**Every wednesday** at **4pm** in **Eckhart 202**.

Practical rare event simulation for small noise diffusions.

I will discuss an importance sampling method for certain rare event problems involving small noise diffusions. Standard Monte Carlo schemes for these problems behave exponentially poorly in the small noise limit. Previous work in rare event simulation has focused on developing, in specific situations, estimators with optimal exponential variance decay rates. I will introduce an estimator related to a deterministic control problem that not only has an optimal variance decay rate under certain conditions, but that can even have vanishingly small statistical relative error in the small noise limit. The method can be seen as the limit of a well known zero variance importance sampling scheme for diffusions which requires the solution of a second order partial differential equation. I will also report on progress toward applying the algorithm within the design of magnetic memory devices.

Protein Folding from a Dynamical Systems Perspective

First we argue that protein folding can be studied via a classical mechanical Hamiltonian system. By applying the Poincare recurrence theorem we show that protein folding is not a feature of the large-time behavior of individual trajectories. Adopting therefore a statistical mechanical point of view, we explain how the results of protein folding experiments can be rationalized using the classical ergodic theorems. Then we define the folded state (apart from an ergodic hypothesis), and discuss issues of (amino acid sequence and thermodynamic parameter dependent) localization and unimodality of the folded state and rates of folding.

Mean Field Games and congestion.

Integration along rough (but not too rough) paths.

Maximum principle and mean field games.

Slender body theory and applications to DNA

We will discuss the Kirchhoff rod model and the incorporation of a first approximation of viscous drag. This model is examined through the most simple case applicable to short segments of DNA in Eukaryotes, the straight rod under tension. We will conclude by briefly discussing application of this theory to DNA.

Singular Perturbation and Bifurcation of Transition Layers

The connection between diffused and sharp interfacial problems in the variational setting are well developed to a large extent by means of Gamma-convergence and also purely analytical techniques such as asymptotic expansion and implicit function theorem. They work well for the understanding of global minimizers and non-degenerate critical points. This talk will describe some results which extend the above framework to analyze the degenerate case, in particular the bifurcation of diffused interface and its connection to sharp interfacial limit. Examples of Allen-Cahn Equation with parameter dependent spatial inhomogeneity are investigated. This is joint work in progress with Chaoqun Huang.

Joint work with Hui Dai and Zachary Geary.

Eigenvalues and Eigenvectors of Large Toeplitz Matrices

We consider singular matrices with Qjk= Q(j-k) and ask for eigenvalues.

If indices in [-∞, -∞] continuum (line) of eigenvalues can be obtained by Fourier transform. They are labeled by real wave vector p.

For the case of singular Q we study, we know:

If indices in [0, ∞] continuum (closed area) of eigenvalues can be obtained by Wiener Hopf method

If indices in [ -∞,0 ] there are no eigenvalues.

I aim to show by analytical and numerical arguments that

If Indices in [0,n- 1] discrete eigenvalues differ from continuum case by O(ln n)/n. They have wave vector with (Im p) = O(ln n)/n.

On the dynamical formation of Black Holes

Regularity results for solutions of some Hamilton-Jacobi equations.

Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi equations with a super-quadratic growth in the gradient variable are proved to be Holder continuous, with a modulus depending only on the growth of the Hamiltonian. The proof involves some representation formula for nonlocal Hamilton-Jacobi equations in terms of controlled jump processes and a weak reverse inequality.

Eigenvalues of non-normal matrices and a single ring theorem

We study the empirical measure L_{An}of the eigenvalues of non-normal square matrices of the form A_{n}=U_{n}D_{n}V_{n}with U_{n},V_{n}independent Haar distributed on the unitary group and D_{n}real diagonal. We show that when the empirical measure of the eigenvalues of D_{n}converges, and D_{n}satisfies some technical conditions, L_{An}converges towards a rotationally invariant measure on the complex plane whose support is a single ring. In particular, we provide a complete proof of Feinberg-Zee single ring theorem [FZ]. We also consider the case where U_{n},V_{n}are independent Haar distributed on the orthogonal group.

Rigidity and flexibility for isometric embeddings in low dimension

A classical theorem in geometry says that the standard 2-sphere is rigid, it is unbendable. On the other hand, the celebrated result of Nash and Kuiper shows that one can nevertheless "crumple" the sphere into an arbitrarily small volume, without creating creases! In more precise terms, whilst there is essentially a unique isometric embedding of the sphere of class C2, there is a huge set of embeddings of class C1. The difference of course lies in the curvature. Borisov proved in the 1950s that the rigidity result can be extended to embeddings with Hölder continuous derivative of order 2/3+, whereas in 1965 he announced that the Nash-Kuiper construction - leading to flexibility- still works with Hölder contiuous derivatives of order 1/7-. In the talk I will revisit these results with a more modern "PDE-approach", based on joint work with Sergio Conti and Camillo De Lellis. At the end of the talk I will comment on the relevance of the latter construction to turbulence.

A Measure Theoretic Computational Approach For Inverse Sensitivity Problems

We describe a computational method for probabilistic inverse sensitivity analysis of a map from a set of parameters and data to a computed quantity of interest. The inverse problem is to describe the random variation in the input and parameters that lead to an imposed or observed random variation on the output quantity. To complicate matters, we are interested in implicitly-defined maps, such as a quantity of interest computed from the solution of a differential equation. We formulate the problem as an ill-posed inverse problem for an integral equation using the Law of Total Probability and then describe a computational method for computing solutions. The method has two stages. In the first part, we approximate the unique set-valued solution to the inverse of the integral equation using derivative information. In the second part, we apply basic ideas from measure theory to compute the approximate probability measure on the parameter and data space that solves the integral equation. We discuss convergence of the method, and explain how to use the method to compute the probability of events in the input (parameter) space. The talk is illustrated with a number of examples. Time permitting, we discuss briefly the numerical analysis (accuracy) of the method and the consideration of multiple quantities of interest and data assimilation.

For questions, contact Luis Silvestre at