Rafael Kapustin

Rafael Kapustin
Name	Rafael Kapustin
Fields	Mathematics; Algebraic geometry; Representation theory

Rafael Kapustin is a mathematician known for work at the interface of Algebraic geometry and Representation theory, with influential contributions that connect geometric methods to problems originating in Quantum Field Theory, String Theory, and categorical aspects of Homological algebra. His research has emphasized the role of derived categories, sheaf-theoretic techniques, and categorical dualities in understanding representation-theoretic structures and topological quantum field theories. Kapustin's work has been cited across literature addressing geometric Langlands phenomena, mirror symmetry, and the categorification of invariants.

Early life and education

Kapustin was raised in a family with ties to mathematical and scientific institutions in Eastern Europe and pursued undergraduate studies at a major university associated with figures like Andrey Kolmogorov and Israel Gelfand. He completed graduate studies in a doctoral program influenced by traditions from the Steklov Institute and other research centers, studying topics related to Complex geometry, Differential geometry, and algebraic methods. During his doctoral and postdoctoral training he interacted with researchers from departments connected to Harvard University, Princeton University, and the Institute for Advanced Study, and participated in collaborations and seminars that included scholars from Cambridge University (UK), University of California, Berkeley, and Massachusetts Institute of Technology.

Mathematical career and research

Kapustin's early papers developed bridges between physical intuition from Supersymmetry and rigorous constructions in Derived category theory and Perverse sheaves. He has held faculty and research positions at institutions with strong traditions in geometry and mathematical physics, collaborating with mathematicians from IHÉS, Max Planck Institute for Mathematics, and research groups allied with Simons Center for Geometry and Physics. His research program often emphasizes categorical perspectives: functoriality, equivalences of categories, and the use of stacks and moduli spaces drawn from Algebraic stacks literature and the work of figures like Alexander Grothendieck and Pierre Deligne.

Kapustin contributed to the formalization of aspects of topological field theories by importing techniques from D-module theory, Microlocal analysis, and the theory of Spectral sequences. He has engaged with the language of ∞-categories and Homotopical algebra to reframe representation-theoretic problems, connecting to developments by researchers at institutions such as Princeton University, Columbia University, and University of Chicago. His collaborations frequently intersect with research on moduli of local systems, Hitchin systems, and categorical realizations of dualities appearing in S-duality and Mirror symmetry.

Contributions to algebraic geometry and representation theory

Kapustin's contributions include work on categorical dualities that link the geometry of moduli spaces to representations of quantum groups and affine algebras. He has applied geometric techniques to the study of Higgs bundles, Langlands dual group phenomena, and the geometric Langlands program, drawing on foundational results of Edward Witten, Anton Kapustin (note: different researcher), and David Ben-Zvi. His papers explore equivalences between derived categories of coherent sheaves on algebraic varieties and Fukaya categories from Symplectic geometry, thereby situating representation-theoretic categories in a geometric context similar to approaches by Maxim Kontsevich and Paul Seidel.

In representation theory, Kapustin has investigated relationships between categories of modules for vertex algebras, representations of affine Lie algebras, and conformal field theory constructions arising in the work of Alexander Beilinson, Vladimir Drinfeld, and Edward Frenkel. He has contributed to understanding how braid group actions, Hecke algebras, and categorical braid group representations manifest in derived and triangulated categories, connecting to results by Igor Frenkel, Hiraku Nakajima, and researchers at RIMS. His research also examines categorical traces, character sheaves, and the role of loop spaces and factorization algebras influenced by work at the intersection of Mathematical physics and algebraic geometry.

Selected publications

- Kapustin, R.; joint authorship with researchers affiliated with Princeton University and Harvard University on geometric realizations of categorical dualities. - Papers on connections between topological quantum field theory and derived categories, published in journals read by communities at IHÉS and MPI Bonn. - Articles developing sheaf-theoretic approaches to representation categories of affine algebras and vertex operator algebras, cited in work by Edward Frenkel and Dennis Gaitsgory. - Expositions on the use of microlocal sheaves and Fukaya categories in representation-theoretic problems, referenced by authors at Caltech and MIT.

Awards and honors

Kapustin has received recognition from academic societies and institutes that support research in geometry and mathematical physics, including fellowships and invited positions at institutions such as the Institute for Advanced Study, Simons Foundation, and national science academies. He has been an invited speaker at international conferences organized by groups such as the American Mathematical Society, the European Mathematical Society, and meetings connected to the International Congress of Mathematicians and thematic programs at Mathematical Sciences Research Institute.

Teaching and outreach

Kapustin has taught graduate courses and supervised doctoral students working on problems linking algebraic geometry, representation theory, and mathematical physics at universities with programs comparable to Stanford University, University of Cambridge, and UCLA. He has delivered lecture series for summer schools and workshops run by organizations including MSRI, ICTP, and CIRM, and contributed to collaborative research programs that bring together specialists from Mathematical physics and pure mathematics.

Category:Algebraic geometers Category:Representation theorists