@article{21718, abstract = {In this paper, we consider the big algebra recently introduced by Hausel for the GLn-action on the coordinate ring of the matrix space Mat(n,r). In particular, we obtain explicit formulas for the big algebra generators in terms of differential operators with polynomial coefficients. We show that big algebras in type A are commutative and relate them to the Bethe subalgebra in the Yangian Y(gln). We apply these results to big algebras of symmetric powers of the standard representation of GLn. .}, author = {Ngo, Nhok T}, issn = {1815-0659}, journal = {Symmetry, Integrability and Geometry: Methods and Applications}, publisher = {National Academy of Science of Ukraine}, title = {{Big algebra in type A for the coordinate ring of the matrix space}}, doi = {10.3842/SIGMA.2026.024}, volume = {22}, year = {2026}, } @article{21931, abstract = {In 1873, James C. Maxwell conjectured that the electric field generated by n point charges in generic position has at most (n-1)^2 isolated zeroes. The first (nonoptimal) upper bound was only obtained in 2007 by Gabrielov, Novikov, and Shapiro, who also posed two additional interesting conjectures. In this article, we give the best upper bound known to date on the number of zeroes of the electric field, and construct a counterexample to Conjecture 1.8 by Gabrielov, Novikov, and Shapiro that the number of equilibria cannot exceed those of the distance function defined by the unit point charges. Finally, we note that it is quite possible that Maxwell's quadratic upper bound is not tight, so it is prudent to find lower bounds. Hence, we also explore examples and construct configurations of charges achieving the highest ratios of the number of electric field zeroes by point charges found to this day.}, author = {Edelsbrunner, Herbert and Fillmore, Christopher D and Oliveira, Goncalo}, issn = {1460-244X}, journal = {Proceedings of the London Mathematical Society}, number = {5}, publisher = {Wiley}, title = {{Counting equilibria of the electrostatic potential}}, doi = {10.1112/plms.70163}, volume = {132}, year = {2026}, } @article{21489, abstract = {We study Kirillov algebras attached to minuscule highest weight representations of semisimple Lie algebras. They can be viewed as equivariant cohomology algebras of partial flag varieties. Real structures on the varieties then induce involutions of these algebras. We describe how these involutions act on the spectra of minuscule Kirillov algebras, and model the fixed points via the equivariant cohomology of real partial flag varieties. We then use this model to characterise freeness of the fixed point coordinate ring over the appropriate base. As an application, we recover a q = -1 phenomenon of Stembridge in the minuscule case by geometric means.}, author = {Elkner, Mischa M}, issn = {1531-586X}, journal = {Transformation Groups}, publisher = {Springer Nature}, title = {{On involutions of minuscule Kirillov algebras induced by real structures}}, doi = {10.1007/s00031-026-09958-y}, year = {2026}, } @phdthesis{22694, abstract = {We develop and employ techniques from equivariant algebraic K-theory and related invariants in the context of geometric representation theory, in both arithmetic and topological situations. We showcase the use of such techniques on the affine Grassmannian Gr, a space of fundamental interest in the geometric Langlands program. It is a deep development of mathematics of the last century that many concrete, yet combina- torially complex algebraic problems may be effectively studied through the lens of algebraic geometry. The objects of interest can be often realized as cohomological invariants of algebraic varieties, and good understanding of their geometry sheds light into the original questions. Such techniques have seen immense applications in the Langlands program, where they go under the label of geometric representation theory. One source of powerful invariants in algebraic geometry comes from algebraic K-theory, Hochschild homology, and their relatives. These localizing invariants contain large amount of information, but are quite hard to compute. For this reason, their usage in geometric representation theory has been limited. The aim of this thesis is to showcase how to control such invariants in the situations of interest and use them to obtain new insights. We start by reinterpreting equivariant Hochschild homology in terms of functions on certain fixed-point schemes, which are of independent interest. We compare it to equivariant K-theory via the trace map. We give new computations and comparisons of such invariants of affine Schubert varieties in Gr, including arithmetic situations. We show that they behave much better than expected. We finally utilize this circle of ideas in a purely topological setting. We describe the varying fixed points of the extended torus action on the affine Grassmannian, and use it to compute its equivariant topological K-theory ring. The answer is nontrivial and verifies an outstanding conjecture in the subject. We compare, partly conjecturally, the resulting K-theory ring to the completed center of an integral even hybrid quantum group and its deformed quantum category O. This gives a genuine application of our computations in pure representation theory.}, author = {Löwit, Jakub}, issn = {2663-337X}, pages = {185}, publisher = {Institute of Science and Technology Austria}, title = {{Equivariant K-theory of affine Grassmannians in representation theory and arithmetic}}, doi = {10.15479/AT-ISTA-22694}, year = {2026}, } @article{21751, abstract = {We define a certain class of simple varieties over a field k by a constructive recipe and show how to control their (equivariant) truncating invariants. Consequently, we prove that on simple varieties: (i) if k = k and char k = p, the p-adic cyclotomic trace is an equivalence; (ii) if k = Q, the Goodwillie–Jones trace is an isomorphism in degree zero; (iii) we can control homotopy invariant K-theory KH, which is equivariantly formal and determined by its topological counterparts. Simple varieties are quite special, but encompass important singular examples appearing in geometric representation theory. We, in particular, show that both finite and affine Schubert varieties for GLn lie in this class, so all the above results hold for them. }, author = {Löwit, Jakub}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {7}, publisher = {Oxford University Press}, title = {{Equivariant localizing invariants of simple varieties}}, doi = {10.1093/imrn/rnag058}, volume = {2026}, year = {2026}, } @article{22693, abstract = {We study torus-equivariant algebraic K-theory of affine Schubert varieties in the perfect affine Grassmannians over Fp. We further compare it to the torus-equivariant Hochschild homology of perfect complexes, which has a geometric description in terms of global functions on certain fixed-point schemes. We prove that Fp-linearly, this comparison is an isomorphism. Our approach is quite constructive, resulting in new computations of these K-theory rings. We establish various structural results for equivariant perfect algebraic K-theory on the way; we believe these are of independent interest.}, author = {Löwit, Jakub}, issn = {1431-0643}, journal = {Documenta Mathematica}, keywords = {equivariant algebraic K-theory, perfection in positive characteristic, affine Grassmannian, affine Schubert varieties, Dennis trace map, equivariant Hochschild homology, fixed-point schemes, toric varieties}, publisher = {EMS Press}, title = {{Equivariant K-theory, affine Grassmannian and perfection}}, doi = {10.4171/dm/1064}, year = {2026}, } @article{17437, abstract = {We prove that the zero-fiber of the moment map of a totally negative quiver has rational singularities. Our proof consists in generalizing dimension bounds on jet spaces of this fiber, which were introduced by Budur. We also transfer the rational singularities property to other moduli spaces of objects in 2-Calabi-Yau categories, based on recent work of Davison. This has interesting arithmetic applications on quiver moment maps and moduli spaces of objects in 2-Calabi-Yau categories. First, we generalize results of Wyss on the asymptotic behaviour of counts of jets of quiver moment maps over finite fields. Moreover, we interpret the limit of counts of jets on a given moduli space as its p-adic volume under a canonical measure analogous to the measure built by Carocci, Orecchia and Wyss on certain moduli spaces of coherent sheaves.}, author = {Vernet, Tanguy}, issn = {1531-586X}, journal = {Transformation Groups}, pages = {1047--1083}, publisher = {Springer Nature}, title = {{Rational singularities for moment maps of totally negative quivers}}, doi = {10.1007/s00031-024-09873-0}, volume = {31}, year = {2026}, } @article{19071, abstract = {An action of a complex reductive group G on a smooth projective variety X is regular when all regular unipotent elements in G act with finitely many fixed points. Then the complex G -equivariant cohomology ring of X is isomorphic to the coordinate ring of a certain regular fixed point scheme. Examples include partial flag varieties, smooth Schubert varieties and Bott-Samelson varieties. We also show that a more general version of the fixed point scheme allows a generalisation to GKM spaces, such as toric varieties.}, author = {Hausel, Tamás and Rychlewicz, Kamil P}, issn = {2491-6765}, journal = {Epijournal de Geometrie Algebrique}, publisher = {EPI Sciences}, title = {{Spectrum of equivariant cohomology as a fixed point scheme}}, doi = {10.46298/epiga.2025.12591}, volume = {9}, year = {2025}, } @article{19621, abstract = {In this paper we obtain a complete description of all indecomposable characters (central positive-definite functions) of inductive limits of the symmetric groups under block diagonal embedding. As a corollary we obtain the full classification of the isomorphism classes of these inductive limits.}, author = {Nessonov, Nikolay and Ngo, Nhok T}, issn = {1088-4165}, journal = {Representation Theory}, number = {8}, pages = {256--288}, publisher = {American Mathematical Society}, title = {{Indecomposable characters of inductive limits of symmetric groups}}, doi = {10.1090/ert/689}, volume = {29}, year = {2025}, } @article{20664, abstract = {Conference travel contributes to the climate footprint of academic research. Here, we provide a quantitative estimate of the carbon emissions associated with conference attendance by analyzing travel data from participants of 10 international conferences in the field of magnetic resonance, namely EUROMAR, ENC and ICMRBS. We find that attending a EUROMAR conference produces, on average, more than 1 t CO2 eq.. For the analyzed conferences outside Europe, the corresponding value is about 2–3 times higher, on average, with intercontinental trips amounting to up to 5 t. We compare these conference-related emissions to other activities associated with research and show that conference travel is a substantial portion of the total climate footprint of a researcher in magnetic resonance. We explore several strategies to reduce these emissions, including the impact of selecting conference venues more strategically and the possibility of decentralized conferences. Through a detailed comparison of train versus air travel – accounting for both direct and infrastructure-related emissions – we demonstrate that train travel offers considerable carbon savings. These data may provide a basis for strategic choices of future conferences in the field and for individuals deciding on their conference attendance.}, author = {Kapoor, Lucky and Ruzickova, Natalia and Zivadinovic, Predrag and Leitner, Valentin and Sisak, Maria A and Mweka, Cecelia N and Dobbelaere, Jeroen A and Katsaros, Georgios and Schanda, Paul}, issn = {2699-0016}, journal = {Magnetic Resonance}, number = {2}, pages = {243--256}, publisher = {Copernicus Publications}, title = {{Quantifying the carbon footprint of conference travel: The case of NMR meetings}}, doi = {10.5194/mr-6-243-2025}, volume = {6}, year = {2025}, } @article{18154, abstract = {In 1976, Deligne and Lusztig realized the representation theory of finite groups of Lie type inside étale cohomology of certain algebraic varieties. Recently, a p-adic version of this theory started to emerge: there are p-adic Deligne–Lusztig spaces, whose cohomology encodes representation theoretic information for p-adic groups – for instance, it partially realizes the local Langlands correspondence with characteristic zero coefficients. However, the parallel case of coefficients of positive characteristic ℓ≠p has not been inspected so far. The purpose of this article is to initiate such an inspection. In particular, we relate cohomology of certain p-adic Deligne–Lusztig spaces to Vignéras's modular local Langlands correspondence for GLn.}, author = {Löwit, Jakub}, issn = {1090-266X}, journal = {Journal of Algebra}, number = {2}, pages = {81--118}, publisher = {Elsevier}, title = {{On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn}}, doi = {10.1016/j.jalgebra.2024.08.033}, volume = {663}, year = {2025}, } @article{20043, abstract = {We establish an isomorphism of complex K-theory of the moduli space M of “SL n​ ”-Higgs bundles of degree d and rank n (in the sense of Hausel–Thaddeus) and twisted complex K-theory of the orbifold M of PGL n​ -Higgs bundles of degree e, where (n,d)=(n,e)=1. Along the way, we prove the vanishing of torsion for H ∗ ( M ) and certain twisted complex K-theory groups of M . We also extend Arinkin’s autoduality of compactified Jacobian to a derived equivalence between SL n​ - and PGL n​ -Hitchin systems over the elliptic locus. In the appendix, we develop a formalism of G-sheaves of spectra, generalising equivariant homotopy theory to a relative setting.}, author = {Groechenig, Michael and Shen, Shiyu}, issn = {1435-9863}, journal = {Journal of the European Mathematical Society}, publisher = {EMS Press}, title = {{Complex K-theory of moduli spaces of Higgs bundles}}, doi = {10.4171/jems/1601}, year = {2025}, } @phdthesis{18443, abstract = {In [KW06] Kapustin and Witten conjectured that there is a mirror symmetry relation between the hyperkähler structures on certain Higgs bundle moduli spaces. As a consequence, they conjecture an equivalence between categories of BBB and BAA-branes. At the classical level, this mirror symmetry is given by T-duality between semi-flat hyperkähler structures on algebraic integrable systems. In this thesis, we investigate the T-duality relation between hyperkähler structures and the corresponding branes on affine torus bundles. We use the techniques of generalized geometry to show that semi-flat hyperkähler structures are T-dual on algebraic integrable systems. We also describe T-duality for generalized branes. Motivated by Fourier-Mukai transform we upgrade the T-duality between generalized branes to T-duality of submanifolds endowed with U(1)-bundles and connections. This T-duality in the appropriate context specializes to T-duality between BBB and BAA-branes. }, author = {Sisak, Maria A}, issn = {2663-337X}, keywords = {hyperkaehler geometry, branes, mirror symmetry, T-duality}, pages = {178}, publisher = {Institute of Science and Technology Austria}, title = {{T-dual branes on hyperkähler manifolds}}, doi = {10.15479/at:ista:18443}, year = {2024}, } @article{14986, abstract = {We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for GLn(k). Let k be an algebraically closed field of characteristic p>n. Let X be a smooth projective curve over k with marked points, and fix a parabolic subgroup of GLn(k) at each marked point. We denote by Bunn,P the moduli stack of (quasi-)parabolic vector bundles on X, and by Locn,P the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category Db(Qcoh(Loc0n,P)) of quasi-coherent sheaves on an open substack Loc0n,P⊂Locn,P, and the bounded derived category Db(D0Bunn,P-mod) of D0Bunn,P-modules, where D0Bunn,P is a localization of DBunn,P the sheaf of crystalline differential operators on Bunn,P. Thus we extend the work of Bezrukavnikov-Braverman to the tamely ramified case. We also prove a correspondence between flat connections on X with regular singularities and meromorphic Higgs bundles on the Frobenius twist X(1) of X with first order poles .}, author = {Shen, Shiyu}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, keywords = {General Mathematics}, number = {7}, pages = {6176--6208}, publisher = {Oxford University Press}, title = {{Tamely ramified geometric Langlands correspondence in positive characteristic}}, doi = {10.1093/imrn/rnae005}, volume = {2024}, year = {2024}, } @article{18108, abstract = {Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig’s q-weight multiplicity polynomials i.e., certain affine Kazhdan–Lusztig polynomials.}, author = {Hausel, Tamás}, issn = {1091-6490}, journal = {Proceedings of the National Academy of Sciences of the United States of America}, number = {38}, publisher = {National Academy of Sciences}, title = {{Commutative avatars of representations of semisimple Lie groups}}, doi = {10.1073/pnas.2319341121}, volume = {121}, year = {2024}, } @article{15248, abstract = {Applying the technique of p-adic integration, we prove the topological mirror symmetry conjecture of Hausel-Thaddeus for the moduli spaces of (strongly) parabolic Higgs bundles for the structure groups SLn and PGLn, building on previous work of Groechenig-Wyss-Ziegler on the non-parabolic case. We also prove the E-polynomial of the smooth moduli space of parabolic GLn-Higgs bundles is independent of the degree of the underlying vector bundles.}, author = {Shen, Shiyu}, issn = {1090-2082}, journal = {Advances in Mathematics}, number = {5}, publisher = {Elsevier}, title = {{Mirror symmetry for parabolic Higgs bundles via p-adic integration}}, doi = {10.1016/j.aim.2024.109616}, volume = {443}, year = {2024}, } @article{14930, abstract = {In this paper we investigate locally free representations of a quiver Q over a commutative Frobenius algebra R by arithmetic Fourier transform. When the base field is finite we prove that the number of isomorphism classes of absolutely indecomposable locally free representations of fixed rank is independent of the orientation of Q. We also prove that the number of isomorphism classes of locally free absolutely indecomposable representations of the preprojective algebra of Q over R equals the number of isomorphism classes of locally free absolutely indecomposable representations of Q over R[t]/(t2). Using these results together with results of Geiss, Leclerc and Schröer we give, when k is algebraically closed, a classification of pairs (Q, R) such that the set of isomorphism classes of indecomposable locally free representations of Q over R is finite. Finally when the representation is free of rank 1 at each vertex of Q, we study the function that counts the number of isomorphism classes of absolutely indecomposable locally free representations of Q over the Frobenius algebra Fq[t]/(tr). We prove that they are polynomial in q and their generating function is rational and satisfies a functional equation.}, author = {Hausel, Tamás and Letellier, Emmanuel and Rodriguez-Villegas, Fernando}, issn = {1420-9020}, journal = {Selecta Mathematica}, number = {2}, publisher = {Springer Nature}, title = {{Locally free representations of quivers over commutative Frobenius algebras}}, doi = {10.1007/s00029-023-00914-2}, volume = {30}, year = {2024}, } @article{15339, abstract = {We define even very stable Higgs bundles and study the Hitchin map restricted to their upward flows. In the GLn case, we classify the type (1,…,1) examples, and find that they are governed by a root system formed by the roots of even height. We discuss how the spectrum of equivariant cohomology of real and quaternionic Grassmannians, 4n-spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.}, author = {González, Miguel and Hausel, Tamás}, issn = {1793-6519}, journal = {International Journal of Mathematics}, number = {09}, publisher = {World Scientific Publishing}, title = {{Hitchin map on even very stable upward flows}}, doi = {10.1142/S0129167X2441009X}, volume = {35}, year = {2024}, } @article{18970, abstract = {Given a smooth projective curve C, nonabelian Hodge theory gives a diffeomorphism between two different moduli spaces associated to C. The first is the moduli space of Higgs bundles on C of rank n, which is equipped with the structure of an algebraic completely integrable Hamiltonian system. The second is the character variety of representations of the fundamental group of C into GL(n). In 2012, de Cataldo, Hausel, and Migliorini [1] proposed the P=W conjecture which identifies the perverse filtration on the cohomology of the Higgs moduli space with the weight filtration on the cohomology of the character variety. Recently, in 2022, two independent proofs of the P=W Conjecture appeared, in work of Maulik &Shen [2] and Hausel, Mellit, Minets &Schiffmann [6]. The aim of the Arbeitsgemeinschaft was to understand the P=W Conjecture and these two recent proofs.}, author = {Hausel, Tamás and Maulik, Davesh and Mellit, Anton and Schiffmann, Olivier and Shen, Junliang}, issn = {1660-8941}, journal = {Oberwolfach Reports}, number = {2}, pages = {949--1004}, publisher = {EMS Press}, title = {{Arbeitsgemeinschaft: Geometry and representation theory around the P=W conjecture}}, doi = {10.4171/owr/2024/16}, volume = {21}, year = {2024}, } @article{17292, abstract = {The Gibbons-Hawking ansatz provides a large family of circle-invariant hyperkähler 4-manifolds, and thus Calabi-Yau 2-folds. In this setting, we prove versions of the Thomas conjecture on existence of special Lagrangian representatives of Hamiltonian isotopy classes of Lagrangians, and the Thomas-Yau conjecture on longtime existence of the Lagrangian mean curvature ow. We also make observations concerning closed geodesics, curve shortening flow and minimal surfaces.}, author = {Lotay, Jason D. and Oliveira, Goncalo}, issn = {0022-040X}, journal = {Journal of Differential Geometry}, number = {3}, pages = {1121--1184}, publisher = {International Press of Boston}, title = {{Special Lagrangians, Lagrangian mean curvature flow and the Gibbons-Hawking ansatz}}, doi = {10.4310/jdg/1717348872}, volume = {126}, year = {2024}, }