On this page, I introduce my research group for prospective graduate students and postdoctoral fellows.
My field of expertise is
Yang–Mills and Seiberg–Witten gauge theory and their applications to 3- and 4-dimensional manifolds.
Below, I briefly describe the background and details of my research.

It is well known that the classification problem of manifolds depends significantly on the dimension. While classification theory in dimensions five and higher has been established, such results do not directly extend to four dimensions. Even for a single smooth 4-manifold, a complete classification of its differentiable structures is still far from being achieved. Nevertheless, gauge theory provides powerful tools that capture uniquely four-dimensional phenomena, and exploring these possibilities has been the central theme of my research. For example, a famous result states that the smooth structures on Euclidean 4-space form an uncountable family.

Research on 4-manifolds extends beyond differentiable structures and also concerns submanifolds and additional geometric structures (such as Riemannian metrics and symplectic structures).

My own research on 4-manifolds so far has addressed topics such as:

• Differentiable structures on 4-manifolds with boundary

• 2- and 3-dimensional submanifolds of 4-manifolds (e.g. knot sliceness, concordance, minimal genus problems, 2-knots)

• Existence problems of positive scalar curvature metrics on 4-manifolds

• Symplectic structures on 4-manifolds with boundary, and contact structures on their boundaries

• Symplectic submanifolds in 4-dimensional symplectic manifolds (e.g. transverse knots)

• Families of 4-manifolds with boundary
  
  

These problems have been studied using gauge theory. In gauge theory on 4-manifolds, one studies moduli spaces of solutions to certain nonlinear PDEs originating in physics, namely the Yang–Mills and Seiberg–Witten equations, modulo the action of the gauge group. By analyzing the geometry of these moduli spaces and integrating appropriate cohomology classes, one defines the Donaldson invariants and Seiberg–Witten invariants. These are powerful differential-topological invariants that can distinguish homeomorphic but non-diffeomorphic 4-manifolds.

These invariants are originally defined for closed smooth 4-manifolds, but natural extensions exist for 4-manifolds with boundary, leading to theories known as instanton Floer homology and monopole Floer homology. Floer homologies are constructed as infinite-dimensional analogues of Morse homology, and admit TQFT-type formulations. A refinement of monopole Floer homology, called the Floer homotopy type, involves equivariant homotopy theory, which has been an active area of study.

The foundations of Yang–Mills and Seiberg–Witten gauge theory (and Floer theory) draw on differential geometry, algebraic topology, and functional analysis, including:

• Principal bundles, connections, holonomy, curvature, associated bundles, flat connections, and twisted cohomology (local coefficient systems); Chern–Weil theory, both abstract and computational aspects

• Vector bundles and characteristic classes, classification of bundles via obstruction theory, topological K-theory, Bott periodicity

• Analytical tools for elliptic operators: Sobolev spaces, Rellich’s lemma, a priori estimates, properties of Fredholm operators, Hodge decomposition (including manifolds with boundary), spectral theory of compact self-adjoint operators

• Spin geometry in dimensions 3 and 4, Dirac and Atiyah–Hitchin–Singer operators, index theorems (including Atiyah–Patodi–Singer), families index theory, excision, localization techniques

• Fredholm maps between Banach manifolds, the implicit function theorem, transversality and Sard’s theorem in infinite dimensions

• Morse theory in infinite dimensions: construction of trajectory spaces, independence of choices, relation to cellular homology, Morse–Bott situations, several constructions of equivariant Morse homology

Because gauge theory applies to 3- and 4-manifolds, it is naturally related to many topics in low-dimensional topology, for example:

• Foundations of 4-manifold topology: intersection forms, orientations, spin structures, Kirby calculus, elliptic surfaces, Lefschetz fibrations, branched covers, logarithmic transforms, Fintushel–Stern knot surgery, surgery in dimension five

• Foundations of 3-manifold and knot theory: classification of Seifert 3-manifolds, constructions via surgery and branched covers, examples of 2-bridge knots, torus knots, hyperbolic knots, satellite knots, algebraic knots; invariants such as the Alexander and Jones polynomials, signatures, algebraic concordance, and Blanchfield pairings

• Constructions of surfaces in 4-manifolds: divisors, blow-ups of immersed curves, ribbon disks, twist-spins, motion pictures, surface diagrams, rim surgery; analysis of branched covers along these surfaces

• Legendrian and transverse knots: Thurston–Bennequin number, self-linking number, inequalities, front projections, braid representations, quasipositive surfaces

• Symplectic (Stein) structures on 4-manifolds, contact structures on 3-manifolds via surgery, branched covers along transverse knots, symplectic surfaces and their branched covers

• Computations in oriented and spin cobordism in dimensions 3 and 4
  
  

Classical 3-manifold invariants related to Yang–Mills theory also form part of the background:

• The Casson invariant of homology 3-spheres (surgery formulas, formulations using SU(2) character varieties)

• The knot signature (formulations using traceless SU(2) character varieties)

• Relations between the Alexander polynomial and SU(2) character varieties
  
  

In the study of Floer homotopy types, additional tools are useful:

• Spectral sequences associated to filtrations (e.g. from fiber bundles) 

• Stable homotopy theory (stable homotopy groups of spheres), equivariant homotopy theory (Borsuk–Ulam theorems, classification of equivariant maps, localization theorems, Sullivan’s conjecture)

• Equivariant K-theory (Bott periodicity, Thom isomorphism in the equivariant setting)
  
  

Floer theories arising in gauge theory, such as instanton and monopole Floer homology, are closely related to Floer theories in symplectic geometry, such as Lagrangian intersection Floer homology, as well as to Heegaard Floer homology. In fact, a deep isomorphism is known:

Monopole Floer homology  ≅  Heegaard Floer homology

Heegaard Floer homology admits surgery formulas, such as the relation between Heegaard Floer homology of Dehn surgeries and knot Floer homology, and has a rich supply of computational techniques. As a result, its structures are much better understood than those of gauge-theoretic Floer homologies. Gauge-theoretic Floer theories also admit partial surgery formulas, organized into sutured instanton and sutured monopole Floer homology, which remain active research areas.

Gauge-theoretic knot Floer theories (such as singular instanton knot homology, equivariant or real Seiberg–Witten knot Floer theories) are also deeply related to Khovanov homology. 

Orbifold techniques provide another framework for studying knots and surfaces in gauge theory, involving:

• Geometry of orbifolds (metrics, differential forms, bundles), orbifold cohomology (relation to the Borel construction), and orbifold fundamental groups.
  
  

The above describes my own research, but prospective Master’s and Ph.D. students are not restricted to these exact topics. Nor is it required that all of this background be mastered before starting graduate study. Students and postdocs with an interest in related fields, who wish to pursue research under my supervision, are encouraged to contact me at an early stage.