Topology, Gauge Theory, and Quantum Field Theory Program

Spring 2029: Categorification, Defects, and Quantum Topology

Categorification replaces numerical or polynomial invariants by richer algebraic objects, while modern field-theoretic formalisms organize how information behaves under cutting, gluing, boundaries, and defects. These ideas now meet low-dimensional topology in many forms: Khovanov-type homologies, skein modules, gauge and Floer theories, Fukaya categories, character varieties, braided and fusion categories, and extended topological field theories. The central opportunity is to determine which of these structures are different expressions of the same locality principles and which carry genuinely distinct geometric information.

  • One cluster concerns functoriality, boundaries, and defects: how do cobordisms act, what data live on interfaces, and when does defect composition encode generalized or non-invertible symmetry?
  • A second concerns categorification and geometric detection, asking whether richer homological or skein-theoretic structures can distinguish manifolds or cobordisms beyond established invariants.
  • A third concerns moduli and representation-theoretic geometry: character varieties, Poisson and quasi-Poisson structures, geometric representation theory, and factorization-type constructions give concrete spaces on which field-theoretic gluing can be tested. Fukaya categories and Lagrangian Floer theory provide a complementary symplectic language for these structures.

At TIAMS, participants will work on common surfaces, three-manifolds, cobordisms, and moduli spaces and make every proposed comparison explicit on those examples. Groups will write down the objects assigned to boundaries, track maps under elementary gluings, and identify the hypotheses under which functoriality or equivalence claims hold. Tutorials will help researchers cross between categorical, symplectic, gauge-theoretic, and representation-theoretic languages. Progress may mean a new invariant, a new computation, a proof of compatibility, a counterexample separating two structures, or a clearer field-theoretic mechanism explaining why a topological construction behaves as it does. Repeated comparison on elementary gluings will be important: equivalence claims that look natural at a formal level should be tested first where every boundary object, map, grading, and coherence condition can be written explicitly. These small tests can also reveal which structures should be promoted to the next, more complicated examples.

 

  • How do gauge/Floer invariants, Khovanov-type theories, skein modules, and extended field theories fit into a common structural picture?
  • Which higher algebraic structures encode locality, boundaries, and defects in quantum field theory?
  • How are generalized and non-invertible symmetries reflected in fusion categories, higher categories, topology, and representation theory?
  • How do character varieties and Poisson or quasi-Poisson structures encode moduli arising from gauge theory and field theory?
  • When do Fukaya-categorical and Floer-theoretic structures provide a bridge between symplectic geometry and categorified topology?
  • Can these interactions produce genuinely new invariants of three- and four-manifolds as well as new interpretations of existing ones?

  • Khovanov-type homology and categorification
  • Skein modules and foam/cobordism structures
  • Extended topological field theory
  • Boundaries, defects, and generalized symmetries
  • Fukaya categories and Lagrangian Floer theory
  • Character varieties and Poisson geometry
  • Geometric representation theory
  • Functoriality and gluing

  • Mikhail Khovanov — intended external scientific leader; Johns Hopkins University
  • Denis Auroux (pending) — intended senior scientific leader; Harvard University

Preliminary Workshops, Schools, or Focused Meetings

Opening tutorials/research school on categorification, quantum topology, and field-theoretic gluing

Dates TBA

Focused workshop on boundaries, defects, and categorical invariants

Dates TBA

Working-group intensive on character varieties, symplectic structures, and categorical field theory

Dates TBA

Semester synthesis meeting

Dates TBA

 

Seminars and Working Groups

  • Categorical detection and cobordism maps: compute on benchmark three- and four-dimensional examples.
  • Defects and generalized symmetries: study composition, locality, and higher categorical structure.
  • Character varieties and quantization: compare Poisson geometry, representation theory, and field-theoretic gluing.
  • Fukaya/Floer bridges: test how Lagrangian and symplectic structures interact with categorified topological invariants.

The semester is intended for researchers in knot and low-dimensional topology, categorification, representation theory, symplectic geometry, Floer theory, gauge theory, mathematical physics, and topological or quantum field theory. Faculty, postdoctoral researchers, and graduate students are invited. Higher-categorical algebraists, experts on Poisson geometry and character varieties, and algebraic geometers should participate when their tools address boundary conditions, defects, moduli spaces, or gluing. The strongest contributions will come from researchers willing to put formal structures onto shared examples and make comparison statements precise. Researchers whose expertise is primarily algebraic or primarily geometric are equally welcome when they are willing to expose definitions and compute across the interface.

  • Extended or month-scale residence around a working group.
  • Short targeted visits for a specific calculation or equivalence problem.
  • Participation in tutorials/research school and focused meetings.
  • Weekly seminars and cross-field working sessions.
  • Graduate/postdoc project participation.
  • Expression of interest through the TIAMS participation form.

Junior researchers will be invited to learn one unfamiliar language well enough to use it on a shared example. Tutorials should make categorical, Floer, symplectic, and field-theoretic conventions explicit; working groups then provide concrete maps, moduli spaces, or diagrams to compute. Students and postdocs should have repeated opportunities to present work in progress and to turn a comparison question into a tractable project with guidance from researchers on both sides of the translation. Short problem sessions should also make it easy to ask basic convention questions that are often barriers to entering an adjacent field.

Possible outcomes include new or refined invariants, computations of skein or homological structures, gluing and functoriality theorems, categorical descriptions of defects, quantization results for character varieties, comparisons with Fukaya/Floer structures, open-problem documents, and collaborations across topology, algebra, geometry, and field theory.

Use the TIAMS Program Participation form to describe the categorical, geometric, or field-theoretic structures you want to study, the examples you can bring to the program, preferred visit dates, and any support request.