Fridays 3:30pm in KAP 414; tea is usually provided at 3:00pm
Organizer: James Zhao <james.zhao@usc.edu>
For seminars/colloquia on other topics, see the Department of Mathematics webpage.
Older seminars: 2013-2014
Polymer pinning with sparse disorder
The standard setup in disordered pinning models is that a polymer configuration is modeled by the trajectory of a Markov chain which is rewarded or penalized by an amount ω_{n} when it returns to a special state 0 at time n. More precisely, for a polymer of length N the Boltzmann weight is e^{βH}, where for a trajectory τ, H(τ) is the sum of the values ω_{n} at the times n≤N of the returns to 0 of τ. Typically the ω_{n} are taken to be a quenched realization of a iid sequence, but here we consider the case of sparse disorder: ω_{n} is 1 at the returns times of a quenched realization of a renewal sequence {σ_{j}}, and 0 otherwise; in the interesting cases the gaps between renewals have infinite mean, and we assume the gaps have a regularly varying power-law tail. For β above a critical point, the polymer is pinned in the sense that τ asymptotically hits a positive fraction of the N renewals in σ. To see the effect of the disorder one can compare this critical point to the one in the corresponding annealed system. We establish equality or inequality of these critical points depending on the tail exponents of the two renewal sequences (that is, σ and the return times of τ.) This is joint work with Quentin Berger.
Probability and the lottery
Various schemes surrounding the lottery, both legal and criminal, both money-losing and money-making, raise interesting probability problems. We explain how to analyze these schemes, how the analysis can distinguish between criminals and problem gamblers, and how the analysis led to lottery operators changing policies and law enforcement cracking down on the criminal schemes.
This talk is about joint work with Aaron Abrams, Richard Arratia, Philip Stark, and others. It will be accessible to a general mathematical audience; the main pre-requisite is something like Math 307 or Math 407.
Normal approximation for Gaussian projections and conic intrinsic volumes: Steining the Steiner formula
Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications. In particular, the discrete probability distribution L(V_{C}) given by the sequence v_{0}, . . . ,v_{d} of conic intrinsic volumes of a closed convex cone C in R^{d} summarizes key information about the success of convex programs used to solve for sparse vectors, and other unknowns, in high dimensional regularized inverse problems. The concentration of V_{C} implies the existence of phase transitions for the probability of recovery of the unknown in the number of observations. Additional information about the probability of recovery success is provided by a normal approximation for V_{C}. Such central limit theorems can be shown by first considering the squared length G_{C} of the projection of a Gaussian vector on the cone C. Applying a second order Poincaré inequality, proved using Stein's method and Malliavin calculus, then produces a non-asymptotic total variation bound to the normal for L(G_{C}). A conic version of the classical Steiner formula in convex geometry shows a normal limit for G_{C} implies one for V_{C}.
This work is joint with Ivan Nourdin and Giovanni Peccati.
Some Thinking about Pathwise Stochastic Calculus
In standard stochastic analysis literature, processes are typically understood in a.s. sense, with notable examples stochastic integration and conditional expectations. Motivated from our study of path dependent PDEs, which is a convenient tool to study non-Markovian (or say path dependent) problems, we are interested in pathwise solutions of certain equations. In particular, we would like to have the regularity of the processes in terms of the underlying paths. For the backward situation, with conditional expectation as a typical example, it seems the Functional Ito formula is the right tool. For the forward situation, with stochastic integration as a simple example, we feel the rough path theory is the convenient tool. With the help of the rough path theory, we will show that a diffusion (and more generally the solution of a stochastic PDE) can be continuous in ω under certain topology.
Conditional random variable approximation with Stein's method
Finding patterns in randomness: A new perspective on the longest increasing subsequence problem
Given a random permutation of 1, . . . , N, how long is the longest subsequence that's monotonically increasing? Decades of work went into solving and understanding this problem, and by now the answer is well-known: the expectation is 2√N and the length is distributed according to the famous Tracy-Widom distribution.
Now, given a random permutation of 1, . . . , N, how long is the longest subsequence that alternates between increasing and decreasing? This problem turns out to be much easier. A few years ago, Stanley showed that this length is normally distributed around a mean of 2N/3.
These two cases have quite different behaviors. We show that they are archetypes of a dichotomy: for any order pattern (of which increasing and alternating are examples), the answer will resemble either the increasing case or the alternating case. The difference is detected by a simple combinatorial feature called drift that arose in some of our earlier work.
(Joint work with E. Babson, H. Landau, Z. Landau, and J. Pommersheim)