University of North Carolina

Arguably some of the most interesting phenomena in fluid dynamics, both from a mathematical and a physical perspective, stem from the interplay between a fluid and its boundaries. This talk will present some examples of how boundary effects lead to remarkable outcomes.  Singularities can form in finite time as a consequence of the continuum assumption when material surfaces are in smooth contact with horizontal boundaries of a fluid under gravity. For fluids with chemical solutes, the presence of boundaries impermeable to diffusion adds further dynamics which can give rise to self-induced flows and the formation of coherent structures out of scattered assemblies of immersed bodies. These effects can be analytically and numerically predicted by simple mathematical models and observed in “simple” experimental setups. 

Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of locally finite affine A_n buildings. From this data, we will show how to construct new fiber functors on the category of tilting modules for SL(n+1) in characteristic p (related to order of the projective geometry) using the web calculus of Cautis, Kamnitzer, Morrison and Brundan, Entova-Aizenbud, Etingof, Ostrik.

Consider a collection of particles whose state evolution is described through a system of interacting diffusions in which each particle
is driven by an independent individual source of noise and also by a small amount of noise that is common to all particles. The interaction between the particles is due to the common noise and also through the drift and diffusion coefficients that depend on the state empirical measure. We study large deviation behavior of the empirical measure process which is governed by two types of scaling, one corresponding to mean field asymptotics and the other to the Freidlin-Wentzell small noise asymptotics. 
Different levels of intensity of the small common noise lead to different types of large deviation behavior, and we provide a precise characterization of the various regimes. We also study large deviation behavior of  interacting particle systems approximating various types of Feynman-Kac functionals. Proofs are based on stochastic control representations for exponential functionals of Brownian motions and on uniqueness results for weak solutions of stochastic differential equations associated with controlled nonlinear Markov processes. 

Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.

A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.

The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.

Consider non-negative integers assigned to the vertexes of an oriented graph. To this combinatorial data we associate a so-called quiver representation. We will study the geometry and the algebra of this representation, when the underlying un-oriented graph is of Dynkin type ADE.

A remarkable object we will consider is Kazarian's equivariant cohomology spectral sequence. The edge homomorphism of this spectral sequence defines the so-called quiver polynomials. These polynomials are generalizations of remarkable polynomials in algebraic combinatorics (Giambelli-Thom-Porteous, Schur, Schubert, their double, universal, and quantum versions). Quiver polynomials measure degeneracy loci of maps among vector bundles over a common base space. We will present interpolation, residue, and (conjectured) positivity properties of these polynomials.

The quiver polynomials are also encoded in the Cohomological Hall Algebra (COHA) associated with the oriented graph. This is a non-commutative algebra defined by Kontsevich and Soibelman in relation with Donaldson-Thomas invariants. The above mentioned spectral sequence has a structure identity expressing the fact that the sequence converges to explicit groups. We will show the role of this structure identity in understanding the structure of the COHA. The obtained identities are equivalent to Reineke's quantum dilogarithm identities associated to ADE quivers and certain stability conditions.