The developments of computing power and of microlocal techniques have led to new progress in the study of trace formulae both in physics and mathematics. Sophisticated tools from dynamics, such as the dynamical zeta functions and Birkhoff normal forms, have played a significant role in this. In my talk I would like describe these concepts and the way in which they help in our understanding of the classical/quantum correspondence.

This geometric model of tree space provides a setting in which questions that have been posed by biologists and statisticians over the last decade can be approached in a systematic fashion. For example, it provides a justification for disregarding portions of a collection of trees that agree, thus simplifying the space in which comparisons are to be made.

This is joint work with Susan Holmes and Karen Vogtmann:

http://www.math.cornell.edu/~vogtmann/Trees/lap.pdf

The Legendre transformation, as a means of dualizing convex functions, was hugely extended by Fenchel around 1950 in work inspired by the surge in interest in optimization that arose with the advent of computers. It was aimed at generalizing patterns such as linear programming duality, however, and its potential in the calculus of variations initially went unrecognized.

This talk will describe how the Legendre-Fenchel transformation and related developments in convex analysis lead to a much broader version of the calculus of variations. As in linear programming duality, problems appear in pairs that are inextricably tied together in optimality. The Euler-Lagrange equations and Hamiltonian equations achieve a generalized formulation in terms of so-called subgradients.

Mina Aganagic is a candidate for an assistant professor position in physics.

This will be an elementary introduction to some of the ideas underlying the theory of stacks. We will try to give the audience some idea of what a stack is and why stacks are very basic mathematical objects, even more basic then spaces or groups.

Results of Maclagan and Thomas relate connectivity of the toric Hilbert scheme to connectivity of the graph of triangulations of the vector configuration given by the columns of $A$. For example, a not connected graph of triangulations containing unimodular triangulations in at least two different components implies that the toric Hilbert scheme is not connected.

Using this property, we present the first known example of such non-connected Hilbert schemes. The example has $n=48$ and $d=6$ and is actually the product of the vertex set of a 24-cell and a segment, suitably homogenized.

We proceed to analyze Renegar's condition number for linear programs under the same perturbation model. Condition numbers measure the ill-posedness of a problem; they are ubiquitous in numerical analysis and scientific computing. Numerous interior point methods have recently been analyzed using Renegar's condition number. We show that the condition number is O~(m^2 d^2 /(sigma^2 delta)) with probability at least 1-delta. This is joint work with Dan Spielman (MIT) and Shang-Hua Teng (UIUC -> BU) submitted to the SIAM Conference on Optimization.

Both of these results can be viewed in the smoothed complexity model introduced by Spielman and Teng. Smoothed analysis interpolates between worst-case and average-case analysis by measuring the running time on an arbitrary instance under a random perturbation. A large perturbation yields traditional average-case analysis; as the perturbation becomes vanishingly small, we obtain traditional worst-case analysis. In this framework, our results are that the perceptron algorithm and condition number have polynomial smoothed complexity with high probability.

Over recent years, it has become evident that equilibrium constraints cannot be solved satisfactorily with standard techniques for Nonlinear Programming (NLP). Both numerical and theoretical evidence has been advanced which support this view.

This talk is aimed at a general mathematical audience and starts by introducing and reviewing equilibrium constraints. We then give some applications which emphasize the usefulness and elegance of equilibrium constraints as a modeling tool. Next, the assertion that standard techniques for NLP cannot be applied to equilibrium constraints is re-examined and some startling numerical evidence is presented using our own NLP solver.

The talk concludes by examining the local convergence properties of certain NLP methods applied to MPECs. It is shown that a simple constraint relaxation strategy allows a proof of second order convergence to be given under reasonable assumptions. A number of illustrative examples are presented which show that some of the assumptions are difficult to relax.

This talk is particulary designed for a general audience. We will discuss results, show some pictures, but provide as few technicalities as possible.