applied mathematics, like the image segmentation theory, or the use of the computer for numerical approximations, can lead to rigorous mathematical proofs of some of those inequalities. I will describe more in detail problems involving the spectrum of the Robin-Laplacian and a Mahler type inequality for the first Dirichlet eigenvalue, and show how those techniques can be applied.

# New Versions of Some Classical Stochastic Inequalities

# On immersion formulas for soliton surfaces

This talk is devoted to a study of the connections between three different analytic descriptions for the immersion functions of 2D-surfaces, derived through the links between the following three types of symmetries: gauge symmetries of the linear spectral problem, conformal transformations of the spectral parameter and generalized symmetries of the related integrable system. We demonstrate that the immersion formulas associated with these symmetries can be linked by gauge transformations. We illustrate the theoretical results by examples involving the sigma model equations.

This is joint work with D. Levi (University of Roma Tre) and L. Martina (University of Salento).

# Riemann-Hilbert problems and large determinants; the example of the Kontsevich integral

Riemann-Hilbert problems are particular boundary value problems in the complex plane. When they depend on parameters one can associate, under certain conditions, a "tau function" whose zeroes indicate the non-solvability of the problem, and hence play the same role as a determinant does for linear systems.

There are cases in which this tau function is literally a determinant (finite or infinite dimensional); in these cases certain asymptotic questions can addressed using analysis of the Riemann-Hilbert problem. I will focus on one example that is related to the Kontsevich matrix integral; this integral gives (in a formal sense) the generating function for certain "intersection numbers" on the moduli space of Riemann surfaces in the limit where the size of the matrix being integrated tends to infinity. The approach via Riemann Hilbert method allows a non-formal treatment of this limit. Based on a joint project with Mattia Cafasso, Angers.

# Trading against disorderly liquidation of a large position under asymmetric information and market impact.

We consider trading against a hedge fund or large trader that must liquidate a large position in a risky asset if the market price of the asset crosses a certain threshold. Liquidation occurs in a disorderly manner and negatively impacts the market price of the asset. We consider the perspective of small investors whose trades do not induce market impact and who possess different levels of information about the liquidation trigger mechanism and the market impact. We classify these market participants into three types: fully informed, partially informed and uninformed investors. We consider the portfolio optimization problems and compare the optimal trading and wealth processes for the three classes of investors theoretically and by numerical illustrations. (joint work with Caroline HILLAIRET Ying JIAO and Renjie WANG)

# Mathematics without apologies. An unapologetic guided tour of the mathematical life

What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers- for the sake of truth, beauty, and practical applications--this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources.

# Outlier Detection for Functional Data Using Principal Components

Principal components analysis is a widely used technique that provides an optimal lower-dimensional approximation to multivariate observations. In the functional case, a new characterization of elliptical distributions on separable Hilbert spaces allows us to obtain an equivalent stochastic optimality property for the principal component subspaces of random elements on separable Hilbert spaces. This property holds even when second moments do not exist.

These lower-dimensional approximations can be very useful in identifying potential outliers among high-dimensional or functional observations. In this talk we propose a new class of robust estimators for principal components, which is consistent for elliptical random vectors, and Fisher-consistent for elliptically distributed random elements on arbitrary Hilbert spaces. We illustrate our method on two real functional data sets, where the robust estimator is able to discover atypical observations in the data that would have been missed otherwise.

# Spatial Branch-and-Cut for Polynomial Optimization

A polynomial optimization problem has an objective function and constraints described by polynomials. Although the problem is computationally challenging (binary integer programming is a special case), polynomial functions provide tremendous modeling flexibility that can be used to capture physical phenomena, uncertainty (e.g. correlated chance constraints), and competition (e.g. bilevel optimization). This talk will focus on an application of scheduling electric generation while modeling steady-state power flows across a network. We will use the branch-and-cut framework for negotiating the trade-off between time, space, and solution quality. The talk will conclude with recent work on strengthening convex relaxations for polynomial optimization by means of convex forbidden zones or S-free sets.