Sami Assaf — Title: Superb Crystals
Abstract: Demazure crystals are certain extremal subsets of highest weight crystals which give a combinatorial skeleton of Demazure modules. In certain special cases, Kashiwara’s tensor product rule for crystals gives a combinatorial realization of excellent filtrations for tensor products of Demazure modules. However, not all tensor products of Demazure crystals result in unions of Demazure crystals. This reflects the fact that not all tensor products of Demazure modules admit excellent filtrations. Polo conjectured tensor products of Demazure modules always admit superb filtrations, wherein successive quotients are Schubert modules. In evidence of this, I recently proved certain products of Demazure characters expand nonnegatively into Schubert characters. In this talk, we introduce a new family of extremal subsets of highest weight crystals and show how they might give a crystal-theoretic model for Polo’s conjecture. 

Pablo Boixeda-Alvarez — Title: Weight modules and Kazhdan-Laumon category $\mathcal{O}$.
Abstract: In work of Mathieu and Fernando the modules of the enveloping algebra with finite dimensional weight spaces are understood. These conditions can be translated into a support condition for the associated graded or some singular support condition for some sheaves on $G/B$. This singular support is given by taking the union of $W$ copies of the conditions for category $\mathcal{O}$. In work of Kazhdan and Laumon the construct a category by glueing $W$ copies of the category of perverse sheaves on $G/U$. This category was studied by Bezrukavnikov, Polishchuk and Morton-Ferguson. In particular some subcategory known as Kazhdan-Laumon category $\mathcal{O}$ was related to the representation theory of the small quantum group $u_q$. In joint work with Morton-Ferguson we relate the Kazhdan-Laumon category $\mathcal{O}$ to some subcategory of weight modules. This connection should explain the relation to the representation theory of $u_q$. In this talk I will discuss Kazhdan-Laumon's category $\mathcal{O}$ and its connection to weight modules and time permitting the connection to $u_q$ via the joint work with Bezrukavnikov, McBreen and Yun and the the geometry of affine Springer fibers. 

Jon Brundan — Title: Categorification of quantum symmetric pairs
Abstract: Given a symmetrizable Cartan matrix $(a_{i,j})_{i,j \in I}$, Lusztig's algebra $\mathbf{f}$, which is half of a Drinfeld-Jimbo quantum group, has generators $\theta_i\:(i \in I)$ subject to the quantum Serre relations. An elementary algebraic way to "categorify" this algebra was introduced nearly 20 years ago by Khovanov, Lauda and Rouquer, based on a new family of algebras called quiver Hecke algebras. The quantum Serre relation gets replaced by a split exact complex whose Euler characteristic recovers the original relation. Another remarkable family of quantum algebras are the quasi-split iquantum groups introduced by Letzter, generalized by Kolb, and studied further by Bao and Wang (who introduced this terminology). They include ordinary Drinfeld-Jimbo quantum groups as a special case. The definition requires the additional data of an involution $\tau$ of the Cartan matrix. When defined by generators and relations, there is again just one required relation, which is similar to the quantum Serre relation but with the $0$ on the right hand side replaced by a complicated lower order term. I will explain how to categorify quasi-split iquantum groups in a uniform way, leading to a categorical explanation for the mysterious extra term in the iSerre relation. This is based on joint work with Weiqiang Wang and Ben Webster.

Ben Elias — Title: Spin link homology and equivariant categorification
Abstract: The categorical dimension of the second fundamental representation of \mathfrak{sl}_4 is q^4 + q^2 + 2 + q^{-2} + q^{-4}, which is the graded dimension of a vector space spanned by all partitions fitting in a 2 \times 2 box. The transpose operation preserves that set of partitions, and if instead of counting partitions (the trace of the identity) one takes the trace of the transpose, one obtains q^4 + q^2 + q^{-2} + q^{-4}, which is the categorical dimension of the spin representation of \mathfrak{so}(5). This is an indication of a relationship between representation theory in types A and B that falls under the rubric of "folding." Folding is not a statement which makes rigorous sense without categorification: the relationship between those Laurent polynomials is merely that they agree modulo 2; only when one constructs the vector space can one see the true relationship. In joint work with Elijah Bodish and David Rose (arXiv '24) we construct a new involution on categorified quantum \mathfrak{sl}_{2n}, which is not itself a Dynkin diagram involution, but corresponds to the type A Dynkin diagram involution under skew Howe duality. This yields an involution on Khovanov-Rozansky homology, whose trace yields the type B spin-colored link homology (modulo some conjectures). In order to prove this, we also must prove new results about the Grothendieck group; we provide a new diagrammatic calculus describing morphisms between tensor powers of the spin representation in type B. Time permitting (and it won't!) we discuss joint work in preparation, which extends this diagrammatic calculus to a full description of type B webs.

Monica Garcia — Title: Semistability and projective presentations
Abstract: Stability conditions are an important tool in algebraic theory for constructing moduli varieties. When applied to the varieties of modules over a finite-dimensional algebra, they give rise to the algebraic notion of semistable modules, which are closely linked to $tau$-tilting theory and cluster algebras. To find these semistable modules, one can compute a special class of regular functions known as determinantal semi-invariants. In this talk, we will revisit the relation of these semi-invariants to projective presentations and explore semistability for varieties of projective presentations. We will recall that determinantal semi-invariants give rise to two interesting types of subcategories, namely, wide subcategories of the module category and thick subcategories of the extriangulated category of projective presentations. Finally, we will introduce an extriangulated version of the correspondences among support $tau$-tilting objects, torsion classes, and wide subcategories. This correspondence extends classical results to the context of projective presentations.

Eduardo Gonzalez — Title: Symplectic shift operators and Coulomb branches
Abstract: We will discuss joint work with D. Pomerleano and C. Y. Mak, on an action via shift operators of the pure Coulomb branch of a Lie group G on the symplectic (quantum) cohomology of a compact hamiltonian G-manifold (M,\omega) under certain conditions.  Using work of C. Teleman and Braverman-Finkelberg-Nakajima this gives a geometric reinterpretation of Coulomb branches in terms of symplectic cohomology.

Cris Negron — Title: $(3-\epsilon)$-dimensional TQFTs from derived quantum group representations
Abstract: I will discuss joint work with Agustina Czenky.  We introduce a $(3-\epsilon)$-dimensional TQFTs which is generated, in some sense, by the derived category of quantum group representations.  This TQFT is valued in the $\infty$-category of dg vector spaces, and the value on a genus $g$ surface is a $g$-th iterate of the Hochschild cohomology for the aforementioned category.  I will explain how this TQFT arises as a derived variant of the usual Reshetikhin-Turaev theory and, if time allows, I will discuss the possibility of introducing local systems into the theory.  Our interest in local systems comes from proposed relationships with $4$-dimensional non-topological QFT.

Khoa Nguyen — Title: On $\mathcal{U}(\mathfrak{h})$-free modules over $\mathfrak{sl} (m|1)$$
Abstract: The study of modules over Lie algebras and superalgebras is usually divided into different categories. The first important category is the category of weight modules. In particular, such a category consists of modules that decompose into direct sums of their weight space with respect to a fixed Cartan subalgebra $\mathfrak{h}$. Free $\mathcal{U}(\mathfrak{h})$-modules are important non-weight modules. Such modules are free of finite rank when restricted to the Cartan subalgebra $\mathfrak{h}$.  In this talk I will present a classification of $\mathcal{U}(\mathfrak{h})$‑free modules of rank $2$ over $\mathfrak{sl}(m|1)$, based on joint work with I. Dimitrov. I will also discuss a construction of families of higher‑rank $\mathcal{U}(\mathfrak{h})$‑free modules, arising from ongoing work with I. Dimitrov, C. Paquette, and D. Wehlau.

Julia Pevtsova — Title: The Balmer spectrum of local dualizable representations of a finite group.
Abstract: Let G be a finite group, and k a field of positive characteristic p. The stable category of representations of G over k is a well-studied entity whose Balmer spectrum was effectively computed by Benson, Carlson and Rickard in 1997. If one fixes a prime ideal in the Balmer spectrum and zooms in to the category of all representations of G supported only at that specific ideal, one discovers many interesting phenomena.  I’ll describe the ``small” (= dualizable) objects in that category, calculate their spectrum, and draw parallels with completions in commutative algebra. Joint work with D. Benson, S. Iyengar, H. Krause.

Raphael Rouquier — Title: Braiding for 2-representations
Abstract: I will explain the constructions of a tensor product and a braiding for 2-representations of Lie algebras. The usual quantum R-matrix appears already in the tensor product, in a way similar to the stable bases construction, while the braiding arises as a Koszul duality.

Eric Rowell — Title: Categorical Perspectives on Classifying Yang-Baxter Matrices
Abstract: Classifying matrix solutions to the Yang-Baxter equation is a problem noted for both its simplicity and a dearth of effective strategies. Recent work with Martin classifying charge conserving solutions to the Yang-Baxter in all dimensions has inspired us to look at the general problem afresh (or possibly poisoned us with hubris).  In particular the categorical perspective suggests that there may be other families of solutions that are amenable to classification.  In work with Martin and Torzewska, we have begun excavating the structure and properties of the category MonFun(B,Mat) of strong monoidal functors from the braid category B to the category Mat of matrices, to see if it will expose a new vein of classification ore. I will discuss our progress so far, with an eye towards putting past success into this broader context and deciding if it can be repeated or if it was just a flash in the pan.

Nick Rozenblyum — Title: Categorification of Deligne-Lusztig theory
Abstract: I will give a brief overview of the theory of traces in higher categories and explain how this gives a new approach to the study of representation of finite groups of Lie type. Specifically, given an algebraic group G over a finite field F_q, I will explain how representations of the finite group G(F_q) arise as traces of categorical representations of G. Moreover, I will explain the higher categorical origin of Deligne-Lusztig representations and give a new conceptual computation of their characters which explains their regularity as a function of q. This is joint work with Gaitsgory and Varshavsky.

Leonid Rybnikov — Title: Bethe subalgebras in Yangians, their degenerations and crystals.
Abstract: Let g be a complex simple finite-dimensional Lie algebra and G be the adjoint Lie group with the Lie algebra g. To every group element C from G, one can assign a commutative subalgebra B(C) in the Yangian Y(g), which is responsible for the integrals of the (generalized) XXX Heisenberg quantum spin chain. For regular semisimple C, the images of Bethe subalgebras in tensor products of fundamental representations are equivariant quantum cohomology rings of Nakajima quiver varieties. We describe all degenerations of Bethe subalgebras and show that they are naturally parametrized by the De Concini -Gaiffi wonderful model for toric arrangement given by the root system of g. Furthermore, using these degenerations, we construct a natural structure of affine crystals on spectra of B(C) in Kirillov-Reshetikhin Y(g)-modules in type A. We conjecture that such a construction exists for arbitrary g and gives Kirillov-Reshetikhin crystals. We also conjecture (and prove in type A) that the monodromy of spectra of B(C) along the real locus of the De Concini-Gaiffi space is given by partial Schutzenberger involutions of the corresponding Kirillov-Reshetikhin crystal. The talk is based on joint results with Aleksei Ilin, Vasily Krylov, and Inna Mashanova-Golikova.

Jeremy Taylor — Title: The universal monodromic Bezrukavnikov equivalence
Abstract: The universal monodromic affine Hecke category is a family of categories over the dual torus. It is obtained by allowing sheaves on the enhanced affine flag variety with arbitrary monodromy along the torus orbits. I will discuss the Langlands dual coherent realization. This is joint work with Dhillon that extends Arkhipov and Bezrukavnikov's results to non-completed families.

Oleksandr Tsymbaliuk — Title: Sevostyanov's approach to qToda and generalizations
Abstract: In the late 90s, the quantum difference Toda lattices were introduced independently in the works of P.Etingof and A.Sevostyanov. The latter approach is important for some other constructions in quantum groups as well as allows natural generalizations.

Yaolong Shen — Title: Coulomb branches of symmetric quivers and truncated shifted twisted Yangians
Abstract: The study of Coulomb and Higgs branches arising from 3d $\mathcal{N} = 4$ gauge theories provides a rich framework for constructing and understanding concepts in modern representation theory. In the classical setting, for framed quivers, the Higgs branch is Nakajima’s quiver variety, which gives a geometric construction of universal enveloping algebras and their representations. The Coulomb branch in this case can be described algebraically by a truncated shifted Yangian, a deformation of the universal enveloping algebra associated with the same quiver. These two structures are related by 3d mirror symmetry, which exchanges the roles of the Coulomb and Higgs branches. In this talk, I will explain how this picture extends to symmetric quivers with framing. For such quivers, we show that the corresponding Coulomb branch can be realized as a truncated shifted twisted Yangian (TruSTY), a loop analogue of the fixed-point subalgebra in symmetric pairs. On the Higgs branch side, we expect the corresponding structure to be closely related to the Cartan-folded enveloping algebra introduced by Enomoto and Kashiwara, which is a folded variant of the universal enveloping algebra associated with the same quiver. This is joint work with Changjian Su and Rui Xiong.

Chelsea Walton — Title: Representations of braided categories.
Abstract: In the talk, I'll discuss recent work with Robert Laugwitz and Milen Yakimov, and with Harshit Yadav, on module categories over braided finite tensor categories. This will be based on the articles, ArXiv/2307.14764 and ArXiv/2411.18453, and I'll aim to keep this down-to-earth.