Tau functions are a central object in the theory of integrable systems. 

So far I have been particularly interested in tau functions of isomonodromic problems on Riemann Surfaces, and their expressions as conformal blocks, Nekrasov functions, or Fredholm determinants of Cauchy operators.

I am currently focusing on the use of Fredholm determinants as a tool to extract detailed information from the tau functions, in particular in their relation to the symplectic geometry of the moduli space of flat connections on Riemann surfaces, and to BPS states and partition functions of class S theories.

Topological String theory and supersymmetric field theories are tightly connected with the rich geometric structures underlying integrable systems.  My work up to now has been concerned with the relation between partition functions and BPS spectrum of such theories with tau functions of differential and q-difference integrable equations.

I am currently working on exploiting the connection with q-Painleve' equations to solve the BPS Riemann-Hilbert Problem of Bridgeland/Gaiotto-Moore-Neitzke type, and relate its solutions to TBA equations and WKB approximation of difference equations.

Cluster algebras make their appearance in many areas of mathematics and physics, like moduli spaces of flat connections on Riemann surfaces, WKB approximation and scattering amplitudes of QFTs. In my research, I used cluster algebras that define discrete integrable systems appearing in the study of Topological Strings on toric Calabi-Yau threefolds. These discrete Integrable Systems are crucial in showing that Topological String (grand-canonical) partition functions are tau functions of q-Painlevé equations, and encode information about the BPS spectrum of the theory on toric Calabi-Yau threefolds.

The representation theory of infinite-dimensional chiral algebras has the crucial role of determining conformal blocks in 2d CFT. In my work I used 2d CFT with W-algebra symmetry,  in the context of free fermions,  to show the relation between free fermion conformal blocks and isomonodromic tau functions for Fuchsian systems on  Riemann Surfaces with arbitrary genus and, providing their explicit series representation in the genus 1 case.