Representations, Moduli and Duality

Titles and abstracts

Francesca Carocci:

Donaldson-Thomas invariants and p-adic integration

In recent years, a series of works of Groechenig, Wyss and Ziegler have showed that p-adic integration can be a powerful tool to study Donaldson-Thomas type invariants in several geometric situations of interests. The p-adic approach to DT theory has been fully developed in the homological dimension one  case by  Groechenig, Wyss and Ziegler; recently they have also obtained results for the case of 2 dimensional Calabi-Yau. In this talk, I will report on a joint work in progress with Dimitri Wyss, in which we extend the p-adic approach to the case of global critical loci and we investigate dimensional reduction in the p-adic setting.

Tudor Dimofte:

CoHA's, Yangians, and Tannakian QFT

I'll describe a setup in 4d N=2 theories whereby "algebras of BPS states" appear explicitly on certain boundary conditions, and represent the category of bulk line operators (related to work of Gaiotto, Grygoryev, and Li). The setup incorporates reconstruction theory for the category of line operators, and helps explain the origin of various algebraic structures in CoHA's and their Yangian-like doubles.

Soheyla Feyzbakhsh:

Hurwitz-Brill-Noether Theory via K3 Surfaces

I will discuss the Brill-Noether theory of a general elliptic K3 surface using wall-crossing with respect to Bridgeland stability conditions. As an application, I will provide an example of a general k-gonal curve from the perspective of Hurwitz-Brill-Noether theory. This is joint work with Gavril Farkas and Andrés Rojas.

Lie Fu:

Hodge conjecture for abelian fourfolds of Weil type with discriminant 1

Based on the work of O'Grady and Voisin, Markman proved the Hodge conjecture for abelian fourfolds of Weil type with discriminant 1 using hyper-Kähler manifolds of generalized Kummer type. I will present a new proof of this theorem recently discovered in my joint work with Salvatore Floccari (arXiv:2504.13607). This new proof is quite elementary and does not rely on the theory of hyper-holomorphic bundles. Our strategy is to use O'Grady's 6-dimensional singular symplectic varieties. 

Tamas Hausel:

Ringifying intersection cohomology

Although there is no natural ring structure on intersection cohomology, in the case of affine and finite Schubert varieties also at their singular points, partly conjecturally, we ringify intersection cohomology using big algebras.

Oscar Kivinen:

Quot schemes on singular curves

The Oblomkov–Rasmussen–Shende conjecture describes the triply graded homology of the link of a plane curve singularity in terms of the Hilbert schemes of points on the singularity. Apart from simple examples, this conjecture is wide open. 

 

In joint work with Trinh, we study Quot schemes of the pushforward of the structure sheaf from the normalization, and prove that an analogue of the ORS conjecture for these Quot schemes does hold for many toric singularities. There are many other ways in which these Quot schemes are easier to work with than the original Hilbert schemes, which we conjecture to be related by a simple yet motivically subtle change of variables ("The Hilb-vs-Quot conjecture”). 

 

Time permitting, I will discuss joint work in progress with Bejleri, where we study more general Quot schemes, possibly on non-reduced curves.

 

Yau Wing Li:

Endoscopy for metaplectic affine Hecke categories

Lusztig (1994) studied sheaves on the enhanced affine flag variety with fixed monodromy along Kac-Moody torus orbits, developing a framework that later became central to the metaplectic and quantum geometric Langlands program. In joint work with Gurbir Dhillon, Zhiwei Yun, and Xinwen Zhu, we show that these monodromic affine Hecke categories are equivalent to non-monodromic affine Hecke categories of smaller groups, constructed via an affine analogue of Langlands’ endoscopy, extending results of Lusztig and Yun for finite Hecke categories. Applications include endoscopic equivalences, confirming a series conjectured by Gaitsgory in quantum geometric Langlands.

Sergej Monavari:

The refined local Donaldson-Thomas theory of curves 

 

The Maulik-Nekrasov-Okounkov-Pandharipande correspondence predicts an equivalence between the partition functions of (numerical) Gromov-Witten and Donaldson-Thomas invariants of smooth projective threefolds. It was recently proposed by Pardon a solution of this conjectural correspondence by reducing to the simpler case of local curves, which are more amenable for computations by means of TQFT methods. Even more recently, inspired by the seminal work of Nekrasov-Okounkov on the index in M-theory, Brini-Schuler proposed a refined GW/DT correspondence. In this talk, I will present a full solution for the Donaldson-Thomas side of the refined GW/DT correspondence in the case of local curves. In particular, I will explain how to derive the refined DT partition function without relying on degeneration techniques, relative invariants and TQFT methods, and how our formulas recover string-theoretic prediction of Aganagic-Schaeffer.

 

Hiraku Nakajima:

Instanton moduli for classical groups, involution on quiver varieties, and quantum symmetric pairs

We consider moduli spaces of instantons on ALE spaces for classical groups. They are examples of fixed point sets of involutions on quiver varieties. Some years ago, Yiqiang Li considered their equivariant cohomology, and by stable envelope technique, constructed representations of coideal subalgebras of Yangian, often called twisted Yangian. I remove unnecessary assumption which Li imposed, and calculate K-matrices as matrices, hence identify which twisted Yangian we talk about. If time permits, I will explain relation to Coulomb branches of type D quiver gauge theories and de Campos Affonso’s symmetric bow varieties. 

David Nadler:

Potent categorical representation theory

I will discuss work with David Ben-Zvi and German Stefanich developing a theory of representations of reductive groups G on categories that treats all parameters equitably. Our inspirations come from genuine equivariant stable homotopy theory and circle-compactification in topological field theory. I will explain how this approach leads to a symmetric form of Langlands duality on the circle relating potent categorical representations of G and its dual group. 

Georg Oberdieck:

Quantum cohomology of the Hilbert scheme of points on an elliptic surface

I will report on a work in progress with Aaron Pixton and Qaasim Shafi, in which we fully compute the quantum cohomology ring (for all curve classes) of the Hilbert scheme of points on an elliptic surface with p_g > 0.

Marco Robalo:

Gluing invariants of Donaldson-Thomas type

In this talk I will explain a general mechanism, based on derived symplectic geometry, to glue local invariants of singularities that appear naturally in Donaldson-Thomas theory.  This mechanism recovers the categorified vanishing cycles sheaves constructed by Brav-Bussi-Dupont-Joyce, and provides a new more evolved gluing of Orlov’s categories of matrix factorisations, answering questions of Kontsevich-Soibelman and Y.Toda. This is a joint work with B. Hennion (Orsay) and J. Holstein (Hamburg).

Francesco Sala:

Cohomological Hall algebras of 1-dimensional sheaves and Yangians

The first part of this talk provides a brief and gentle introduction to the theory of 2-dimensional cohomological Hall algebras. The second part focuses on the introduction of the nilpotent cohomological Hall algebra COHA(S, Z) of coherent sheaves on a smooth quasi-projective complex surface S set-theoretically supported on a closed subscheme Z. When S is the minimal resolution of an ADE singularity and Z is the exceptional divisor, I will describe how to characterize COHA(S, Z) via the Yangian of the corresponding affine ADE quiver (based on arXiv:2502.19445, co-authored with Emanuel Diaconescu, Mauro Porta, Oliver Schiffmann, and Eric Vasserot, as well as ongoing work with Parth Shimpi and Oliver Schiffmann).

 

Junliang Shen:

The intrinsic cohomology ring for universal compactified Jacobians

Compactifications of the universal Jacobian over the Deligne–Mumford moduli space of stable marked curves have been studied intensively since the work of Caporaso and Pandharipande in the 90s. However, the cohomology rings of these spaces have not been explored much, and a key difficulty is the absence of a canonical compactification. The purpose of this talk is to address this issue by introducing the intrinsic cohomology ring of the universal compactified Jacobian. This new invariant is defined as a degeneration of the cohomology ring of a chosen compactification, and we show that it is, in fact, independent of that choice. Our main tool combines Ngô’s support theorems with a theory of Fourier transform for dualizable abelian fibrations. This is based on joint work with Younghan Bae, Davesh Maulik, and Qizheng Yin.

Andrey Smirnov:

Quantum K-theory at roots of unity

 

In this talk, I will discuss a version of quantum K-theory introduced by A.Okounkov, which can be defined through quasimap counts. In this framework, the quantum K-theory ring is obtained as a specialization of the equivariant quasimap count at q=1, where q is the equivariant parameter associated with the torus action on the source of the quasimaps. A related, but less explored, structure emerges when $q$ is specialized at the roots of unity. I will outline the key ideas behind this construction and its implications. As an application, I’ll also describe the spectrum of $p$-curvature for the quantum connection, which offers a new proof of a recent result by P.Etingof and A.Varchenko. This talk is based on joint work with P. Koroteev.

 

Jeremy Taylor:

Tame local Betti Langlands

The universal monodromic affine Hecke category is a family of categories over a torus. It is composed of sheaves on the enhanced affine flag variety with arbitrary monodromy along the torus orbits. I will discuss its Langlands dual realization as coherent sheaves on the Steinberg stack. This universal monodromic enhancement of Bezrukavnikov's equivalence is joint work with Gurbir Dhillon.

 

David Treumann:

Profinite tensor powers

I'll discuss the problem of defining a tensor product of profinitely many copies of a vector space V, and propose a definition $\bigotimes_X^{mcc} V$ in the special situation that (1) V is finite-dimensional over F_2, and (2) the profinite X indexing the tensor factors is acted on with finitely many orbits by a pro-2-group. The "mcc" on the tensor sign stands for "magnetized and conditionally convergent." A variant construction makes sense when V is a bimodule over a semisimple F_2-algebra, and the index set X has the profinite version of a cyclic order. The definition organizes some computations in Heegard Floer homology: it can be pitched as a computation of the HF of some pro-3-manifolds, though we do not know how to define such a thing. This is joint work with CM Michael Wong.