(Distinguished Lecture)

Non-self-adjoint operators appear in many settings, from kinetic theory 
and quantum mechanics to linearizations of equations of mathematical 
physics. The spectral analysis of such operators, while often notoriously 
difficult, reveals a wealth of new phenomena, compared with their 
self-adjoint counterparts. Spectra for non-self-adjoint operators display 
fascinating features, such as lattices of eigenvalues for operators of 
Kramers-Fokker-Planck type, say, and eigenvalues for operators with 
analytic coefficients in dimension one, concentrated to unions of curves 
in the complex spectral plane. In this talk, after a general introduction, 
we shall discuss spectra for non-self-adjoint perturbations of 
self-adjoint operators in dimension two, under the assumption that the 
classical flow of the unperturbed part is completely integrable.
The role played by the flow-invariant Lagrangian tori of the completely 
integrable system, both Diophantine and rational, in the spectral analysis 
of the non-self-adjoint operators will be described. In particular, we 
shall discuss the spectral contributions of rational tori, leading to 
eigenvalues having the form of the "legs in a spectral centipede". This 
talk is based on joint work with Johannes Sj\"ostrand.

Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. Therefore, it is desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that balance these criteria. This talk offers an invitation to the field of matrix concentration inequalities and their applications. This talk is designed for a general audience in mathematics and related fields.

Associativity is ubiquitous in mathematics.  Unlike commutativity, its more popular cousin, associativity has for the most part taken a backseat in importance.  But over the past few decades, associativity has blossomed and matured, appearing in theories of particle collisions, elliptic curves, and enumerative geometry.  We start with a brief look at this history, and then explore the visualization of associativity in the forms of polytopes, manifolds, and complexes. This talk is heavily infused with imagery and concrete examples.

Linear programs (LPs) are, without any doubt, at the core of both the theory and the practice of mondern applied and computational Optimization (e.g., in discrete optimization LPs are used in practical computations using branch-and-bound, and in approximation algorithms, e.g., in rounding schemes). At the same time  Dantzig’s Simplex method is one of the most famous algorithms to solve LPs and SIAM elected it as one of the top 10 most influential algorithms of the 20th Century.

But despite its key importance, many simple easy-to-state mathematical properties of the Simplex method and its geometry remain unknown. The geometry of the simplex method is very much the convex-combinatorial geometry of polyhedra (e.g., cubes, simplices, etc). Perhaps the most famous geometric-combinatorial challenge is to determine a worst-case upper bound for the graph diameter of polyhedra.  Although a lot of progress has been made, today even for the most elementary textbook linear programs we remain ignorant as to what the exact diameter upper bounds are. In this talk, I will discuss this  geometric problem and present the key ideas for proving that the diameter of graphs of all network-flow polytopes satisfy the Hirsch linear bound. This is joint work with S. Borgwardt (Univ of Colorado) and E. Finhold (Fraunhofer Institut).