Topology, Gauge Theory, and Quantum Field Theory
Topology, Gauge Theory, and Quantum Field Theory Program
2028 - 2029
Low-dimensional topology now has several powerful languages for the same underlying question: how can one construct geometric objects and determine which features of them are genuinely smooth, symplectic, or quantum?
Four-manifold topology supplies explicit surgeries, sums, fibrations, and regluings; gauge and Floer theories supply analytic invariants; categorification and skein theories create algebraic structures sensitive to cobordisms and boundaries; quantum field theory organizes locality, gluing, defects, and symmetry. The 2028-2029 TIAMS program will bring these languages into sustained contact around examples concrete enough to compute and structures general enough to compare.
Fall 2028
Smooth Four-Manifolds: Constructions and Invariants
The semester program will ask how construction technologies such as knot and torus
surgery, Luttinger surgery, symplectic sums, rational blowdowns, Lefschetz fibrations,
cork/regluing methods, and trisections can be compared systematically with Seiberg-Witten,
Floer, gauge-theoretic, and newer categorical invariants.
Spring 2029
Categorification, Defects, and Quantum Topology
The semester program will develop the algebraic and field-theoretic side: Khovanov-type theories, skein modules, Fukaya categories, character varieties, Poisson geometry, higher categories, extended field theories, generalized symmetries, and defects. Common boundary and gluing examples provide a bridge between semesters.
The year-long bridge
TIAMS will use residence to let researchers carry the same manifolds, cobordisms, moduli spaces, and algebraic structures through several formalisms. Working groups can compare what each invariant sees, where functoriality or naturality is available, and which extra boundary or defect data preserve information under gluing. Junior researchers should enter through focused tutorials and an embedded research-school component tied to active examples. Progress includes new constructions, new calculations of invariants, clearer comparison theorems, counterexamples that separate formalisms, and precise open problems identifying where present tools fail. Modern diagrammatic and computational tools can help participants move quickly among handle descriptions, algebraic data, and categorical constructions, while explicit functoriality and gluing hypotheses will govern every translation between languages. Shared notation and small benchmark computations should make disagreements visible early.
Program Details
The year is built around two linked residential semesters, with explicit examples serving as the shared research currency. Opening instruction establishes common conventions and background; focused workshops bring in broader communities; long working periods allow calculations and gluing arguments to mature; and closing synthesis meetings identify which correspondences are proved, which are conditional, and which fail.
Preliminary Workshops, Schools, or Focused Meetings
Fall opening tutorials on four-manifold constructions and invariant technologies
Dates TBA
Fall focused workshop on constructions, geography, and detection
Dates TBA
Spring opening tutorials / research school on categorification, field theory, and
defects
Dates TBA
Spring focused workshop on categorical and symplectic structures in quantum topology
Dates TBA
Year-end synthesis on gluing, functoriality, and comparison of invariants
Dates TBA
Seminars and Working Groups
- Construction dictionaries for four-manifolds
- Small exotica and detection
- Boundary-sensitive/categorical invariants
- Fukaya and Floer structures
- Character varieties and geometric representation theory
- Defects/generalized symmetries in field theory
Groups should maintain common examples and a shared notation sheet so that genuine mathematical differences are not obscured by convention changes.
The program is intended for low-dimensional topologists, gauge theorists, Floer theorists, symplectic geometers, representation theorists, researchers in categorification and knot homology, mathematical physicists, and specialists in topological or quantum field theory. Faculty, postdoctoral researchers, and graduate students are invited. Researchers in higher category theory, Poisson geometry, geometric representation theory, algebraic geometry, and related areas should participate when their structures address gluing, moduli spaces, defects, or invariant construction. The year will be especially productive for researchers willing to bring one explicit manifold, cobordism, moduli space, or algebraic object that can serve as a common test across several formalisms. Researchers who contribute a precise example, even a deliberately simple one, can give several communities a shared place to compare what their methods actually detect.
- Semester-scale or extended residence around one of the two program themes.
- Month-scale visits overlapping with active examples and working groups.
- Short targeted visits for a calculation, comparison, or focused meeting.
- Tutorial/research-school and workshop participation.
- Weekly seminar and working-group participation.
- Expression of interest through the TIAMS participation form.
Junior researchers will be encouraged to join groups organized around explicit examples and contribute to calculations from the beginning. Tutorials should provide the construction languages, gauge/Floer background, categorical formalisms, or field-theory conventions needed to enter those groups. Repeated informal presentations will let students and postdocs test calculations early, learn what different communities mean by the same words, and develop projects with more than one technical viewpoint available. They should also have structured opportunities to carry a Fall example into Spring and see how a different invariant or categorical language changes the question.
Expected outcomes include new or streamlined four-manifold constructions, computations and comparisons of invariants, gluing or functoriality results, categorical or field-theoretic structures attached to boundaries/defects, open-problem documents, software or diagrammatic calculation tools, and collaborations connecting geometric and algebraic methods.
Researchers may apply for one semester or indicate interest in the full year through the TIAMS Program Participation form.
Describe the examples, invariants, or structures you want to study, preferred dates, potential working groups, and any support request.