L3

I will talk about the fluid equations used to model pedestrian motion and traffic. I will present the compressible-incompressible Navier-Stokes two phase system describing the flow in the free and in the congested regimes, respectively. I will also show how to approximate such system by the compressible Navier-Stokes equations with singular pressure for the fixed barrier densities and also some recent developments for the barrier densities varying in the space and time.
This is a talk based on several papers in collaboration with: D. Bresch, C. Perrin, P. Degond, P. Minakowski, and L. Navoret.
 

Transport properties of liquids and gases in the regime of weak coupling (or effective weak coupling) are determined by the solutions of relevant kinetic equations for particles or quasiparticles, with transport coefficients being proportional to the minimal eigenvalue of the linearized kinetic operator. At strong coupling, the same physical quantities can sometimes be determined from dual gravity, where quasinormal spectra enter as the eigenvalues of the linearized Einstein's equations. We discuss the problem of interpolating between the two regimes using results from higher derivative gravity.

Understanding the structure of hadrons in terms of their fundamental constituents requires an understanding of QCD at large distances, a vastly complex and unsolved dynamical problem. I will discuss in this talk a new approach to hadron structure based on superconformal quantum mechanics in the light-front and its holographic embedding in a higher dimensional gravity theory. This approach captures essential aspects of the confinement dynamics which are not apparent from the QCD Lagrangian, such as the emergence of a mass scale and confinement, the occurrence of a zero mode: the pion, universal Regge trajectories for mesons and baryons and precise connections between the light meson and nucleon spectra. This effective semiclassical approach to relativistic bound-state equations in QCD can be extended to heavy-light hadrons where heavy quark masses break the conformal invariance but the underlying dynamical supersymmetry holds.
 

The chromatic number of a graph is trivially bounded from above by the maximum degree plus one, and from below by the size of a largest clique. Reed proved in 1998 that compared to the trivial upper bound, we can always save a number of colors proportional to the gap between the maximum degree and the size of a largest clique. A key step in the proof deals with how to spare colors in a graph whose every vertex "sees few edges" in its neighborhood. We improve the existing approach, and discuss its applications to Reed's theorem and strong edge coloring.  This is joint work with Thomas Perrett (Technical University of Denmark) and Luke Postle (University of Waterloo).

The joint spectral radius (JSR) of a set of matrices characterizes the maximum growth rate that can be achieved by multiplying them in arbitrary order. This concept, which essentially generalizes the notion of the "largest eigenvalue" from one matrix to many, was introduced by Rota and Strang in the early 60s and has since emerged in many areas of application such as stability of switched linear systems, computation of the capacity of codes, convergence of consensus algorithms, tracability of graphs, and many others. The JSR is a very difficult quantity to compute even for a pair of matrices. In this talk, we present optimization-based algorithms (e.g., via semidefinite programming or dynamic programming) that can either compute the JSR exactly in special cases or approximate it with arbitrary prescribed accuracy in the general case.

Based on joint work (in different subsets) with Raphael Jungers, Pablo Parrilo, and Mardavij Roozbehani.
 

A heterotic $G_2$ system is a quadruple $([Y,\varphi], [V, A], [TY,\theta], H)$ where $Y$ is a seven dimensional manifold with an integrable <br /> $G_2$ structure $\varphi$, $V$ is a bundle on $Y$ with an instanton connection $A$, $TY$ is the tangent bundle with an instanton connection $\theta$ and $H$ is a three form on $Y$ determined uniquely by the $G_2$ structure on $Y$. Further, H  is constrained so that it satisfies a condition that involves the Chern-Simons forms of $A$ and $\theta$, thus mixing the geometry of $Y$ with that of the bundles (this is the so called anomaly cancelation condition).  In this talk I will describe the tangent space of the moduli space of these systems. We first prove that a heterotic system is equivalent to an exterior covariant derivative $\cal D$ on the bundle ${\cal Q} = T^*Y\oplus {\rm End}(V)\oplus {\rm End}(TY)$ which satisfies $\check{\cal D}^2 = 0$ for some appropriately defined projection of the operator $\cal D$.  Remarkably, this equivalence implies the (Bianchi identity of) the anomaly cancelation condition. We show that the infinitesimal moduli space is given by the cohomology group $H^1_{\check{\cal D}}(Y, {\cal Q})$ and therefore it is finite dimensional.   Our analysis leads to results that are of relevance to all orders in $\alpha’$.  Time permitting, I will comment on work in progress about the finite deformations of heterotic $G_2$ systems and the relation to differential graded Lie algebras.

The gauged linear sigma model (GLSM) is a supersymmetric gauge theory in two dimensions which captures information about Calabi-Yaus and their moduli spaces. Recent result in supersymmetric localization provide new tools for computing quantum corrections in string compactifications. This talk will focus on the hemisphere partition function in the GLSM which computes the quantum corrected central charge of B-type D-branes. Several concrete examples of GLSMs and the application of the hemisphere partition function in the context of transporting D-branes in the Kahler moduli space will be given.

Calabi-Yau 3-folds play a large role in string theory.  Cohomology of sheaves on such varieties has many uses in string theory, including counting the number of particles or fields in a theory, as well as to help identify terms in the superpotential that determines the equations of motion of the corresponding string theory, and many other uses as well.  As a computational algebraic geometer, string theory provides a rich source of new computational problems to solve.

In this talk, we focus on the search for rigid divisors on these Calabi-Yau hypersurfaces of toric varieties.  We have had methods to compute sheaf cohomology on these varieties for many years now (Eisenbud-Mustata-Stillman, around 2000), but these methods fail for many of the examples of interest, in that they take a very long time, or the software (wisely) refuses to try!

We provide techniques and formulas for the sheaf cohomology of certain divisors of interest in string theory, that other current methods cannot handle.  Along the way, we describe a Macaulay2 package for computing with these objects, and show its use on examples.

This is joint work with Andreas Braun, Cody Long, Liam McAllister, and Benjamin Sung.