The classical Furstenberg Theorem states that the norm of a product of random i.i.d. matrices (under very mild assumptions) grows exponentially: almost surely one has
\lim_n (1/n) \log |A_n…A_1| = \lambda > 0.
My talk will be devoted to our recent work with Anton Gorodetski, where we have considered an analogous setting with A_j being independent, but no longer identically distributed. In such a setting it is natural to expect an (accordingly modified) version of the Furstenberg Theorem to hold. And indeed, we show that (again, under some mild assumptions on the distributions of A_j) there exists a deterministic sequence L_n such that

\liminf (1/n) L_n >0

 and almost surely

\lim (1/n) [\log |A_n…A_1| - L_n] = 0.

Moreover, there is an analogue of the Large Deviations Theorem.

The difficulty here is that in the nonstationary setting one cannot use the usual tools of the dynamical systems theory (stationary measure, ergodic theorem, etc.). I will discuss the proof of above results, as well as the general intuition of survival in the nonstationary world.

Stein's method for distributional approximation has become a valuable tool in probability and statistics by providing finite sample distributional bounds for a wide class of target distributions in a number of metrics. A key step in popular versions of the method involves making couplings constructions, and a family of couplings of Chen and Roellin vastly expanded the range of applications for which Stein's method for normal approximation could be applied. This family subsumes both Stein's classical exchangeable pair, and the size bias coupling. A further simple generalization includes zero bias couplings, and also allows for situations where the coupling is not exact. The zero bias versions result in bounds for which often tedious computations of a variance of a conditional expectation is not required. An example to the Lightbulb process shows that even though the method may be simple to apply, it may yield improvements over previous results that had achieved bounds with optimal rates and small, explicit constants. 

A numerical semigroup is a subset S of the natural numbers that is closed under addition.  One of the primary attributes of interest in commutative algebra are the relations (or trades) between the generators of S; any particular choice of minimal trades is called a minimal presentation of S (this is equivalent to choosing a minimal binomial generating set for the defining toric ideal of S).  In this talk, we present a method of constructing a minimal presentation of S from a portion of its divisibility poset.  Time permitting, we will explore connections to polyhedral geometry.  

A linear code is called self-orthogonal if it is contained in its dual. Self-orthogonal codes are theoretically interesting and have application to the construction of quantum error-correcting codes. They include self-dual codes which are closely related to combinatorial designs, secret sharing schemes, unimodular lattices, sphere packing, and groups.

In this talk, we introduce a series of recent papers on binary optimal self-orthogonal codes with small dimensions. In particular, we solve two conjectures on the largest minimum distance of a binary self-orthogonal code with dimension 5.