ICTS

This mini-course aims first at presenting an infinite set of partial differential equations known as the KP hierarchy and, second, explaining its relevance in enumerative geometry, including some applications. I will follow the approach of Sato's school to the KP hierarchy, which can be summarized as: the KP hierarchy is the image of the Plücker relations for the orbit of an action of GL(∞) on the semi-infinite wedge via the boson-fermion correspondence. If you find this sentence scary (as is expected), fear not my friend for the purpose of this course is to make sense of it. I will try to explain every bit of this sentence to hopefully make sense of the full hierarchy.

After explaining this theory, we will make progress towards the study of Hurwitz numbers, which count branched covers of the sphere according to the profiles of the ramifications and the genus of the cover. This problem, which emanates from enumerative geometry, can be translated into combinatorics as counting the number of factorizations of the identity in the symmetric group. In the case of the so-called double weighted Hurwitz numbers, the generating function satisfies several sets of partial differential equations that have different origins, some combinatorial and some algebraic including the KP hierarchy! It turns out that combining these equations from different origins is known to lead to strikingly beautiful and efficient recurrence relations, as originally shown by Goulden and Jackson in 2008. I will focus on providing methods that prove that the generating function of dessins d'enfants, aka bipartite maps, satisfies the KP hierarchy. If time permits I will go all the way and derive the recurrence relations of the Goulden-Jackson type for dessins d'enfants.

This mini-course aims first at presenting an infinite set of partial differential equations known as the KP hierarchy and, second, explaining its relevance in enumerative geometry, including some applications. I will follow the approach of Sato's school to the KP hierarchy, which can be summarized as: the KP hierarchy is the image of the Plücker relations for the orbit of an action of GL(∞) on the semi-infinite wedge via the boson-fermion correspondence. If you find this sentence scary (as is expected), fear not my friend for the purpose of this course is to make sense of it. I will try to explain every bit of this sentence to hopefully make sense of the full hierarchy.

After explaining this theory, we will make progress towards the study of Hurwitz numbers, which count branched covers of the sphere according to the profiles of the ramifications and the genus of the cover. This problem, which emanates from enumerative geometry, can be translated into combinatorics as counting the number of factorizations of the identity in the symmetric group. In the case of the so-called double weighted Hurwitz numbers, the generating function satisfies several sets of partial differential equations that have different origins, some combinatorial and some algebraic including the KP hierarchy! It turns out that combining these equations from different origins is known to lead to strikingly beautiful and efficient recurrence relations, as originally shown by Goulden and Jackson in 2008. I will focus on providing methods that prove that the generating function of dessins d'enfants, aka bipartite maps, satisfies the KP hierarchy. If time permits I will go all the way and derive the recurrence relations of the Goulden-Jackson type for dessins d'enfants.

This mini-course aims first at presenting an infinite set of partial differential equations known as the KP hierarchy and, second, explaining its relevance in enumerative geometry, including some applications. I will follow the approach of Sato's school to the KP hierarchy, which can be summarized as: the KP hierarchy is the image of the Plücker relations for the orbit of an action of GL(∞) on the semi-infinite wedge via the boson-fermion correspondence. If you find this sentence scary (as is expected), fear not my friend for the purpose of this course is to make sense of it. I will try to explain every bit of this sentence to hopefully make sense of the full hierarchy.

After explaining this theory, we will make progress towards the study of Hurwitz numbers, which count branched covers of the sphere according to the profiles of the ramifications and the genus of the cover. This problem, which emanates from enumerative geometry, can be translated into combinatorics as counting the number of factorizations of the identity in the symmetric group. In the case of the so-called double weighted Hurwitz numbers, the generating function satisfies several sets of partial differential equations that have different origins, some combinatorial and some algebraic including the KP hierarchy! It turns out that combining these equations from different origins is known to lead to strikingly beautiful and efficient recurrence relations, as originally shown by Goulden and Jackson in 2008. I will focus on providing methods that prove that the generating function of dessins d'enfants, aka bipartite maps, satisfies the KP hierarchy. If time permits I will go all the way and derive the recurrence relations of the Goulden-Jackson type for dessins d'enfants.

The theory of the topological vertex provides an algorithm to explicitly calculate Gromov--Witten invariants of toric Calabi--Yau threefolds. It was originally inspired by large $N$ duality in physics and has now been established as a mathematically rigorous framework.

The theory of the topological vertex produces an infinite family of partition functions with remarkable properties: they are tau functions for integrable hierarchies, enumerate plane partitions, satisfy the topological recursion, possess quantum mirror curves, and encode non-trivial integer invariants. The study of such partition functions has motivated work on Hurwitz numbers, intersection theory on moduli spaces of curves, and the notion of refinement.

In this series of lectures, we will take a leisurely tour through these topics. By the end, you should be able to calculate examples of the aforementioned partition functions and observe their remarkable properties.

The theory of the topological vertex provides an algorithm to explicitly calculate Gromov--Witten invariants of toric Calabi--Yau threefolds. It was originally inspired by large $N$ duality in physics and has now been established as a mathematically rigorous framework.

The theory of the topological vertex produces an infinite family of partition functions with remarkable properties: they are tau functions for integrable hierarchies, enumerate plane partitions, satisfy the topological recursion, possess quantum mirror curves, and encode non-trivial integer invariants. The study of such partition functions has motivated work on Hurwitz numbers, intersection theory on moduli spaces of curves, and the notion of refinement.

In this series of lectures, we will take a leisurely tour through these topics. By the end, you should be able to calculate examples of the aforementioned partition functions and observe their remarkable properties.

The theory of the topological vertex provides an algorithm to explicitly calculate Gromov--Witten invariants of toric Calabi--Yau threefolds. It was originally inspired by large $N$ duality in physics and has now been established as a mathematically rigorous framework.

The theory of the topological vertex produces an infinite family of partition functions with remarkable properties: they are tau functions for integrable hierarchies, enumerate plane partitions, satisfy the topological recursion, possess quantum mirror curves, and encode non-trivial integer invariants. The study of such partition functions has motivated work on Hurwitz numbers, intersection theory on moduli spaces of curves, and the notion of refinement.

In this series of lectures, we will take a leisurely tour through these topics. By the end, you should be able to calculate examples of the aforementioned partition functions and observe their remarkable properties.

The theory of the topological vertex provides an algorithm to explicitly calculate Gromov--Witten invariants of toric Calabi--Yau threefolds. It was originally inspired by large $N$ duality in physics and has now been established as a mathematically rigorous framework.

The theory of the topological vertex produces an infinite family of partition functions with remarkable properties: they are tau functions for integrable hierarchies, enumerate plane partitions, satisfy the topological recursion, possess quantum mirror curves, and encode non-trivial integer invariants. The study of such partition functions has motivated work on Hurwitz numbers, intersection theory on moduli spaces of curves, and the notion of refinement.

In this series of lectures, we will take a leisurely tour through these topics. By the end, you should be able to calculate examples of the aforementioned partition functions and observe their remarkable properties.

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 this talk I will report new results on BPS counting for local Calabi-Yau threefolds that permit the complete solution of the BPS spectral problem, i.e. the computation of all semistable states and their refined BPS invariants (motivic DT) for infinite classes of cases, at the same time challenging some widespread beliefs regarding M-theoretic geometric engineering. 

First, I will present a theorem on how to induce stability conditions on (resolved) orbifolds of local CY3s and its implications for the spectrum, as wel as relations with the BPS Riemann-Hilbert problem and WKB for q-difference equations. As an application, I will give a closed formula for the spectrum of stable BPS states and for the Kontsevich–Soibelman wall-crossing invariant for the local Calabi–Yau threefolds Y^{(N,0)} for any N, geometrically engineering 5d SU(N) super Yang-Mills.

I will conclude by applying the same techniques to orbifold of nontoric CY3s, and describe some entirely novel aspects of the resulting physical theories and their geometric origin. These include a surprising infinite family of rank-1 theories that evades all known classifications.

In 2022, G. Chapuy and M. Dołęga introduced the b‑version of dessins d’enfants. In my talk, following the ideas discussed in Fesler, Hahn, and K.‑Markwig (2025), I will revisit their construction and discuss the algebraic setup in which refined dessins d’enfants arise naturally.

Hurwitz theory provides a large variety of enumerative problems related to algebraic geometry, mathematical physics, and combinatorics. We give a general framework to approach the large genus asymptotics of Hurwitz theory using only elementary methods and apply it to several types of Hurwitz numbers. We also apply our method to b-content Hurwitz numbers. As a specialisation, we recover some previously known about the large genus asymptotics of Hurwitz theory, namely classical results by Hurwitz and recent results of Do-He-Robertson, C. Yang, and results connected to recent work of X. Li. Join work with Davide Accadia and Giulio Ruzza.

In recent work with Nick Kuhn and Felix Thimm, we proved a Joyce-style "universal" wall-crossing formula for certain equivariant moduli problems of 3-Calabi-Yau type. This provides a refinement (in many senses) of the celebrated wall-crossing formulas of Joyce-Song and Kontsevich-Soibelman, and leads immediately to solutions of many basic open problems in refined enumerative geometry. For example, we obtain a Donaldson-Thomas/Pandharipande-Thomas correspondence for both K-theoretic primary vertices and descendent vertices, as well as a rigorous construction of refined Vafa-Witten invariants. Moreover, I will outline how the same techniques are applicable to similar moduli problems of 4-Calabi-Yau type. From this perspective, it becomes mathematically clear why the Nekrasov insertion is required in 4-fold DT theory.

In a joint work with A. Garbali [2308.16583], we introduced a (q,t)-deformation of the 2D Toda hierarchy by replacing the underlying $gl(\infty)$ symmetry algebra with the quantum toroidal gl(1) algebra. The resulting hierarchy is governed by a family of (q,t)-bilinear identities whose expansion yields the associated difference-differential equations, including two (q,t)-deformations of the 2D Toda equation. A distinctive feature of this construction is that the non-trivial coproduct structure of the quantum toroidal algebra naturally leads to families of tau functions rather than a single tau function. In this talk, I will review the construction the hierachy, and present new preliminary results on polynomial solutions of the corresponding (q,t)-deformed KP hierarchy.

We begin by writing down a certain q,t-partition function (or tau function) involving Macdonald polynomials. This can be seen as a deformation of the corresponding sum with Schur functions, typically considered in Hurwitz theory. Then we look for geometric interpretations of our partition function. One such interpretation is well-understood and comes from Haiman's computation of global sections of a certain vector bundle on the Hilbert scheme of C^2. A path to another interpretation is not so well understood, and goes via a conjecture of Hausel-Letellier-Rodriguez-Villegas and its interpretation by Chuang-Diaconescu-Donagi-Pantev. It involves the moduli space of twisted Higgs bundles on P^1. Our result makes this precise. We compute the Borel-Moore homology of the moduli stack of twisted Higgs bundles on P^1 and relate it to the local cohomology of a certain line bundle on the Hilbert scheme of C^2. This can be seen as an instance of 3d-mirror symmetry.