Gauge theory is a subject that has emerged from theoretical physics. It has deep links with many areas of mathematics (including partial differential equations, representation theory, algebraic geometry, differential geometry and topology). Most of the mathematical work on gauge theory has focused on low dimensions, where one can exploit the anti-self-dual Yang-Mills equation and the analytic difficulties are still quite tractable.
In this talk I will discuss three concrete questions that arise in gauge theory in higher dimensions. First, I will discuss gauge theory on Kähler manifolds, with a particular focus on singular Hermitian Yang-Mills connections. Afterwards, I will move on to the more exotic topic of gauge theory on G2-manifolds. I will discuss a method to construct solutions of the Yang-Mills equation on a class of G2-manifolds called twisted connected sums. Finally, I will talk about the prospects of defining enumerate gauge theoretical invariants for G2-manifolds and the difficulties arising from codimension four bubbling.

Title: Free dynamics of a tracer particle in a Fermi sea

Date: 01/19/2017

Time: 11:00 AM - 11:50 AM

Place: C304 Wells Hall

Speaker: Soeren Petrat, IAS and Princeton

The talk is about the dynamics of a tracer particle coupled strongly to a dense non-interacting electron gas in one or two dimensions. I will present a recent result that shows that for high densities the tracer particle moves freely for very long times, i.e., the electron gas becomes transparent. However, the correct phase factor is non-trivial. To leading order, it is given by mean-field theory, but one also has to include a correction coming from immediate recollision diagrams.

Title: Turaev-Viro invariants of links and the colored Jones polynomial

Date: 01/19/2017

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

Speaker: Renaud Detcherry, MSU

In a recent work by Tian Yang and Qingtao Chen, it has been observed that one can recover the hyperbolic volume from the asymptotic of Turaev-Viro invariants of 3-manifolds at a specific root of unity. This is reminiscent of the volume conjecture for the colored Jones polynomial.
In the case of link complements, we derive a formula to express Turaev-Viro invariants as a sum of values of colored Jones polynomial, and get a proof of Yang and Chen's conjecture for a few link complements. We also discuss the link between this conjecture and the volume conjecture. This is joint work with Effie Kalfagianni and Tian Yang.