This is a thesis defense.

Classical theta functions are a special class of holomorphic functions on 𝘊ᵍ that are quasi-periodic with respect to a lattice. However, theta functions can also be viewed as sections of a line bundle on an abelian variety and by fixing a bit of data, a canonical basis can be obtained. Classically, the construction of canonical theta functions depended heavily on the group structure of abelian varieties, making generalizations difficult. We will look at how recent use of toric varieties and mirror symmetry have helped provide a generalized notion of theta functions to other varieties, namely K3 surfaces.

I will describe Kevin Costello's approach to constructing integrable lattice models from supersymmetric gauge theory. The basic point is the commutativity of OPE of line operators in more than 2 dimensions. When applied to a version of N=1 super Yang-Mills in 4d this recovers the original Heisenberg spin chains whose study led to the invention of quantum groups.

The question addressed here is how fast a front will propagate when a line, having a strong diffusion of its own, exchanges mass with a reactive medium. More precisely, we wish to know how much the diffusion on the line will affect the overall front propagation. This setting was proposed (collaboration with H. Berestycki and L. Rossi) as a model of how biological invasions can be enhanced by network transportations. In a previous series of works, we were able to show that the line could speed up propagation indefinitely with its diffusivity. For that, we used a special type of nonlinearity that allowed the reduction of the problem to explicit computations. In the work presented here, the reactive medium is governed by nonlinearity that does not allow explicit computations anymore. We will explain how propagation speed-up still holds. In doing so, we will discuss a new transition phenomenon between two speeds of different orders of magnitude. Joint work with L. Dietrich.

The (2,3,7)-triangle group is the minimal co-area Fuchsian group, i.e. the quotient of the hyperbolic plane by this group is the hyperbolic 2-orbifold of smallest area. Also, torsion-free normal subgroups of this group correspond to the Hurwitz surfaces. The Hurwitz surfaces are the compact hyperbolic surfaces with maximal number of automorphisms: 84(g-1). I will give an introduction to these important results. Note that next week's Colloquium will discuss the 3d analogue.

In the 60's, Harish-Chandra studied certain systems of differential equations on a reductive group, of which the characters of irreducible representations were solutions. These ideas were framed in the language of D-modules by Hotta and Kashiwara twenty years later, relating the Harish-Chandra systems to the subject of Springer theory - a geometric construction of representations of the Weyl group. At around the same time, Lusztig showed how Springer theory fit in to a bigger picture (which came to be known as the generalized Springer correspondence), introducing the the key notion of a cuspidal local system. I will give an overview of these ideas from the perspective of my own work, which unifies Lusztig's generalized Springer theory with Hotta and Kashiwara's study of the Harish-Chandra system.

In this talk, we present a sparse grid discontinuous Galerkin (DG) scheme for transport equations and applied it to kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG schemes for hyperbolic problems and is proven to be stable and convergent. A major advantage of the scheme lies in its low computational and storage cost due to the employed sparse finite element approximation space. This attractive feature is explored in simulating Vlasov and Boltzmann transport equations. We also discuss extension of the scheme to adaptive sparse grid methods.

The thin obstacle problem is a variation of the classical obstacle problem in which the solution is constrained to lie above the obstacle only on an (n-1)-dimensional (or "thin") set. C^{1,\alpha} regularity for solutions in arbitrary dimensions was proved by Caffarelli in '79. Different proofs were given by Kinderlehrer in '81 and Uralsteva in '85. In '04, Athanasopoulos and Caffarelli proved C^{1,1/2} regularity of solutions, which is optimal. In this talk, we will follow the approach of Caffarelli-Salsa-Silvestre '08 to get the optimal regularity. We will introduce Almgren's Frequency formula that leads to a specific notion of blowups. Then, the study of homogeneous global solutions will give the C^{1,1/2} regularity.

One way to solve the Boltzmann equation numerically is to use spectral methods. Recently, a spectral method has been created which not only agrees with past approximations to the Boltzmann equation, but also preserves many important properties of the equation (in particular, conservation of mass, momentum and energy). This talk will give a derivation of the method and explain how to implement it, along with some description of it's many powerful properties and a couple of examples to see it action.