Andrei Agrachev
Floer homology is a good example of homological invariants living in the infinite dimension. We suggest a way to construct this kind of invariants using only soft essentially finite-dimensional tools; no hard analysis or PDE is involved. This work is partially inspired by the M. Gromov's survey ``Soft and hard symplectic geometry'' at ICM-86.
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Floer invented his theory in the mid eighties in order to prove the Arnol'd conjectures on the number of fixed point of Hamiltonian diffeomorphisms and Lagrangian intersections. Over the last thirty years, many versions of Floer homology have been constructed. In symplectic and contact dynamics and geometry they have become a principal tool, with applications that go far beyond the Arnol'd conjectures: The proof of the Conley conjecture and of many instances of the Weinstein conjecture, rigidity results on Lagrangian submanifolds and on the group of symplectomorphisms, lower bounds for the top…
Topological 8d $\mathcal{N}=1$ Gauge Theory: Novel Floer Homologies, and $A_\infty$-categories of Six, Five, and Four-Manifolds
从拓扑扭曲的8维N=1规范理论出发，定义七、六、五维流形的新型规范理论Floer同调，并导出六、五、四维流形的Fukaya-Seidel型A_∞-范畴，实现Donaldson-Thomas等构型的范畴化，建立Atiyah-Floer型对偶。
In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold $M$ with codimension-$3$ oriented Riemannian foliation $F$. Under a certain topological condition, we construct the basic Seiberg-Witten invariant and the monopole Floer homologies $\bar{HM}(M,F,\mfs;Γ),~\hat{HM}(M,F,\mfs;Γ), ~\widecheck{HM}(M,F,\mfs;Γ)$, for each transverse \spinc structure $\mfs$, where $Γ$ is a complete local system. We will show that these homologies are independent of the bundle-like metric and generic perturbation. The major difference between the basic monopole Floer homologie…
Topological 5d $\mathcal {N} = 2$ Gauge Theory: Novel Floer Homologies, their Dualities, and an $A_\infty$-category of Three-Manifolds
本文利用拓扑扭曲 5d $\mathcal{N}=2$ 规范理论构造多流形新型 Floer 同调，揭示其 Atiyah-Floer、S-对偶与 Langlands 关系，并给出三流形 Hitchin 构型的 Fukaya-Seidel 型 $A_\infty$-范畴。
Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture, (ii) Munoz's theorem relating quantum and instanton Floer cohomology, (iii) their monopole counterparts, and (iv) their higher rank generalizations. In the case where the boundary is a Seifert manifold, one can also relate its instanton Floer homology to modules of an affine algebra via a 2d A-model with target the based loop group. As an offshoot, we will be able to demonstrate a…
Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification (String Math 2022 Proceedings)
本文是Vafa-Witten理论相关研究论文的精简版，将刊载于美国数学学会出版的《弦数学2022会议论文集》，原论文对应arXiv:2203.17115。
This is an expansion on my talk at the Geometry and Topology conference at McMaster University, May 2004. We outline a program to relate the Heegaard Floer homologies of Ozsvath-Szabo, and Seiberg-Witten-Floer homologies as defined by Kronheimer-Mrowka. The center-piece of this program is the construction of an intermediate version of Floer theory, which exhibits characteristics of both theories.
Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification
该研究在完整Vafa-Witten方程的模空间设定下重新研究Vafa-Witten理论，从物理推导新型不变量、Floer同调及几何Langlands对应等结果，揭示了数学多分支与物理拓扑场论的联系。
Topological 5d $\mathcal{N} = 2$ Gauge Theories: Mirror Symmetry and Langlands Duality of $A_\infty$-categories of Floer Homologies
研究五维空间中拓扑 $\mathcal{N} = 2$ 诺特定理理论的镜像对称与 $A_\infty$-范畴的朗兰兹对称性，通过流形的 gauge 理论解释，揭示 Floer 同调范畴的对偶关系及 symplectic 分类对应。
Topological Gauge Theories with Sixteen Supercharges: Higher $A_\infty$-categorification of Floer Homologies
本文通过3d和2d有质量Landau-Ginzburg模型解释，推导出新型A∞-范畴，2-范畴化Floer同调，并提供物理证明和数学猜想的推广。
In this paper we investigate the properties of a semi-linear problem on a spin manifold involving the Dirac operator, through the construction of Rabinowitz-Floer homology groups. We give several existence results for sub-critical and critical non-linearities as application of the computation of the different homologies.
For a rational homology 3-sphere $Y$ with a $\spinc$ structure $\s$, we show that simple algebraic manipulations of our construction of equivariant Seiberg-Witten Floer homology lead to a collection of variants which are topological invariants. We establish exact sequences relating them, we show that they satisfy a duality under orientation reversal, and we explain their relation to ou previous construction of equivariant Seiberg-Witten Floer (co)homologies. We conjecture the equivalence of these versions of equivariant Seiberg-Witten Floer homology with the Heegaard Floer invariants introduce…
This paper is concerned with the rational symplectic field theory in the Floer case. For this observe that in the general geometric setup for symplectic field theory the contact manifolds can be replaced by mapping tori of symplectic manifolds with symplectomorphisms. While the cylindrical contact homology is given by the Floer homologies of powers of the symplectomorphism, the other algebraic invariants of symplectic field theory provide natural generalizations of symplectic Floer homology. For symplectically aspherical manifolds and Hamiltonian symplectomorphisms we study the moduli spaces o…
Surgery exact triangles in various 3-manifold Floer homology theories provide an important tool in studying and computing the relevant Floer homology groups. These exact triangles relate the invariants of 3-manifolds, obtained by three different Dehn surgeries on a fixed knot. In this paper, the behavior of $SU(N)$-instanton Floer homology with respect to Dehn surgery is studied. In particular, it is shown that there are surgery exact tetragons and pentagons, respectively, for $SU(3)$- and $SU(4)$-instanton Floer homologies. It is also conjectured that $SU(N)$-instanton Floer homology in gener…
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