First Week
Pierrick Bousseau: Donaldson--Thomas Theory, Flow Trees and Log Gromov--Witten Invariants
Abstract: In this lecture series, we will begin with an introduction to Donaldson–Thomas (DT) theory, which concerns the enumeration of stable objects in three-dimensional Calabi–Yau categories. Key examples of such categories include derived categories of coherent sheaves, Fukaya categories on Calabi–Yau threefolds, and the category of representations of quivers with potential. Focusing on the latter, we will explore the wall-crossing formula for quiver DT invariants and demonstrate how attractor flow trees provide a powerful computational framework for these invariants. Building on this, we will establish a correspondence between quiver DT invariants and log Gromov–Witten invariants of toric and cluster varieties, offering new insights into their interplay. Finally, we will delve into holomorphic Floer theory, proposing a broader perspective on why such a correspondence should hold in more general settings.
Lectures:
Donaldson-Thomas (DT) theory
Wall crossing in DT theory
Quiver DT invariants from attractor flow trees
The quiver DT / log Gromov--Witten correspondence
Holomorphic Floer theory
Hulya Arguz's Exercise Sessions: Calculating quiver DT/log Gromov-Witten invariants in higher dimensions
Abstract: We will go through some concrete examples calculating quiver DT invariants using attractor flow trees, and log Gromov--Witten invariants of higher dimensional toric and cluster varieties using wall crossing.
Hulya Arguz's Research Session: Non-toric brane webs, Calabi-Yau 3-folds, and 5d SCFTs
Abstract: One of the most remarkable predictions of string/M-theory is the existence of 5-dimensional superconformal field theories (5d SCFTs). There are two main approaches for constructing these 5d SCFTs, using either M-theory on canonical 3-fold singularities, or intersecting branes in Type IIB string theory. A natural question is to compare these two approaches. The answer is well-known for webs of 5-branes in Type IIB string theory, where the M-theory dual canonical 3-fold singularity is a toric Calabi–Yau 3-fold. In this talk, building on recent advances in mirror symmetry and enumerative geometry, I will provide an answer for the more general case of webs of 5-branes with 7-branes and explain how to construct the M-theory dual non-toric Calabi-Yau 3-fold. This is joint work with Valery Alexeev and Pierrick Bousseau.
Joel Kamnitzer: From quantum cohomology of symplectic resolutions to compactifications of families of Gaudin algebras
Abstract: In recent years, symplectic resolutions have been a highly active area of study in algebraic geometry and representation theory. I will introduce this area, including the quantum cohomology of symplectic resolutions. Quantum cohomology will naturally lead us toward the theory of quantum integrable systems, which are large commutative subalgebras inside non-commutative algebras. In particular, we will consider Bethe subalgebras in Yangians, and Gaudin algebras inside tensor products of universal enveloping algebras. These algebras are related (respectively) to the quantum cohomology of quiver varieties and affine Grassmannian slices. Finally, we will switch gears and study the compactification of the parameter space of these algebras. Remarkably, this will lead us to the definition of new moduli spaces of curves.
Dinakar Muthiah’s Exercise Sessions: The BFN construction of Coulomb branches, symplectic duality, and affine Grassmannians slices
Abstract: I will discuss the Braverman-Finkelberg-Nakajima construction of Coulomb branches. Their construction gives rise to many examples of symplectic duality. In particular, their construction gives rigorous meaning to the statement that finite-type Nakajima quiver varieties are symplectically dual to affine Grassmannian slices.
Dinakar Muthiah’s Research Session: Kac-Moody affine Grassmannian slices
Abstract: I will explain how to use the BFN construction to define Kac-Moody affine Grassmannian slices, which allow us to approach the Geometric Satake Correspondence in Kac-Moody types. This final talk will include discussion of joint work with Alex Weekes.
Bernd Siebert: Logarithmic Gromov-Witten invariants and its application to mirror symmetry
Abstract: The purpose of this lecture series is a gentle introduction to the subject of logarithmic Gromov-Witten invariants and its application to mirror symmetry. Logarithmic geometry in the sense of Illusie and Kato is a version of algebraic geometry adapted to situations relative a divisor with normal or even toroidal crossings, including central fibers of normal crossing degenerations. Logarithmic Gromov-Witten theory enhances Gromov-Witten theory with contact conditions with these divisors, even when the curve has components mapping into the divisor. Apart from a richer theory coming from the added information, such contact conditions naturally come up in degeneration and gluing situations.
A remarkable feature of the theory is that the usual dual intersection graphs of the domain curves arising in ordinary Gromov-Witten theory become promoted to tropical curves. In Calabi-Yau situations arising in mirror symmetry, these tropical curves sweep out the wall structures carrying the quantum corrections in the construction of mirror geometries.
Lectures:
I. Geometric introduction to logarithmic geometry
II. Kato-Nakayama spaces of log spaces, tropicalization, stable log maps
III. Logarithmic Gromov-Witten invariants, Artin fans, punctured Gromov-Witten invariants
IV. The gluing formalism for log Gromov-Witten theory
V. Intrinsic mirror symmetry
Samuel Johnston's Exercise Session I: The tropical geometry of the moduli space of log smooth curves
Abstract: We will explore the relationship between tropical geometry and logarithmic geometry through examples. Specific attention will be given to the tropical moduli theory behind the identification of moduli spaces of log smooth curves, thought of as moduli problems over log schemes, with algebraic stacks with log structure whose underlying stacks are moduli spaces of pointed prestable curves.
Samuel Johnston's Exercise Session II: The tropical geometry of the moduli space of stable log maps
Abstract: Continuing on from Tuesday's session, we will study the logarithmic and tropical geometry of the moduli space of log stable maps to a toric variety. We will explore the relationship between moduli of tropical curves in R^n and the moduli of log stable maps to an n-dimensional projective toric variety, and how study of the former yield insight into the geometry of the latter.
Samuel Johnston's Research Session: Quantum periods, toric degenerations and intrinsic mirror symmetry
Abstract: One half of mirror symmetry for Fano varieties is typically stated as a relation between the symplectic geometry of a Fano variety Y and the complex geometry of a Landau-Ginzburg model, realized as a pair (X,W) with X a quasi-projective variety and W a regular function on X. The pair (X,W) itself is expected to reflect a pair on the Fano side, namely a decomposition of Y into a disjoint union of an affine log Calabi-Yau and an anticanonical divisor D, thought of as mirror to W. We will discuss recent work which shows how the intrinsic mirror construction of Gross and Siebert naturally produce LG models from a pair (Y,D), assuming milder conditions on the singularities of D than typically required for the intrinsic mirror construction. In particular, we show that the classical periods of the LG models recover the quantum periods of Y. In the setting when Y\D is an affine cluster variety, we will describe how these LG models naturally give rise to Laurent polynomial mirrors and encode certain toric degenerations of Y. As an example, we consider Y = Gr(k,n), D a particular choice of anticanonical divisor with affine cluster variety complement and give an explicit description of the intrinsic LG model in terms of Plücker coordinates on Gr(n-k,n), recovering mirrors constructed and investigated by Marsh-Rietsch and Rietsch-Williams.
Second Week
Junliang Shen: Decomposition Theorem for abelian fibrations
Abstract: The decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber allows us to compute the “relative cohomology” of a proper map in terms of “simple objects”; however, the “simple objects” are not really simple in general, so it is very difficult to carry out explicit calculations. In his proof of the fundamental lemma of the Langlands program, Ngô initiated a systematic study of the decomposition theorem for abelian fibrations (i.e. a proper map whose general fibers are abelian varieties and special fibers are their degenerations); this package is nowadays referred to as the Ngô support theorem. In the last two decades, these techniques have been further developed by many people, and have rich applications in several directions, including the topology of Hitchin systems, BPS invariants in enumerative geometry, the study of compact hyper-Kähler manifolds, algebraic cycles etc. In the lectures, I will start with the decomposition theorem, then I will discuss ideas of the Ngô support theorem, and present some recent applications.
Camilla Felisetti's Exercise Session I: Intersection cohomology and decomposition theorem, examples and exercises
Camilla Felisetti's Exercise Session II: Weak abelian fibrations, examples and exercises
Camilla Felisetti's Research Session: Decomposition theorem for abelian fibrations, examples and exercises
Olivier Schiffmann: Cohomological Hall algebras of sheaves on surfaces and applications
Abstract: The aim of these lectures is to explain the construction of an associative algebra structure on the Borel-Moore homology of the stack of properly supported coherent sheaves on a smooth complex surface (the COHA) and to give some motivations and applications. More precisely, we hope to touch upon the following topics:
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Definition of the COHA of the preprojective algebra of a quiver, and the relation to Kac-Moody (and larger!) algebras and Kac polynomials.
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Definition of the COHA of zero-dimensional sheaves on S, and computation of that COHA (at least under some assumption on S, for instance if S is projective). Relation to W-algebras.
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Action of the above COHA of zero-dimensional sheaves on suitable moduli stacks / spaces of coherent sheaves on S. Examples related to Hilbert schemes on S, instanton spaces for the special case $S=\mathbb{A}^2$, moduli stacks of Higgs sheaves for $S=T^*C$, $C$ a smooth projective complex curve.
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Definition of the COHA for arbitrary (properly supported) coherent sheaves on a surface S, and some examples / applications (depending on time)
Lucien Hennecart's Exercise Session I: Examples of cohomological Hall algebras
In this first session, I will begin by reviewing the six operations on the constructible derived category of an algebraic variety. I will then present the construction of the CoHA product at the sheafified level, and illustrate this with several examples of cohomological Hall algebras associated to quivers and preprojective algebras.
Lucien Hennecart's Exercise Session II: Tools to study cohomological Hall algebras
In this session, I will introduce some of the key tools used in the study of cohomological Hall algebras: dimensional reduction, the BPS Lie algebra, and the Poincaré--Birkhoff--Witt (PBW) theorem for cohomological Hall algebras.
Lucien Hennecart's Research Session: BPS Lie algebras and generalized Kac--Moody Lie algebras
I will present the BPS Lie algebra associated to a preprojective algebra and explain how it can be realized as a generalized Kac–Moody Lie algebra. This is based on joint work with Davison and Schlegel Mejia. I will then describe the structure of the full cohomological Hall algebra of the affine space, and discuss conjectural structures for the CoHA in more general settings.
Yukinobu Toda: Categorical Donaldson-Thomas theory and the Dolbeault geometric Langlands conjecture
Abstract: Donaldson–Thomas (DT) invariants virtually count stable coherent sheaves on Calabi–Yau threefolds. In the rank one case, they are curve-counting invariants and are related to other curve- counting theories such as Gromov–Witten invariants and Gopakumar–Vafa invariants. The DT invariants admit further refinements, to motivic and cohomological DT theories, which connect DT theory to broader areas of mathematics including geometric representation theory.
In this lecture series, I will introduce DT theory and explain the motivation toward its categorification via dg-categories. While the global construction of these dg-categories remains open in general, they are expected to be constructed by gluing locally defined dg-categories of matrix factorizations. The resulting dg-categories are expected to recover the original DT invariants by taking suitable categorical invariants.
I will focus on two main examples: quivers with potentials and Higgs bundles. In both cases, the notion of quasi-BPS categories plays a central role as a building block of categorical DT theory. I will also introduce some key tools, including the window theorem for GIT quotient stacks and the construction of categorical Hall products, and show how these lead to a categorical wall-crossing formula.
I will then turn to the Dolbeault geometric Langlands conjecture. This conjecture, proposed by Donagi and Pantev, states an equivalence between certain derived categories of moduli stacks of Higgs bundles. It can be viewed as a classical limit of the geometric Langlands correspondence and as a categorical version of the Hausel–Thaddeus mirror symmetry for Higgs bundles. I will explain how ideas from categorical DT theory lead to a new formulation of the Dolbeault Langlands conjecture, highlighting a symmetry between quasi-BPS categories. This part is based on joint work (partly in progress) with Tudor Pădurariu.
Tudor Pădurariu's Exercise Session I: Vanishing cycles and cohomology of stacks
Abstract: We first discuss explicit computations of vanishing cycle cohomology. Second, we discuss an example of the decomposition of singular cohomology of a quotient stack in terms of parabolic induction of intersection cohomology of certain associated moduli spaces, which is an instance of the Davison-Meinhardt theorem.
Tudor Pădurariu's Exercise Session II: Semiorthogonal decompositions
Abstract: I plan to go over explicit examples of semiorthogonal decompositions for quotient stacks where the group is G_m. First, we see an example where the stack is smooth. Second, we discuss a singular example, obtained using matrix factorizations. These examples are local models of the semiorthogonal decompositions of the moduli of semistable (twisted or not) Higgs bundles on a curve. Time permitting, I will also talk about the derived equivalence between an abelian variety and its dual.
Tudor Pădurariu's Research Session: Quasi-BPS categories and Lie theory
Abstract: I will discuss potential future applications of quasi-BPS categories. For finite or affine quivers, quasi-BPS categories are expected to be equivalent to categories of representations of KLR (Khovanov-Lauda-Rouquier) algebras. However, quasi-BPS categories are defined in much larger generality, and thus we might expect the definition of new KLR-type algebras (for general cotangent stacks) for which the above statement remains true.
We expect that quasi-BPS categories are closely related to categories of line defects in 4d N=2 field theories. Thus they should also have a description in terms of the Coulomb branch of such theory. I will speculate on this relation, which seems to imply that these new KLR-type algebras admit (for some cotangent stacks) monoidal categorifications.
For quivers, these expectations are a step in extending various constructions in geometric representation theory (such as KLR algebras, or symplectic duality) from finite or affine Lie algebras to Maulik-Okounkov Lie algebras.
These expectations and potential future applications are based on discussions with Sabin Cautis and Yukinobu Toda, and are also inspired by work of Okounkov et.al.
