Andreas Floer

Bio: Andreas Floer is an academic researcher from Courant Institute of Mathematical Sciences. The author has contributed to research in topics: Symplectic geometry & Symplectomorphism. The author has an hindex of 16, co-authored 20 publications receiving 3880 citations. Previous affiliations of Andreas Floer include ETH Zurich.

Morse theory for Lagrangian intersections

Abstract: Soit P une variete symplectique compacte et soit L⊂P une sous-variete lagrangienne avec π 2 (P,L)=0. Pour un diffeomorphisme exact φ de P avec la propriete que φ(L) coupe L transversalement, on demontre une inegalite de Morse reliant l'ensemble φ(L)∩L a la cohomologie de L

Symplectic fixed points and holomorphic spheres

Abstract: LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism onP is nondegenerate, then it satisfies the Morse inequalities onP. We also discuss fixed point estimates for general exact diffeomorphisms.

An instanton-invariant for 3-manifolds

Abstract: To an oriented closed 3-dimensional manifoldM withH1(M, ℤ)=0, we assign a ℤ8-graded homology groupI*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on the space of instantons onM×ℝ.

The unregularized gradient flow of the symplectic action

Abstract: The symplectic action can be defined on the space of smooth paths in a symplectic manifold P which join two Lagrangian submanifolds of P. To pursue a new approach to the variational theory of this function, we define on a subset of the path space the flow whose trajectories are given by the solutions of the Cauchy-Riemann equation with respect to a suitable almost complex structure on P. In particular, we prove compactness and transversality results for the set of bounded trajectories.

Transversality in elliptic Morse theory for the symplectic action

Abstract: Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder These results play a central role in the denition of symplectic Floer homology and hence in the proof of the Arnold conjecture There is currently no other reference to these transversality results in the open literature Our approach does not require Aronszajn’s theorem Instead we derive the unique continuation theorem from a generalization of the Carleman similarity principle

Quantum field theory and the Jones polynomial

Abstract: It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS 3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

Topological quantum field theory

Abstract: A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments in topology of low dimensional manifolds; the Donaldson polynomial invariants of four manifolds and the Floer groups of three manifolds appear naturally. The model may also be interesting from a physical viewpoint; it is in a sense a generally covariant quantum field theory, albeit one in which general covariance is unbroken, there are no gravitons, and the only excitations are topological.

Pseudo holomorphic curves in symplectic manifolds

Abstract: Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.

Topological sigma models

Abstract: A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surfaceΣ to an arbitrary almost complex manifoldM. It possesses a fermionic BRST-like symmetry, conserved for arbitraryΣ, and obeyingQ 2=0. In a suitable version, the quantum ground states are the 1+1 dimensional Floer groups. The correlation functions of the BRST-invariant operators are invariants (depending only on the homotopy type of the almost complex structure ofM) similar to those that have entered in recent work of Gromov on symplectic geometry. The model can be coupled to dynamical gravitational or gauge fields while preserving the fermionic symmetry; some observations by Atiyah suggest that the latter coupling may be related to the Jones polynomial of knot theory. From the point of view of string theory, the main novelty of this type of sigma model is that the graviton vertex operator is a BRST commutator. Thus, models of this type may correspond to a realization at the level of string theory of an unbroken phase of quantum gravity.

A categorification of the Jones polynomial

Abstract: Author(s): Khovanov, Mikhail | Abstract: We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.