Abstract:
We give an introduction to the physics and mathematics involved in the recently observed relation between topological string theory and knot contact homology and then discuss this relation. The note is based on two lectures given at the Gokova Geometry and Topology Conference, 2013 and reports on joint work by Aganagic, Ng, Vafa, and the author.

Abstract:
We relate the version of rational Symplectic Field Theory for exact Lagrangian cobordisms introduced in [5] with linearized Legendrian contact homology. More precisely, if $L\subset X$ is an exact Lagrangian submanifold of an exact symplectic manifold with convex end $\Lambda\subset Y$, where $Y$ is a contact manifold and $\Lambda$ is a Legendrian submanifold, and if $L$ has empty concave end, then the linearized Legendrian contact cohomology of $\Lambda$, linearized with respect to the augmentation induced by $L$, equals the rational SFT of $(X,L)$. Following ideas of P. Seidel, this equality in combination with a version of Lagrangian Floer cohomology of $L$ leads us to a conjectural exact sequence which in particular implies that if $X=\C^{n}$ then the linearized Legendrian contact cohomology of $\Lambda\subset S^{2n-1}$ is isomorphic to the singular homology of $L$. We outline a proof of the conjecture and show how to interpret the duality exact sequence for linearized contact homology of [6] in terms of the resulting isomorphism.

Abstract:
Regular homotopy classes of immersions of a 3-sphere in 5-space constitute an infinite cyclic group. The classes containing embeddings form a subgroup of index 24. The obstruction for a generic immersion to be regularly homotopic to an embedding is described in terms of geometric invariants of its self intersection. Geometric properties of self intersections are used to construct two invariants J and St of generic immersions which are analogous to Arnold's invariants of plane curves. We prove that J and St are independent first order invariants and that any first order invariant is a linear combination of these. As by-products, some invariants of immersions of 3-spheres in 4-space are obtained. Using them, we find restrictions on the topology of self intersections.

Abstract:
A natural oriented (2k+2)-chain in CP^{2k+1} with boundary twice RP^{2k+1}, its complex shade, is constructed. Via intersection numbers with the shade, a new invariant, the shade number of k-dimensional subvarieties with normal vector fields along their real part, is introduced. For an even-dimensional real variety, the shade number and the Euler number of the complement of the normal vector field in the real normal bundle of its real part agree. For an odd- dimensional orientable real variety, a linear combination of the shade number and the wrapping number (self-linking number) of its real part is independent of the normal vector field and equals the encomplexed writhe as defined by Viro. Shade numbers of varieties without real points and encomplexed writhes of odd- dimensional real varieties are, in a sense, Vassiliev invariants of degree 1. Complex shades of odd-dimensional spheres are constructed. Shade numbers of real subvarieties in spheres have properties analogous to those of their projective counterparts.

Abstract:
We consider properties of the total absolute geodesic curvature functional on circle immersions into a Riemann surface. In particular, we study its behavior under regular homotopies, its infima in regular homotopy classes, and the homotopy types of spaces of its local minima. We consider properties of the total curvature functional on the space of 2-sphere immersions into 3-space. We show that the infimum over all sphere eversions of the maximum of the total curvature during an eversion is at most 8\pi and we establish a non-injectivity result for local minima.

Abstract:
We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$ and $L$ have cylindrical ends. The theory associates to $(X,L)$ a $\Z$-graded chain complex of vector spaces over $\Z_2$, filtered with $k$ filtration levels. The corresponding $k$-level spectral sequence is invariant under deformations of $(X,L)$ and has the following property: if $(X,L)$ is obtained by joining a negative end of a pair $(X',L')$ to a positive end of a pair $(X'',L'')$, then there are natural morphisms from the spectral sequences of $(X',L')$ and of $(X'',L'')$ to the spectral sequence of $(X,L)$. As an application, we show that if $\Lambda\subset Y$ is a Legendrian submanifold of a contact manifold then the spectral sequences associated to $(Y\times\R,\Lambda_k^s\times\R)$, where $Y\times\R$ is the symplectization of $Y$ and where $\Lambda_k^s\subset Y$ is the Legendrian submanifold consisting of $s$ parallel copies of $\Lambda$ subdivided into $k$ subsets, give Legendrian isotopy invariants of $\Lambda$.

Abstract:
Let $L\subset J^1(M)$ be a Legendrian submanifold of the 1-jet space of a Riemannian $n$-manifold $M$. A correspondence is established between rigid flow trees in $M$ determined by $L$ and boundary punctured rigid pseudo-holomorphic disks in $T^\ast M$, with boundary on the projection of $L$ and asymptotic to the double points of this projection at punctures, provided $n\le 2$, or provided $n>2$ and the front of $L$ has only cusp edge singularities. This result, in particular, shows how to compute the Legendrian contact homology of $L$ in terms of Morse theory.

Abstract:
Loose Legendrian n-submanifolds, for n at least 2, were introduced by Murphy and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial. We show that the converse is not true by constructing non-loose Legendrian n-spheres in standard contact (2n+1)-space, for n at least 2, with trivial DGA.

Abstract:
First order invariants of generic immersions of manifolds of dimension nm-1 into manifolds of dimension n(m+1)-1, m,n>1 are constructed using the geometry of self-intersections. The range of one of these invariants is related to Bernoulli numbers. As by-products some geometrically defined invariants of regular homotopy are found.

Abstract:
We construct a Legendrian 2-torus in the 1-jet space of $S^1\times\R$ (or of $\R^2$) from a loop of Legendrian knots in the 1-jet space of $\R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit any augmentation (linearization) but which still has non-trivial homology, as well as two Legendrian tori with isomorphic linearized contact homologies but with distinct contact homologies.