I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.
I plan to cover the following topics: Euler equations; Burger's equation; p-system; symmetric hyperbolic PDE's; shock formation; Lax method of solving Riemann problems; Glimm's method for solving Cauchy problems; Entropy solutions; artificial viscosity.
We present an overview of analytical tools in random matrix theory and related areas, involving Toeplitz/ Hankel determinants and symmetric functions, with an emphasis on their relevance in the study of topological gauge theory and focussing on some specific Chern-Simons theories and 2d Yang-Mills theories. We will also explain how these methods and results are intertwined with localization results in supersymmetric gauge theories.
We present microstate counting formulae for BPS black holes in an $N=2$ STU-model.
It is well known that ordinary bicategories can always be replaced by bi-equivalent strict 2-categories. Special cases of this are the strictification of monoidal categories and categorical groups. We give an abstract strictification construction for pseudo-monoids in a monoidal 2-category. It is easy to see that bicategories internal to an appropriate category are such pseudo-monoids, and can hence be strictified. (Joint work with Nelson Martins-Ferreira)
One of the most beautiful objects of classical geometry is the Kummer Surface, that was studied by Kummer in the 19th century. In a celebrated paper of 1969 Narasimhan and Ramanan studied the moduli space of vector bundles of rank 2 and trivial determinant over a curve of genus 2, proving that this space is isomorphic to projective space of dimension 3. In this space the moduli space of non-stable bundles is parameterized by a Kummer Surface.
I'll discuss a joint work with Sergiu Klainerman on the stability of Schwarzschild as a solution to the Einstein vacuum equations with initial data subject to a certain symmetry class.
We present a recent result establishing a bridge between noncommutative scalar field theory in $2$ dimensions and topological field theory in $3$ dimensions.
I will discuss the application of Siegel modular forms for extracting the degeneracy of states of symmetric orbifold CFTs. These modular forms are closely related to the generating function for the elliptic genera of such CFTs and I will present an efficient technic for extracting their Fourier coefficients. I will then discuss to what extent symmetric orbifold CFTs can admit nice gravity duals and thus make an interesting connection between number theory and quantum gravity.
Recently an interesting connection between topological string theory and lattice models in condensed matter physics was discussed by several authors. In this talk, we will focus on the Harper-Hofstadter Hamiltonian. For special values of the magnetic flux, its energy spectrum can be exactly solved and its graph has a beautiful shape known as Hofstadter's butterfly. We are interested in the non-perturbative information inside the spectrum. First we consider the weak magnetic field limit and write down a trans-series ansatz for the energies.