We construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two differe
We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, omega, tau)$ with nonempty and compact real part $L={rm Fix}(tau)$. For given $Lambdain (0, +infty]$ and $minNcup{0}$ we show the equivalence of
the following two claims: (i) $sharp(Lcapphi^H_1(L))ge m$ for any Hamiltonian function $Hin C_0^infty([0, 1]times M)$ with Hofers norm $|H|<Lambda$; (ii) $sharp {cal P}(H,tau)ge m$ for every $Hin C^infty_0(R/Ztimes M)$ satisfying $H(t,x)=H(-t,tau(x));forall (t,x)inmathbb{R}times M$ and with Hofers norm $|H|<2Lambda$, where ${cal P}(H, tau)$ is the set of all $1$-periodic solutions of $dot{x}(t)=X_{H}(t,x(t))$ satisfying $x(-t)=tau(x(t));forall tinR$ (which are also called brake orbits sometimes). Suppose that $(M, omega)$ is geometrical bounded for some $Jin{cal J}(M,omega)$ with $tau^ast J=-J$ and has a rationality index $r_omega>0$ or $r_omega=+infty$. Using Hofers method we prove that if the Hamiltonian $H$ in (ii) above has Hofers norm $|H|<r_omega$ then $sharp(Lcapphi^H_1(L))gesharp {cal P}_0(H,tau)ge {rm Cuplength}_{F}(L)$ for $F=Z_2$, and further for $F=Z$ if $L$ is orientable, where ${cal P}_0(H,tau)$ consists of all contractible solutions in ${cal P}(H,tau)$.
We revisit Atiyah and Botts study of Morse theory for the Yang-Mills functional over a Riemann surface, and establish new formulas for the minimum codimension of a (non-semi-stable) stratum. These results yield the exact connectivity of the natural m
ap (C_{min} E)//G(E) --> Map^E (M, BU(n)) from the homotopy orbits of the space of central Yang-Mills connections to the classifying space of the gauge group G(E). All of these results carry over to non-orientable surfaces via Ho and Lius non-orientable Yang-Mills theory. A somewhat less detailed version of this paper (titled On the Yang-Mills stratification for surfaces) will appear in the Proceedings of the AMS.
We prove that for regular contact forms there exists a bijective correspondence between the $C^0$ limits of sequences of smooth strictly contact isotopies and the limits with respect to the contact distance of their corresponding Hamiltonians.
We prove a quantum version of the localization formula of Witten that relates invariants of a git quotient with the equivariant invariants of the action. Using the formula we prove a quantum version of an abelianization formula of S. Martin relating
invariants of geometric invariant theory quotients by a group and its maximal torus, conjectured by Bertram, Ciocan-Fontanine, and Kim. By similar techniques we prove a quantum Lefschetz principle for holomorphic symplectic reductions. As an application, we give a formula for the fundamental solution to the quantum differential equation (qde) for the moduli space of points on the projective line and for the smoothed moduli space of framed sheaves on the projective plane (a Nakajima quiver variety).
We use the technique of stabilizing divisors introduced by Cieliebak-Mohnke to construct finite dimensional, strictly unital Fukaya algebras of compact, oriented, relatively spin Lagrangians in compact symplectic manifolds with rational symplectic cl
asses. The homotopy type of the algebra and the moduli space of solutions to the weak Maurer-Cartan equation are shown to be independent of the choice of perturbation data. The Floer cohomology is the cohomology of a complex of vector bundles over the space of solutions to the weak Maurer-Cartan equation and is shown to be independent of the choice of perturbation data up to gauge equivalence.
We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians i
n simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was also obtained by Nadler). The ideas in that paper were extended by Abouzaid who proved that vanishing Maslov class alone implies homotopy equivalence. In this paper we present a short proof of the fact that any exact Lagrangian with vanishing Maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism. When the fundamental group of the base is pro-finite this implies homotopy equivalence.
We establish an existence $h$-principle for symplectic cobordisms of dimension $2n>4$ with concave overtwisted contact boundary.
We obtain a formula for the Heegaard Floer homology (hat theory) of the three-manifold $Y(K_1,K_2)$ obtained by splicing the complements of the knots $K_isubset Y_i$, $i=1,2$, in terms of the knot Floer homology of $K_1$ and $K_2$. We also present a
few applications. If $h_n^i$ denotes the rank of the Heegaard Floer group $widehat{mathrm{HFK}}$ for the knot obtained by $n$-surgery over $K_i$ we show that the rank of $widehat{mathrm{HF}}(Y(K_1,K_2))$ is bounded below by $$big|(h_infty^1-h_1^1)(h_infty^2-h_1^2)- (h_0^1-h_1^1)(h_0^2-h_1^2)big|.$$ We also show that if splicing the complement of a knot $Ksubset Y$ with the trefoil complements gives a homology sphere $L$-space then $K$ is trivial and $Y$ is a homology sphere $L$-space.
We show that the Siegel upper half space $Sigma_{d}$ is identified with the Marsden-Weinstein quotient obtained by symplectic reduction of the cotangent bundle $T^{*}mathbb{R}^{2d^{2}}$ with $mathsf{O}(2d)$-symmetry. The reduced symplectic form on $S
igma_{d}$ corresponding to the standard symplectic form on $T^{*}mathbb{R}^{2d^{2}}$ turns out to be a constant multiple of the symplectic form on $Sigma_{d}$ obtained by Siegel. Our motivation is to understand the geometry behind two different formulations of the Gaussian wave packet dynamics commonly used in semiclassical mechanics. Specifically, we show that the two formulations are related via the symplectic reduction.
