Topology, Gauge Theory, and Quantum Field Theory Program

Fall 2028: Smooth Four-Manifolds: Constructions and Invariants

Smooth four-manifold topology has accumulated a rich collection of construction methods and an equally rich collection of invariants, yet the relation between the two remains difficult to organize. Knot and torus surgery, Luttinger surgery, symplectic sums, rational blowdowns, Lefschetz fibrations and pencils, cork or plug regluings, surface-bundle building blocks, Kirby calculus, and trisections each expose different geometric features. Gauge theory, Seiberg-Witten theory, Heegaard and knot Floer theories, symplectic invariants, and newer categorical constructions detect overlapping but distinct information. The semester will ask how to make construction and detection speak to one another systematically.

  • A first cluster concerns construction dictionaries: when do distinct surgery or fibration descriptions encode the same operation, and which presentations make fundamental group, homology, symplectic structure, or canonical class easiest to control?
  • A second concerns small exotic manifolds and geography, where explicit constructions press against tight topological constraints and the available invariants may become less discriminating.
  • A third concerns boundary-sensitive and categorified detection. Recent work showing that a Khovanov-Rozansky skein lasagna module can distinguish an exotic pair of compact knot traces demonstrates that categorical data can detect smooth structure in a boundary setting, raising precise questions about what information survives gluing to closed manifolds.

At TIAMS, groups will carry selected examples through several construction languages and invariant packages. The goal is to identify which features are robust, which computations are genuinely comparable, and where extra boundary or functorial data are essential. Tutorials and problem sessions will make Kirby diagrams, mapping-class factorizations, gauge/Floer invariants, and categorical tools accessible across subcommunities. Progress may be a new example, a no-go result, a comparison theorem, a gluing formula, a computation that separates two manifolds, or a clean obstruction showing why a proposed detector cannot see the phenomenon in question. The semester will value negative information when it is precise: knowing that a construction cannot meet a fundamental-group, geography, or invariant constraint can redirect a working group as effectively as a successful example.

  •  Can surgery, symplectic-sum, Lefschetz-fibration, rational-blowdown, and regluing methods be translated into a usable common toolkit?
  • How systematically can new building blocks and monodromy structures produce simply connected or small exotic four-manifolds?
  • Which geometric data survive in Seiberg-Witten, Heegaard Floer, gauge-theoretic, symplectic, or Bauer-Furuta-type invariants?
  • Can categorical or skein-theoretic invariants detect smooth phenomena invisible to established analytic invariants?
  • What boundary or cobordism data enable categorical detection, and what is lost when pieces are glued into a closed manifold?
  • Can trisections, Kirby calculus, symplectic geometry, and mapping-class factorizations provide a common language for comparing constructions?

  • Knot, torus, and Luttinger surgery
  • Symplectic and surface sums
  • Rational blowdown and plumbing replacement
  • Lefschetz fibrations and pencils
  • Corks, plugs, and regluing
  • Gauge, Seiberg-Witten, and Floer invariants
  • Categorified and skein-theoretic detection
  • Trisections and Kirby calculus

  • Mikhail Khovanov  — intended year-level external leader and categorical bridge; Johns Hopkins University
  • András Stipsicz  — intended external scientific leader; HUN-REN Alfréd Rényi Institute of Mathematics

The semester should begin with shared examples and conventions, then alternate between concentrated meetings and protected research residence. A focused workshop will bring construction experts and invariant specialists together around a small set of benchmark manifolds. Working groups will continue those calculations through October and November, with a later intensive aimed at a specific bottleneck—gluing, fundamental groups, mapping-class data, or invariant computation. A closing synthesis will record successful comparisons and sharp open problems for Spring 2029.

Preliminary Workshops, Schools, or Focused Meetings

Opening tutorials on four-manifold construction languages and invariant packages

Dates TBA

 
Focused workshop on smooth constructions, geography, and detection

Dates TBA

 
Working-group intensive on gluing and boundary-sensitive invariants

Dates TBA

 
Semester synthesis meeting

Dates TBA

 

Seminars and Working Groups

  • Construction dictionaries: translate among surgeries, sums, fibrations, Kirby diagrams, and trisections
  • Small exotica: identify tractable target manifolds and the strongest available construction/detection packages
  • Boundary data and gluing: test what information survives from compact pieces to closed manifolds
  • Invariant comparison: compute gauge/Floer, symplectic, and categorical data on the same examples

The semester is intended for four-manifold topologists, gauge theorists, Floer theorists, symplectic geometers, experts in Lefschetz fibrations and mapping class groups, Kirby-calculus and trisection specialists, and researchers developing categorical or skein invariants. Faculty, postdoctoral researchers, and graduate students are welcome. Researchers from adjacent areas should consider participating when they bring a technique relevant to explicit constructions, gluing, moduli spaces, smooth invariants, or computable boundary data. Specialists in group presentations, mapping class groups, contact topology, or computational topology may also provide decisive tools on the benchmark examples even when four-manifolds are not their primary field. Researchers who can make difficult examples computable or convert a geometric construction into explicit algebraic data will be especially useful to the working groups.

  • Extended residence around one of the benchmark construction/detection problems.
  • Month-scale visits timed to overlap with a working group.
  • Short technical visits for a calculation or comparison.
  • Participation in tutorials, the focused workshop, or the working-group intensive.
  • Weekly research seminar and informal blackboard sessions.
  • Expression of interest through the TIAMS participation form.

Junior researchers will have a direct role in carrying examples between formalisms. Tutorials should give enough command of construction diagrams and major invariant technologies to make seminar discussions usable, while working groups provide narrower tasks that can develop into research projects. Students and postdocs will be encouraged to present partial calculations early, compare conventions with experts, and learn which hypotheses are essential before investing in a long computation. They should also have access to diagram-checking and computation sessions where technical mistakes can be found before they become embedded in a larger argument.

Possible outcomes include new constructions, simplifications or equivalences among known constructions, invariant computations, gluing and functoriality results, obstructions to proposed constructions, comparison tables for benchmark examples, open-problem documents, and collaborations that continue into the Spring categorification program.

Use the TIAMS Program Participation form to describe the constructions, invariants, or benchmark examples you want to study, preferred dates, and possible working-group contribution.