Applying ideas of fractional analogue of Monge-Amp\'ere equations, we consider an analogue of fractional k-Hessian operators and intend to reproduce the regularity results of global solutions of Dirichlet problems with prescribed data at infinity and global barriers when k=2.

I will introduce Nakajima quiver varieties and describe some examples. Quiver varieties are quasi projective varieties; I will explain that they admit modular compactifications with good properties. As a consequence, the cohomology of quiver varieties is generated by tautological classes, i..e, they satisfy “hyperkahler Kirwan surjectivity.” Similar statements hold for generalized cohomology theories and derived categories. The preceding is an instance of a general story about moduli of objects in certain categories, that I hope to discuss as time permits (as well as, possibly, what one can say about the hyperkahler Kirwan surjectivity problem in general). The talk is based on joint work with K. McGerty.

In this talk, I'll provide an introduction to the world of moduli spaces starting with the most primitive of examples and working up to the moduli spaces of stable maps. Along the way we will brush up against topics from string theory and will attempt to answer questions from classical geometry.

The world is filled with hierarchical structures. Atoms join together to form molecules, molecules form bigger molecules, which form cell parts, which form cells, tissues, and people. People form communities, states, and countries. Where did all this structure come from? The laws of physics are mostly short-range, and do not directly explain these large-scale groupings. Instead, using non-periodic tilings as a model, we'll explore how short-range rules can lead to an infinite hierarchy of structures. We will also see how tiling spaces can help us understand chaos.

We solve the Cauchy problem for the space homogeneous Boltzmann equation with an scaled angle-potential concentrated singular mass that becomes concentrated in the grazing collision limit. In addition, we show the solutions of this model converge to a bounded solution of the Landau equation for Coulomb interactions in the sense of distributions. More specifically, our result works for intramolecular potentials that range from Coulomb to moderately soft ones. We show that such Boltzmann solutions propagate norms for global in time and uniformly in the grazing parameter. Furthermore, the sequence of such Boltzmann solutions converges to a weak solution of the Landau equation weakly star in and in for r slightly bigger than 1 and . These Landau solutions and conserve mass, momentum and energy.

The space homogeneous Boltzmann equation models the mass density and it can be written as where , the quadratic integral operator, expresses the change of f due to instantaneous binary collisions of particles. In the talk, I will give a brief introduction to the Boltzmann Equation and its notation and present the gain of integrability property of the gain part of the Boltzmann collision operator , i.e., that the Lebesgue (Sobolev resp.) norm of the collision operator is controlled by a strictly lower order Lebesgue (Sobolev resp.) norm of the input functions.

I will revisit the structure of gauge and global G symmetries in supersymmetric QFT, focusing on the simplest case, supersymmetric quantum mechanics. While G gauge symmetry in supersymmetric quantum mechanics is well known to lead to equivariance, G global symmetry has a much subtler effect: the Hilbert space of supersymmetric ground states becomes a trivial representation of G but a (potentially) nontrivial representation of the cohomology H_1(G). The relation between gauged and ungauged Hilbert spaces is captured by a version of Koszul duality familiar from mathematical work of Goresky-Kottwitz-MacPherson. As an application, we construct models of supersymmetric quantum mechanics with a deformation that a) breaks supersymmetry and lifts the classical vacuum energy at order one; but b) lifts the quantum vacuum energy at arbitrarily high order. The effect is topologically protected, and comes from an equivariant version of the Hodge theorem. (Joint work with C. Beem, D. Ben-Zvi, M. Bullimore, A. Neitzke.)

We will introduce the notion of a bounded weight structure on a stable oo-category and prove a generalization of Waldhausen’s sphere theorem for the algebraic K-theory of higher categories. The algebraic K-theory of a stable oo-category with a bounded non-degenerate weight structure will be proven to be equivalent to the algebraic K-theory of the heart of the weight structure. We will relate this theorem to previous results as well as new applications. This talk constitutes the oral defense of the speaker's dissertation.

I'll discuss some WENO reconstructions for nonlinear conservation laws.

Stochastic networks arise in a variety of applications, ranging from communication and service networks to biochemical reaction networks. In many cases, these networks are too complex to be amenable to an exact analysis, and so it is useful to identify tractable approximations that provide qualitative insight into the dynamics, and whose accuracy can be rigorously justified via limit theorems in a suitable asymptotic regime. The scaling limits that arise are quite different for different classes of networks, ranging from reflected diffusions to interacting measure-valued processes. In this talk I will give a survey of some mathematical methods that have been developed to establish such approximations as well as to analyze the limit processes in a variety of settings. I will also provide examples to illustrate the insight that these scaling limits provide into network performance and design.

In this talk, I will provide an overview of solution techniques and iterative techniques employed to solve the first-order form of the radiation transport equation on massively parallel machines. A review of scaling efficiency for transport sweeps (up to order 1-million processes) will be provided for logically Cartesian grids. Challenges posed by the need to move to unstructured (load-unbalanced) grids and ongoing research will be discussed. Diffusion-based synthetic accelerators for the one-speed (within-group) and multigroup transport equations will be presented and issues related to massively parallel diffusion-accelerated transport sweeps be analyzed.