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2026 FRC: Representation Theory

May 18-29, 2026
Louisiana State University

LSU's Focused Research Communities (FRCs) are intensive, collaborative 2-week learning workshops funded by the NSF RTG Representation Theory, Topology and Mathematical Physics at Louisiana State University. FRC participants work together in small groups, under the guidance of faculty mentors, to study research-level topics and to present those topics to their peers. The workshops are accessible to graduate students who have completed a typical first-year graduate curriculum. The NSF grant will cover travel, lodging, and per diem expenses for all participants.

The theme of the 2026 FRC is geometric representation theory. There will be three working groups, listed below. Each participant will be assigned to one of these groups. You can indicate your preferences among these groups on the application form.

To apply, fill out the application form. Then, have your advisor or faculty mentor complete the recommendation form.

Priority will be given to applications received by March 25, 2026.

Working Group 1: Symplectic duality and Coulomb branches

  • Mentors: Ana Balibanu, Tom Gannon

  • Summary: Symplectic duality predicts that many of the important results concerning the geometry of the nilpotent cone of a semisimple Lie algebra extend to a broad class of varieties with symplectic singularities in the sense of Beauville. As the theory of symplectic duality developed, it became clear that it naturally admits a physical interpretation, namely via 3d mirror symmetry between the Higgs and Coulomb branches of a 3d $\mathcal N = 4$ supersymmetric gauge theory. After the Coulomb branches of a certain class of theories were placed on firm mathematical foundations by Braverman–Finkelberg–Nakajima, part of symplectic duality was reformulated as a duality between the Higgs branch and the Coulomb branch associated with a finite-dimensional representation of a complex reductive group.

    After reviewing some of the necessary background, we'll survey some important theorems concerning the geometry of the nilpotent cone. Then we'll introduce the notion of symplectic dual pairs and explain how those theorems generalize to this setting. We'll also discuss the precise construction of the Coulomb branch associated to a finite dimensional representation of a reductive group. If time permits and depending on participant demand, we'll discuss some more modern literature in symplectic duality.

  • Prerequisites:
    • Algebraic Geometry (as in Hartshorne Chapters 1-2)
    • Representation Theory of Lie algebras (as in Fulton and Harris)

Working Group 2: Localization and representations of Lie algebras in positive characteristic

  • Mentors: Pramod Achar, Pablo Boixeda Alvarez

  • Summary: The representation theory of complex semisimple Lie algebras has been studied since the beginning of the 20th century. In 1981, Beilinson-Bernstein and Brylinski-Kashiwara proved a breakthrough result, called the Localization Theorem, relating representations of complex semisimple Lie algebras to differential operators on flag manifolds and thus providing geometric tools to the study of these representations. The representation theory of Lie algebras in characteristic $p$ behaves very differently, but in some ways it is simpler. In particular, all irreducible representations are finite-dimensional. However, the naive analogue of the Localization Theorem is false. Nevertheless, in 2008, Bezrukavnikov-Mirkovic-Rumynin discovered and proved a modified version of the Localization Theorem that is valid in positive characteristic and thus provides similar geometric tools for understanding the representation theory in characteristic $p$. This course will be about this BMR localization theorem. A rough outline of topics to be covered is:

    1. Introduction to Lie algebra representations in characteristic $p$
    2. Geometry of flag varieties
    3. Differential operators and $D$-modules in characteristic $p$
    4. Derived localization
    5. The Azumaya property and coherent sheaves

    We'll also work out hands-on examples for the Lie algebra $\mathfrak{sl}_2$.

Working Group 3: Quantum groups

  • Mentors: Cris Negron, Thibault Décoppet, Alexandra Utiralova

  • Summary: Quantum groups are representation theoretic devices which provide a mathematical entry point into quantum field theories in low-dimensions. They play a role, for example, in mathematical formalizations of Chern-Simons topological field theories in dimension 3, and in WZW conformal field theories in dimension 2. Quantum groups also sit as an intermediate point between classical representations for algebraic groups over the complex numbers, and modular representations in finite characteristic.

    In this program we will focus on the case of $\mathrm{SL}_2$. We’ll classify simple and indecomposable tilting representations for quantum $\mathrm{SL}_2$ via the standard theory of dominant weights, and also introduce a ribbon structure on the category of representations. We will discuss the process of semi-simplification, in which one annihilates so-called negligible objects in order to produce finite “fusion” categories from infinite ribbon tensor categories. If time allows, we will present diagrammatics for the tiltings, and discuss phenomena in higher rank.

  • Prerequisites: One year of graduate algebra (groups, rings, modules, semisimple modules, Artin-Wedderburn, radicals, Jordan-Holder, etc.)

  • Preliminary readings: Representation theory for classical $\mathfrak {sl}_2(\mathbb C)$. See for example