ICTS

The talk is devoted to several real and tropical enumerative problems. We suggest new invariants of the projective plane (and, more generally, of certain toric surfaces) that arise from appropriate signed enumeration of real algebraic curves of genus 1 and 2. These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the genus zero case. We also suggest an extension of the invariants under consideration to a non-toric setting.
This is a joint work with Eugenii Shustin.

Over the last years, several "quadratic enrichments" of enumerative invariants have been proposed, in particular, the quadratic Gromov-Witten invariants of planar rational curves by constructed by Kass, Levine, Solomon, and Wickelgren. These invariants can be defined over (almost) any base field and generalize classical Gromov-Witten and Welschinger invariants of rational curves. In our work, we use the framework of Witt-invariants (here, invariance refers to the behaviour under base change) to study the relationship between the quadratic and the classical enumerative invariants. In particular, using a crucial integrality  condition, we show that in many cases, the classical invariants completely determine the quadratic ones. (joint work with Erwan Brugallé and Kirsten Wickelgren)

I will show that symmetric encode observables of 4d N=2 theories related to wall-crossing phenomena, observables in 3d Chern-Simon theory, and characters of 2d CFTs. On the other hand, the same quivers encode 3d N=2 theories and their associated BPS invariants. I will argue that these latter BPS invariant provide refinements of various quantities in the aforementioned theories in 2, 3 and 4 dimensions, and all these theories form a duality web worth further exploration.

I will report on joint work with Penka Geogieva on a structure theorem for real Gromov-Witten invariants of Calabi-Yau 3-folds with an anti-symplectic involution. Our results were motivated by the Gopakumar-Vafa and Walcher conjectures, and generalize earlier joint work with Thomas Parker, Aleksander Doan and Thomas Walpuski proving the integrality and respectively finiteness part of the Gopakumar-Vafa conjecture for the Gromov-Witten invariants of 3-folds.

In joint work with Shivang Jindal, we construct a surjective homomorphism from the loop nilpotent cohomological Hall algebra of a tripled quiver (a Higgs branch like object) to the Coulomb branch algebra of the same quiver gauge theory. We emphasize the shuffle algebra which represents the combinatorial underpinning of both branches.