Mathematical and Theoretical Physics Seminar

Abstract: We study the dynamics of a system of N interacting bosons in a cigar- or disc-shaped trap, which initially exhibit Bose-Einstein condensation and interact via a non-negative interaction potential in the Gross-Pitaevskii scaling regime. The trap is realized by an external potential, which confines the bosons in two/one spatial dimensions to a region of order $\varepsilon$. We study the simultaneous limit $(N,\varepsilon)\to(\infty,0)$ and show that the N-body dynamics preserve condensation. The time-evolved condensate wave function is the solution of a 1D/2D Gross-Pitaevskii equation. Joint work with Stefan Teufel.

Abstract: The notions of Spin and Dirac Operator are well-known in a physics framework. In this lecture we introduce certain mathematical tools which are involved in spin geometry and indicate some applications.

Abstract: Symplectic structures with an internal integer grading on a manifold M can accommodate a metric on TM. This opens up the possibility to construct gravitational theories of very general kind starting with a graded Poisson algebra. In this talk we will review some aspects of graded geometry, hint at relations with Courant algebroids and mention how it can be applied to yield a Hilbert-Einstein-like action functional.

Abstract: In the last couple of years it was discovered that some 4d N=1 quantum field theories flow in the IR to 4d N=2 superconformal field theories (often of generalized Argyres-Douglas type), therefore showing a phenomenon of Supersymmetry Enhancement at the IR fixed point. The N=2 IR theory is often non-lagrangian while the N=1 UV theory is lagrangian, therefore such flows are extremely useful to learn features of the IR non-lagrangian theory, by using the UV formulation to compute RG-flow protected quantities as for example the superconformal index. However, up to date it is not completely clear why such flows exist, and how the SUSY enhancement happens. Limiting ourself to the case of rank one theories, we show how it is possible to understand the enhancement phenomenon in a geometric way, by condering a D3 brane probing a local singularity in F-theory corresponding to a T-brane of seven-branes. It is also possible to understand the enhancement as an hyperkahler restoration on the moduli space of solutions of the (generalized) Hitchin system associated to such theories.

Abstract: The Index Theorem is a general result which has its roots in classical geometry. In the first of the two talks we will explain its statement and relevant ingredients in the special case of the Hirzebruch-Riemann-Roch Theorem which was extended from algebraic geometry to an arbitrary analytic setting by the Index Theorem.
In the second lecture we will introduce the essential tools for stating the Index Theorem, e.g., the necessary PDE background, and give a very rough idea of its proof.

Abstract: The Index Theorem is a general result which has its roots in classical geometry. In the first of the two talks we will explain its statement and relevant ingredients in the special case of the Hirzebruch-Riemann-Roch Theorem which was extended from algebraic geometry to an arbitrary analytic setting by the Index Theorem.
In the second lecture we will introduce the essential tools for stating the Index Theorem, e.g., the necessary PDE background, and give a very rough idea of its proof.