Speaker: Pamela Harris (Williams College)

Title: Kostant’s partition function

Abstract: In this talk we introduce Kostant’s partition function which counts the number of ways to represent a particular weight (vector) as a nonnegative integral sum of positive roots of a Lie algebra (a finite set of vectors). We provide two fundamental uses for this function. The first is associated with the computation of weight multiplicities in finite-dimensional irreducible representations of classical Lie algebras and the second is in the computation of volumes of flow polytopes. We provide some recent results in the representation theory setting and state a direction of ongoing research related to the computation of the volume of a new flow polytope associated to a Caracol diagram.

Speaker: Lee Altenberg (ICS UH Mānoa)

TITLE: Application of Dual Convexity to the Spectral Bound of Resolvent Positive Operators

ABSTRACT:

Donsker and Varadhan (1975) developed a variational expression for the spectral bound of generators of strongly continuous positive semigroups, which Karlin (1981) applied to finite matrices to show that the spectral radius of product [(1-m) I + m P]D decreases monotonically in m, where P is a stochastic matrix and D a positive diagonal matrix. This result has key applications in evolutionary and population dynamics. Simultaneously, Cohen (1981) showed that the spectral bound of essentially nonnegative matrices is convex in the diagonal elements. Kato (1982) extended Cohen’s result to resolvent positive operators. A “dual convexity” lemma shows that Cohen’s and Karlin’s results are actually equivalent, and via Kato’s theorem, allows extension of Karlin’s theorem to all resolvent positive operators, showing that d/dm s(L m + V) <= s(L), where s is the spectral bound, L is a resolvent positive operator, and V is an operator of multiplication. The motivation behind this work is that it unifies many results in reaction diffusion theory and shows generally that increased diffusion in the presence of local variation in decay or growth rates will decrease asymptotic growth rates.

Title: Active matter invasion of a viscous fluid and a no-flow theorem

Abstract: Suspensions of swimmers or active particles in fluids exhibit incredibly rich behavior, from organization on length scales much longer than the individual particle size to mixing flows and negative viscosities. We will discuss the dynamics of hydrodynamically interacting motile and non-motile stress-generating particles as they invade a surrounding viscous fluid, modeled by equations which couple particle motions and viscous fluid flow. Depending on the nature of their self-propulsion, colonies of swimmers can either exhibit a dramatic splay, or instead a cascade of transverse concentration instabilities, governed at small times by an equation which also describes the Saffman-Taylor instability in a Hele-Shaw cell, or Rayleigh-Taylor instability in two-dimensional flow through a porous medium. Analysis of concentrated distributions of particles matches the results of full numerical simulations. Along the way we will prove a very surprising “no-flow theorem”: particle distributions initially isotropic in orientation lose isotropy immediately but in such a way that results in no fluid flow anywhere and at any time.

Devin Murray

Title: Introduction to right-angled Artin groups

Speaker: John Calabrese (Rice)

Title:

From Hilbert’s Nullstellensatz to quotient categories

Abstract:

A common theme in algebraic geometry is the interplay between algebra and geometry. In this talk I will discuss a few “reconstruction theorems”, in which the algebra determines the geometry.

Keller Hall 301

Abstract: My plan is to go through (as much as time will allow of) Measure and Integrals in Conditional Set Theory by Jamneshan et al. with the goal of getting to at least one theorem there that witnesses the merits of conditional set theory.

Speaker: Dusko Pavlovic (UMH ICS)

Title: From Data Analysis to Dedekind-MacNeille Completions of Categories

Abstract: Solutions of mathematical problems are well-known to help with practical applications (to the point of being “unreasonably effective”, as Wigner put it). It is less well-known that practical applications sometimes help solving long standing mathematical problems. I will tell a story of this second kind.

The Dedekind-MacNeille completion of a poset is the smallest complete lattice that contains it, or equivalently the largest complete lattice where each element is both a meet and a join of the elements of the poset. Dedekind devised it to reconstruct the reals as a completion the rationals, and MacNeille generalized it to arbitrary posets. When posets are generalized to categories (so that the partial ordering a<b is expanded into the morphisms a->b), then meets and joins become limits and colimits, and the obvious task arises: generalize the Dedekind-MacNeille completion to categories. The task is thus to embed any given category into a category with all small limits and colimits, in such a way that any limits and colimits that already existed are preserved, and that any new objects that are added are both limits and colimits from the original category. This task was formulated already in the 50s, and it was listed as the most important open problem in Lambek’s 1966 “Completions of Categories” (volume #24 of Springer LNM). Stunningly, in 1972, Isbell proved that already the group Z_4, viewed as a one-object category, cannot be embedded into a bicomplete category where each object is both a limit and a colimit of diagrams built from copies of Z_4. But since the inductive process of adjoining limits to a category obviously settles at various bicompletions, and since it is easy to see that some of these bicompletions must be minimal, Isbell’s negative result just expanded the question: What are minimal bicompletions of categories, and which properties make them minimal? The question remained open for more than 40 years, or almost 60, depending how you count.

In this talk I will sketch the answer at which we arrived in 2015. It emerged as a special case of a matrix bicompletion construction, developed in a data analysis project. In the meantime, the practical applications of the result have expanded, but some of the mathematical repercussions, and most of the algorithmic issues, have not been settled.

Keller Hall 301

Abstract: My plan is to go through (as much as time will allow of) Measure and Integrals in Conditional Set Theory by Jamneshan et al. with the goal of getting to at least one theorem there that witnesses the merits of conditional set theory.