Georgia Institute of Technology College of Sciences

This talk is about an application of complex function theory to inverse spectral problems for differential operators. We consider the Schroedinger operator on a finite interval with an L^1-potential. Borg's two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function. After a brief review of inverse spectral theory of one dimensional regular Schroedinger operators, we will discuss complex analytic methods for the following problem: Can one spectrum together with subsets of another spectrum and norming constants recover the potential?

In this talk, I will discuss from a mathematical viewpoint some sufficient conditions that guarantee the energy equality for weak solutions. I will mainly focus on a fluid equation example, namely the inhomogeneous Euler equations. The main tools are the commutator Lemmas.  This is a joint work with Ming Chen.

I will consider the isotropic XY chain with a transverse magnetic field acting on a single site, and analyze the long time behaviour of the time-dependent state of the system when a periodic perturbation drives the impurity. I will show that, under some conditions, the state approaches a periodic orbit synchronized with the forcing. Moreover I will provide the explicit rate of convergence to the asymptotics. This is a joint work with G. Genovese.

As countless examples show, sequences of complicated objects should be studied all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of (orbit-)configuration spaces: using the notion of twisted commutative algebras, which categorify exponential generating functions. With this idea the configuration space “generating function” factors into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it also gives rise to representation stability - a notion of homological stability for sequences of representations of differing groups.