Donaldson-Thomas invariants of a Calabi-Yau 3-fold Y are related to Pandharipande-Thomas invariants via a wall-crossing formula known as the DT/PT correspondence, proved by Bridgeland and Toda. The same relation holds for the “local invariants”, those encoding the contribution of a fixed smooth curve in Y. We show how to lift the local DT/PT correspondence to the motivic level and provide an explicit formula for the local motivic invariants, exploiting the critical structure on certain Quot schemes acting as our local models. Our strategy is parallel to the one used by Behrend, Bryan and Szendroi in their definition and computation of degree zero motivic DT invariants. If time permits, we discuss a further (conjectural) cohomological upgrade of the local DT/PT correspondence.
Joint work with Ben Davison.
 

Given a smooth variety X with a normal crossings divisor D (or more generally a smooth log variety) we consider the ring of logarithmic differential operators: the subring of differential operators on X generated by vector fields tangent to D. Modules over this ring are called logarithmic D-modules and generalize the classical theory of regular meromorphic connections. They arise naturally when considering compactifications.

We will discuss which parts of the theory of D-modules generalize to the logarithmic setting and how to overcome new challenges arising from the logarithmic structure. In particular, we will define holonomicity for log D-modules and state a conjectural extension of the famous Riemann-Hilbert correspondence. This talk will be very example-focused and will not require any previous knowledge of D-modules or logarithmic geometry. This is joint work with Mattia Talpo.
 

Stability conditions on triangulated categories were introduced by Bridgeland, based on ideas from string theory. Conjecturally, they control existence of solutions to the deformed Hermitian Yang-Mills equation and the special Lagrangian equation (on the A-side and B-side of mirror symmetry, respectively). I will focus on the symplectic side and sketch a program which replaces special Lagrangians by "spectral networks", certain graphs enhanced with algebraic data. Based on joint work in progress with Katzarkov, Konstevich, Pandit, and Simpson.

In the talk I will discuss the  following result and related analytic and geometric questions:   On the boundary of any convex body in the Euclidean space there exists at least one infinite geodesic.