arXiv research

Paper detects duality obstruction in smooth calibrations.

We find the T-duality transformation rules for 2-dimensional (2,1) supersymmetric sigma-models in (2,1) superspace. Our results clarify certain aspects of the (2,1) sigma model geometry relevant to the discussion of T-duality. The complexified duality transformations we find are equivalent to the usual Buscher duality …

The technique of generating families produces obstructions to the existence of embedded Lagrangian cobordisms between Legendrian submanifolds in the symplectizations of 1-jet bundles. In fact, generating families may be used to construct a TQFT-like theory that, in addition to giving the aforementioned obstructions, yi…

A space is created to realize a specific cohomology module, showing PL structure but not smoothability.

New proof of chain duality for simplicial complexes.

Geometric framework for inverse problems using foliations and dual connections.

For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's result…

We observe that the Poincare duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod algebra. In particular, the pattern of the operations Sq^1, Sq^2, and Sq^4 on the cohomology of a string manifold has a symmetry around the middle dimension. We character…

This thesis explores Hamiltonian systems and Kähler structures on complex coadjoint orbits.

The A-B slice problem is a reformulation of the topological 4-dimensional surgery conjecture in terms of decompositions of the 4-ball and link homotopy. We show that link groups, a recently developed invariant of 4-manifolds, provide an obstruction for the class of model decompositions, introduced by M. Freedman and X.…

Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension S^j K so that the resulting space satisfies Poincare duality. When this is possible, we say that S^j K is a spine. We introduce the notion of quadratic self duality and show that if K is quad…

Identifies a mod-$p$ triple cup product for rational homology 3-spheres with specific first homology.

The author connects Poincaré embeddings to Reidemeister traces and diagonal maps.

Invariant $r^\sharp$ predicts H-flux behavior under T-duality.

The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard…

The paper refines Mather-Thurston theorems for flat connections in manifolds.

This paper initiates the study of topological arbiters, a concept rooted in Poincare-Lefschetz duality. Given an n-dimensional manifold W, a topological arbiter associates a value 0 or 1 to codimension zero submanifolds of W, subject to natural topological and duality axioms. For example, there is a unique arbiter on $…

We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of th…

Study topological correlators for $SU(2)$ SYM on four-manifolds, deriving explicit formulae and confirming S-duality.

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …

A long-standing conjecture due to Michael Freedman asserts that the 4-dimensional topological surgery conjecture fails for non-abelian free groups, or equivalently that a family of canonical examples of links (the generalized Borromean rings) are not A-B slice. A stronger version of the conjecture, that the Borromean r…

We combine Gromov's amenable localization technique with the Poincaré duality to study the traversally generic vector flows on smooth compact manifolds $X$ with boundary. Such flows generate well-understood stratifications of $X$ by the trajectories that are tangent to the boundary in a particular canonical fashion. Sp…

Let $M$ be a compact smooth Riemannian $n$-manifold with boundary. We combine Gromov's amenable localization technique with the Poincaré duality to study the {\sf traversally generic} geodesic flows on $SM$, the space of the spherical tangent bundle. Such flows generate stratifications of $SM$, governed by rich univers…

We consider geometric and analytical aspects of M-theory on a manifold with boundary Y. The partition function of the C-field requires summing over harmonic forms. When Y is closed Hodge theory gives a unique harmonic form in each de Rham cohomology class, while in the presence of a boundary the Hodge-Morrey-Friedrichs…

Proves Poincaré duality for Hopf algebroids with bijective antipode.

Introduces Kähler duality between domains in complex space.

Research on dualities in geometric stereotypes.

This paper studies torsion obstructions to complex sections on manifolds.

Duality restored in gauge theory, gravity, and string theory models.

Unified proof of four Bavard dualities and new results on quasimorphisms.

Verma Howe duality connects tensor products of Verma modules to LKB representations.

Cohomological and homological spectral sequences are shown to be isomorphic.

The paper proves T-duality and Hori formulae for winding loop spaces.

Equivariant T-duality connects bundles with twists.

We give the definition of a duality that is applicable to arbitrary $k$-forms. The operator that defines the duality depends on a fixed form $Ω$. Our definition extends in a very natural way the Hodge duality of $n$-forms in $2n$ dimensional spaces and the generalized duality of two-forms. We discuss the properties of …

The paper establishes T-duality for 2D σ-models with H-flux.

QP perspective on Poisson-Lie T-duality topology changes.

Unified framework for T-duality in both trivial and non-trivial topologies.

Geometric duality connects graph isomorphism and knot equivalence.

Koszul duality for manifold modules proven.

Fewer obstructions for small graphs in knotless embedding.

This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.

Maps self-duality in little disks operad to framed manifolds.

3028 obstructions found for embedding without knots.

We study generalized complex structures and $T$-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal $T$-duality". As an application we deal with the problem of finding symplectic stru…

We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich's stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced b…

New spherical T-duality for higher degree forms in fiber bundles.

We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…