2013 in Glacier National Park  2012 at IAS 
Symplectic Geometry: 
A proof of the Arnold conjecture by polyfold techniques, with P.Albers, B.Filippenko, J.Fish, almost preprint. The (weak) Arnold conjecture is a lower bound on the number of periodic orbits of any periodic Hamiltonian system in terms of the rank of homology of the underlying phase space. Floer's proof obtains this bound by constructing Floer homology and identifying it with Morse homology via S^{1}equivariant transversality. This proof was extended to general symplectic manifolds by various refinements of algebraic and transversality techniques, the latter of which are currently undergoing fundamental revision. An alternative approach outlined by PiunikhinSalamonSchwarz is to construct morphisms between the Morse and Hamiltonian Floer complexes but again requires S^{1}equivariant transversality to prove the isomorphism property. We refine the latter approach algebraically by constructing PSS and SSP morphisms and proving that SSP∘PSS is an isomorphism of Morse homology that factors through the Floer complex. This implies the weak Arnold conjecture without requiring equivariant transversality or even Floer homology. To obtain a proof for general symplectic manifolds, we instead build on the expected HoferWysockiZehnder construction of SFT moduli spaces by polyfold Fredholm sections. Assuming this, we can construct all moduli spaces needed for the modified PSS program by fiber products of SFTtype pseudoholomorphic curve moduli spaces with smooth structures on compactified spaces of halfinfinite Morse trajectories. 
Floer cohomology and multiply covered composition of Lagrangian correspondences, almost preprint and on video (starting at 20:00). We prove a bijection between moduli spaces of Floer trajectories under geometric composition of Lagrangian correspondences, relaxing previous monotonicity assumptions and generalizing to multiply covered composition. As sample application, we calculate the Floer cohomology between ℝP^{n} and Clifford torus in ℂP^{n}, and between the Chekanov torus and a product torus in S^{2}×S^{2}. 

Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts, with N.Bottman preprint and on video. We establish a Gromov compactness theorem for strip shrinking in pseudoholomorphic quilts when composition of Lagrangian correspondences is immersed. In particular, we show that figure eight bubbling occurs in the limit, argue that this is a codimension0 effect, and predict its algebraic consequences  geometric composition extends to a curved A∞bifunctor, in particular the associated Floer complexes are isomorphic after a figure eight correction of the bounding cochain. An appendix with Felix Schm"aschke provides examples of nontrivial figure eight bubbles. 

A_{∞} functors for Lagrangian correspondences, with S.Ma'u, C.Woodward, preprint with disclosure. We construct A_{∞} functors associated to monotone Lagrangian correspondences and show that the composition of A_{∞} functors is homotopic to the functor for the geometric composition. By constructing associated natural transformations, we obtain an A_{∞} functor on extended Fukaya categories Fuk(M_{0},M_{1}) → Funk(Fuk(M_{0}),Fuk(M_{1})), which is the symplectic analogue of a quasiequivalence D^{b}_{∞}(X ×X) ≅ Fun(D^{b}_{∞}(X),D^{b}_{∞}(X)) for certain derived categories of coherent sheaves on a projective variety X. 

Exact triangle for fibered Dehn twists, with C.Woodward, preprint with disclosure. We use quilted Floer theory to construct categoryvalued field theories for tangles and links using Lagrangian correspondences arising from moduli spaces of flat connections on punctured surfaces. The formal properties of the field theories are similar to those of KhovanovRozansky homology and the gaugetheoretic invariants developed by KronheimerMrowka. As an application, we show the nontriviality of certain elements in the symplectic mapping class groups of moduli spaces of flat connections on punctured spheres. 

Functoriality for Lagrangian correspondences in Floer theory, with C.Woodward, Quantum Topology . Using quilted Floer cohomology and relative quilt invariants, we associate a functor on DonaldsonFukaya categories of generalized Lagrangian submanifolds, Φ(L_{01}) : Don(M_{0}) → Don(M_{1}) to any Lagrangian correspondence L_{01} ⊂ M_{0}^{} × M_{1}. We show that ``categorification commutes with composition'', Φ(L_{01}) ∘Φ(L_{12})≅Φ(L_{01}∘ L_{12}), if the geometric composition is embedded. These constructions are extended to a symplectic 2category which contains all Lagrangian correspondences as morphisms, and a categorification 2functor. 

Orientations for pseudoholomorphic discs and quilts, with C.Woodward, preprint . We construct coherent orientations on moduli spaces of pseudoholomorphic quilts and determine the effect of various gluing operations on the orientations. We also investigate the behavior of the orientations under composition of Lagrangian correspondences. 

Pseudoholomorphic quilts, with C.Woodward, J.Symp.Geom. We define relative Floer theoretic invariants arising from moduli spaces of pseudoholomorphic quilts: Collections of pseudoholomorphic maps to various symplectic target spaces with seam conditions on Lagrangian correspondences. As application we construct a morphism on quantum homology associated to any monotone Lagrangian correspondence and show how it interacts with the ring structure. 

Quilted Floer trajectories with constant components, with C.Woodward, Geom.&Topol. We fill a gap in the construction of quilted Floer cohomology by addressing trajectories for which some but not all components are constant. Namely, we show that for generic sets of split Hamiltonian perturbations and split almost complex structures, the moduli spaces of parametrized quilted Floer trajectories of a given index are smooth of expected dimension. An additional benefit of the generic split Hamiltonian perturbations is that they perturb the given cyclic Lagrangian correspondence such that any geometric composition of its factors is transverse and hence immersed. 

Quilted Floer cohomology, with C.Woodward, Geom.&Topol. . We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences and establish gradings. For sequences related by geometric composition, we establish an isomorphism of the graded Floer cohomologies. We give applications to calculations of Floer cohomology, nondisplaceability of Lagrangian correspondences, and transfer of nondisplaceability under geometric composition. 

Floer cohomology and geometric composition of Lagrangian correspondences, with C.Woodward, Advances in Math. . We prove an isomorphism between Floer cohomologies HF(L_{0} × L_{12},L_{01}× L_{2}) ≅ HF(L_{0} × L_{2},L_{01} ∘ L_{12}) under geometric compositionin exact and monotone settings. The proof is by an adiabatic limit, shrinking the relative width of a pseudoholomorphic strip to zero, and excluding a novel type of "figure eight" bubbling. 

Asymptotic operators in symplectizations and the ConleyZehnder index, diploma thesis . The main part concerns the topological structure of Sp(n), gives a spectral flow construction of the ConleyZehnder index, and compares it to an axiomatic definition and a construction in terms of winding numbers. An extra chapter provides an improvement of H.Hofer, K.Wysocki, E.Zehnder: Properties of Pseudoholomorphic Curves in Symplectizations III: Fredholm theory. 

Topology: 
Floer Field Philosophy, AWM Symposium Proceedings . Floer field theory is a construction principle for e.g. 3manifold invariants via decomposition in a bordism category and a functor to the symplectic category, and is conjectured to have natural 4dimensional extensions. This survey provides an introduction to the categorical language for the construction and extension principles and provides the basic intuition for two gauge theoretic examples which conceptually frame AtiyahFloer type conjectures in Donaldson theory as well as the relations of Heegaard Floer homology to SeibergWitten theory. 
Connected Cerf theory, with D.Gay, C.Woodward, preprint . We review the theory of Cerf describing the decomposition of cobordisms into elementary cobordisms, and the Cerf moves between different decompositions. We put special emphasis on connectedness and define Cerf decompositions as decompositions of morphisms in the category of connected manifolds and connected cobordisms. In addition, we discuss the cyclic Cerf theory of Morse functions to S^{1}. 

Floer Field Theory for coprime rank and degree, with C.Woodward, preprint with disclosure. We construct functorvalued invariants of threedimensional cobordisms using quilted Floer theory in moduli spaces of bundles with fixed determinant and degree coprime to the rank. The underlying categories are the DonaldsonFukaya categories of the moduli spaces of bundles over the boundary components. The resulting threemanifold invariants are symplectic versions of instanton Floer homology in the reduciblefree case. 

Floer Field Theory for tangles, with C.Woodward, preprint with disclosure. We use quilted Floer theory to construct categoryvalued field theories for tangles and links using Lagrangian correspondences arising from moduli spaces of flat connections on punctured surfaces. The formal properties of the field theories are similar to those of KhovanovRozansky homology and the gaugetheoretic invariants developed by KronheimerMrowka. As an application, we show the nontriviality of certain elements in the symplectic mapping class groups of moduli spaces of flat connections on punctured spheres. 

Gauge Theory: 
An openclosed isomorphism for instanton Floer homology, with D.Salamon, preprint. Given a splitting Z=YuH of an integral homology 3sphere into a 3manifold Y and a handlebody H, we construct a natural isomorphism from the instanton Floer homology of Z to the instanton Floer homology of Y with Lagrangian boundary conditions L_{H}. This proves in particular a well defined part of the AtiyahFloer conjecture. 
L^{2}topology and Lagrangians in the space of connections over a Riemann surface, with T.Mrowka, Geom.&Func.Analysis. We examine the L^{2}topology of the gauge orbits over a closed Riemann surface. We prove a subtle local slice theorem based on the divcurl Lemma of harmonic analysis, and deduce local connectedness of the gauge orbits, up to an action of the stabilizer group at reducible connections. Based on this, we generalize bubbling and removable singularity results for instantons to general (not geometrically constructed) gauge invariant Lagrangian boundary conditions, thus extending instanton Floer homology to this class. 

Instanton Floer homology with Lagrangian boundary conditions, with D.Salamon, Geom.&Topol.. We define instanton Floer homology groups for a compact oriented 3manifold with boundary and a gauge invariant ``irreducible monotone'' Lagrangian submanifold of the space of SU(2)connections over the boundary (arising from a handle body with the same boundary, or general by the above additional analysis). This invariant is built to interpolate between the closed instanton Floer theory and the Lagrangian Floer theory in the AtiyahFloer conjecture. 

Energy identity for antiselfdual instantons on ℂ×Σ, Math.Res.Lett. We show that SU(2)connections of finite energy on the product of complex plane and Riemann surface have integral charge. This supports a conjecture that finite energy ASD instantons on ℂ×Σ extend to stable holomorphic bundles over ℂP^{1}×Σ, whose first Chern number is this charge. A similar charge identity for a larger class of principal bundles over Riemann surfaces. 

Lagrangian boundary conditions for antiselfdual instantons and the AtiyahFloer conjecture, J.Symp.Geom. We explain an approach to the AtiyahFloer conjecture via the instanton Floer homology with Lagrangian boundary conditions. The paper also provides a guide to the general construction of Floer homologies. 

Energy quantization and mean value inequalities for nonlinear boundary value problems, J.EMS We give a unified statement and proof of a class of well known mean value inequalities for nonnegative functions with a nonlinear bound on the Laplacian. We generalize these to domains with boundary by requiring a nonlinear bound on the normal derivative. This explains the bubbling analysis for holomorphic curves and instantons with Lagrangian boundary conditions in terms of an εregularity result. 

Antiselfdual instantons with Lagrangian boundary conditions II: Bubbling, Comm.Math.Phys. We prove energy quantization and removal of singularity results for instantons on the product ${\mathbb H}^2\times\S$ of half space and a closed Riemann surface, which satisfy a global Lagrangian boundary condition on each boundary slice $\{z\}\times\S$, arising from a handle body bounding $\S$. For instantons on $({\mathbb R}^2\setminus\{0\})\times\S$, our methods provide a new approach to the analysis of codimension $2$ singularities. 

Antiselfdual instantons with Lagrangian boundary conditions I: Elliptic theory, Comm.Math.Phys. We study nonlocal Lagrangian boundary conditions for instantons on 4manifolds with a spacetime splitting of the boundary. We establish the basic regularity and compactness properties as well as the Fredholm theory in a compact model case. 

Banach space valued CauchyRiemann equations with totally real boundary conditions, Comm.Cont.Math. We prove elliptic regularity result for CauchyRiemann equations in complex Banach spaces with totally real boundary conditions. Secondly, we construct examples of such totally real submanifolds in the space of L^{p}connections over a Riemann surface Σ, arising from handle bodies bounding Σ. 

Uhlenbeck Compactness, EMS Series; Introduction, Erratum, Erratum 2. This textbook is a selfcontained exposition of Uhlenbeck compactness with all analytic details. It covers the Neumann problem with inhomogeneous boundary conditions, Uhlenbeck gauge, patching, weak compactness, local slice theorems, and strong compactness for YangMills connections. 

Antiselfdual instantons with Lagrangian boundary conditions, Ph.D. thesis: Beyond the publications Antiselfdual ... I and Banach space …, this thesis contains the partial desciption of a bubbling phenomenon leading to holomorphic halfplanes in the Banach space of connections over a Riemann surface, satisfying Lagrangian boundary conditions. 

Regularization: 
A_{∞}structures from Morse trees with pseudoholomorphic disks, with Jiayong Li, preliminary draft . Floer homology for Lagrangian submanifolds of symplectic manifolds is in general not well defined due to the fact that pseudoholomorphic disk bubbles appear in codimension 1. FukayaOhOhtaOno captured these obstructions in an A_{∞}structure generated by a certain geometric subclass of cochains on the Lagrangians and an inverse limit involving integrations against virtual fundamental chains. In order to construct finitely generated A_{∞}algebras for any compact Lagrangian submanifold, we extrapolate previous ideas of clusters and pearly strings to obtain moduli spaces of pseudoholomorphic disk trees, a special case of which are Morse trees with ghost bubbles at the vertices. We sketch the construction and explain why Novikov rings over ℤ or even ℤ_{2} suffice if sphere bubbling is a priori excluded. Towards a rigorous construction in terms of Mpolyfold Fredholm sections, we construct the appropriate generalization of a DeligneMumford space and use it to prove injectivity of the future Mpolyfold charts. 
Smooth Kuranishi atlases with isotropy, with D.McDuff, preprint . Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions, given by smooth sections that are equivariant under a finite isotropy group. Building on our notions of topological Kuranishi atlases and perturbation constructions in the case of trivial isotropy, we develop a theory of Kuranishi atlases and cobordisms that transparently resolves the challenges posed by nontrivial isotropy. We assign to a cobordism class of weak Kuranishi atlases both a virtual moduli cycle (VMC  a cobordism class of weighted branched manifolds) and a virtual fundamental class (VFC  a Cech homology class). 

The fundamental class of smooth Kuranishi atlases with trivial isotropy, with D.McDuff, preprint . Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions. The first sections of this paper discuss topological, algebraic and analytic challenges that arise in this program. (See additional literature annotation for more details.) We then develop a theory of Kuranishi atlases and cobordisms that transparently resolves these challenges, for simplicity concentrating on the case of trivial isotropy. In this case, we assign to a cobordism class of additive weak Kuranishi atlases both a virtual moduli cycle (VMC  a cobordism class of smooth manifolds) and a virtual fundamental class (VFC  a Cech homology class). We moreover show that such Kuranishi atlases exist on simple GromovWitten moduli spaces and develop the technical results in a manner that easily transfers to more general settings. 

The topology of Kuranishi atlases, with D.McDuff, preprint. Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Starting from the same core idea (patching local finite dimensional reductions) we develop a theory of topological Kuranishi atlases and cobordisms that transparently resolves algebraic and topological challenges in this virtual regularization approach. It applies to any Kuranishitype setting, e.g. atlases with isotropy, boundary and corners, or lack of differentiable structure. 

How not to construct obstruction bundles, with D.Salamon, note. As illustration of obstacles to regularizations defying intuition we debunk a folk understanding of how obstruction bundles might be constructed. Let H be a Banach space and V_{i}(s) ⊂ H for i=1,2 two smooth families of finite dimensional subspaces. Then it is not necessarily true that there exists a smooth family W(s) ⊂ H of finite dimensional subspaces such that V_{i}(s) ⊂ W(s) for i=1,2. In fact, there exists a smooth family of unit vectors in the sequence space v(s) ∈ l^{2} such that any continuous family of subspaces W(s)⊂ l^{2} which contain both v(0) and v(s) is the entire space. 

Polyfolds: A first and second look, with R.Golovko, O.Fabert, J.Fish, preprint . Polyfold theory was developed by HoferWysockiZehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs. It aims to systematically address the common difficulties of ``compactification'' and ``transversality'' with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. Since both theory and applications are only slowly becoming available in formidable length and technicality, we try to shine metamathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and theorems of polyfold theory into a streamlined exposition, and outline their application at the example of Morse theory. 

Fredholm notions in scale calculus and Hamiltonian Floer theory, preprint . We give an equivalent definition of the Fredholm property for linear operators on scale Banach spaces and introduce a nonlinear scale Fredholm property with respect to a splitting of the domain. The latter implies the Fredholm property defined by HoferWysockiZehnder in terms of contraction germs, but is easier to check in practice and holds in applications to holomorphic curve moduli spaces. We demonstrate this at the example of trajectory breaking in Hamiltonian Floer theory, thus providing the analytic core for a polyfold construction of Hamiltonian Floer homology. 

Kuranishi structures with trivial isotropy  the 2013 state of affairs, with D.McDuff, preprint . We give a survey of regularization techniques in symplectic topology, pointing to some general analytic issues, and discussing topological issues of the Kuranishi structure approach. To focus on the most fundamental issues, we restrict our discussion to moduli spaces of holomorphic spheres without nodes or nontrivial isotropy. We provide a new abstract framework of (weak) Kuranishi structures satisfying an a priory transversality condition (additivity), including notions of orientations and cobordisms, and show how such (additive weak) structures can be obtained for GromovWitten moduli spaces. By developing categorical refinement techniques and a subtle perturbation scheme for sections in categorical bundles with nonmetrizable realization, we then associate a well defined \v{C}ech homology class and cobordism class of closed manifolds to any cobordism class of additive weak Kuranishi structures with trivial isotropy on a compact metrizable space. 

Smooth structures on Morse trajectory spaces, featuring finite ends and associative gluing, Proceedings of the Freedman Fest, Geom.Topol.Monogr. We construct natural manifold with corner structures and associative gluing maps on compactifications of spaces of infinite, half infinite, and finite Morse flow lines. In the case of a Euclidean normal form near critical points, we prove smoothness of evaluation maps, which allows for fiber product constructions combining Morse trajectory spaces and polyfold Fredholm sections. 

The area of squares and rectangles, work in progress 