Schedule

We review three perspectives on topological intersection theory for singular spaces: Cheeger's L2 cohomology, the intersection cohomology of Goresky and MacPherson, and a more recent homotopy theoretic perspective. The latter will constitute the focus of the talk. The associated cohomology theory is not isomorphic to intersection cohomology and we will provide several alternative descriptions of it, such as a de Rham picture and an L2-Hodge description. Classical intersection homology is invariant under small resolutions; the new theory is invariant under "small" deformations and sometimes carries a mixed Hodge structure. We will illustrate the theory on examples including toric varieties and the Calabi-Yau quintic. We conclude with observations on conifold transitions and mirror symmetry.

I will report on joint work in progress with Luca Giovenzana. After giving a very brief introduction to the theory of pseudolattices, I will describe how they show up naturally in the context of Type II degenerations of K3 surfaces, and show how this can be used to recover some classical results on Type II degenerations due to Friedman. I will then describe how the same picture appears in the context of elliptically fibred K3 surfaces, and discuss how these structures seem to provide a natural setting to study mirror symmetry for Type II degenerations of K3 surfaces. 

Recently, certain widely accepted beliefs about the string theory landscape have been challenged. In part, this is was possible because important parametric constraints obeyed by the relevant compact geometries were overlooked. In part, overly optimistic assumptions about the multitude of Calabi-Yau-based geometries may have been made. I review some of the key challenges, explaining in particular the Singular Bulk Problem, the Tadpole Constraint and the Tadpole Conjecture. I will also discuss which concrete geometrical questions may help to overcome those challenges or, on the contrary, to remove large parts of the string landscape.

We discuss how to extend the Doran-Harder-Thompson (DHT) mirror construction in case of some higher type degenerations of a Calabi-Yau manifold $X$. One of the key ingredients is a higher rank Landau-Ginzburg (LG) model, a multi-potential analogue of the ordinary LG model that will describe a mirror of each component in the degeneration fiber. They can be glued to produce a mirror Calabi-Yau $Y$ with a Calabi-Yau fibration structure. Moreover, the perverse Leray filtration associated to this fibration is predicted to be mirror to the monodromy weight filtration in the degeneration of $X$. We explain how this “P=W” type phenomenon is related with another mirror P=W, introduced by Harder-Katzarkov-Przyjalkowski, for the mirrors of components of the degeneration fiber.

As a result of work of Bloch-Esnault-Kreimer, and subsequent work of Brown, one can reinterpret Feynman integrals as periods of mixed Hodge structures attached to pairs of hypersurfaces in certain toric varieties. The hypersurfaces taking part in this construction are built from a Feynman graph. This is essentially the data of a graph decorated with mass parameters attached to each edge and momentum parameters attached to each vertex. Despite the combinatorial origin of their geometry, the motives of Bloch-Esnault-Kreimer and Brown are only understood in a few basic cases. In this talk, I will discuss results which show that for a simple but infinite class of graphs, the mixed Hodge structures controlling the corresponding Feynman integrals are built from hyperelliptic curves. This generalizes results of Bloch and Kerr and helps clarify many computations appearing in the physics literature. I'll demonstrate this in some examples and discuss the relationship to computational work of Lairez-Vanhove.

The partition function of topological string theory is an asymptotic series in the topological string coupling and provides in a certain limit a generating function of Gromov-Witten (GW) invariants of an underlying Calabi-Yau threefold X. I will discuss how the resurgence analysis of the partition function for a class of non-compact geometries allows one to extract the Donaldson-Thomas (DT) or BPS invariants of the same underlying geometry X. I will further discuss how the analytic functions in the topological string coupling obtained by Borel summation admit a dual expansion in the inverse of the topological string coupling leading to another asymptotic series at strong coupling and to the notion of topological string S-duality. I will further discuss how this S-duality leads to a new modular structure in the topological string coupling which is captured in terms of elliptic gamma functions. I will also discuss relations to refined topological string theory, difference equations and the exact WKB analysis of the mirror geometry. This is based on various joint works with Lotte Hollands, Arpan Saha, Iván Tulli and Jörg Teschner as well as on work in progress.

Donaldson-Thomas (DT) invariants of a quiver with potential can be expressed in terms of simpler attractor DT invariants by a universal formula. The coefficients in this formula are calculated combinatorially using attractor flow trees. In joint work with Bousseau (arXiv:2302.02068), we prove that these coefficients are genus 0 log Gromov--Witten invariants of d-dimensional toric varieties, where d is the number of vertices of the quiver. This result follows from a log-tropical correspondence theorem which relates (d-2)-dimensional families of tropical curves obtained as universal deformations of attractor flow trees, and rational log curves in toric varieties.

We recall some relations between the Grothendieck Ring of Varieties and the Grothendieck Witt Ring of Quadratic forms and use them to get refinements of Donaldson-Thomas (DT) invariants taking values in the latter starting from the motivic DT invariants. Our main example is the case of degree zero invariants of $\mathbb A^3$, from which we can recover real and complex invariants. We raise some questions regarding the generality of these refinements and their relationships with the literature in both $\mathbb A^1$-enumerative geometry and in motivic DT theory. This is joint work in progress with Johannes Walcher.

With Spencer Bloch and Robin de Jong, we recently proved that in a nodal degeneration of smooth curves, the periods of the resulting limit mixed Hodge structure (LMHS) contain arithmetic information. In particular, if the nodal fiber is identified with a smooth curve $C$ glued at two points $p$ and $q$ then the LMHS relates to the Neron-Tate height of $p-q$ in the Jacobian of $C$. In making this relation precise, we observed that a "tropical correction term" is required that is based on the finite reductions of the degenerate fiber. In this talk, I will explain this circle of ideas with the goal of arriving at the tropical correction term.

Airy structures were introduced by Kontsevich and Soibelman in 2017 as an algebraic reformulation and extension of the Chekhov-Eynard-Orantin topological recursion, which appears in many contexts, from enumerative geometry to mathematical physics. In this talk I will introduce the concept of Airy structures as particular left ideals in the Rees Weyl algebra, such that the quotient of the algebra by the ideal is canonically isomorphic to a Rees polynomial module twisted by an automorphism of a simple type. This approach provides a clean reformulation of the theory in the language of D-modules, in which the foundational existence and uniqueness theorem of Kontsevich and Soibelman follows naturally. I will then mention some recent applications of the theory to enumerative geometry, VOAs and gauge theories, and discuss potential generalizations. My hope with this talk is to convey why I believe that the formalism of Airy structures (and topological recursion) should be in the toolbox of all geometers and mathematical physicists!