Supergroupoids, double structures, and equivariant cohomology by Rajan Amit Mehta Doctor of Philosophy in Mathematics University of California, Berkeley Professor Alan Weinstein, Chair Q-groupoids and Q-algebroids are, respectively, supergroupoids and superalgebroids that are equipped with compatible homological vector fields. These new objects are closely related to the double structures of Ma...

The ‘Folk Theorem’ that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type is proved in the context of manifolds with corners. This procedure is shown to capture the simultaneous resolution of all isotropy types in a ‘resolution tower’ which projects to a resolution, with iterated boundary fibration, of the quotient. Equivaria...

Short-range entangled topological phases of matter are closely connected to Topological Quantum Field Theory. We use this connection to classify Symmetry Protected Topological Phases in low dimensions, including the case when the symmetry involves time-reversal. To accomplish this, we generalize Turaev’s description of equivariant TQFT to the unoriented case. We show that invertible unoriented ...

We study D-branes and Ramond-Ramond fields on global orbifolds of Type II string theory with vanishing H-flux using methods of equivariant K-theory and K-homology. We illustrate how Bredon equivariant cohomology naturally realizes stringy orbifold cohomology. We emphasize its role as the correct cohomological tool which captures known features of the low-energy effective field theory, and which...

Modern Schubert calculus has been mostly concerned with the study of the cohomology andK-theory (including their equivariant and quantum generalizations) of flag manifolds. The basic results for other cohomology theories have only been obtained recently; additional complexity is due to the dependence of the geometrically defined classes on reduced words for the corresponding Weyl group elements...

1. Interval positroid varieties 1 1.1. Schubert varieties 1 1.2. Schubert calculus 2 1.3. First positivity result 3 1.4. Interval rank varieties 5 2. Vakil’s Littlewood-Richardson rule 7 2.1. Combinatorial shifting 7 2.2. Geometric shifting 7 2.3. Vakil’s degeneration order 9 2.4. Partial puzzles 10 3. Equivariant and Kextensions 11 3.1. K-homology 11 3.2. K-cohomology 12 3.3. Equivariant K-the...

We define the category of manifolds with extended tangent bundles, we study their symmetries and we consider the analogue of equivariant cohomology for actions of Lie groups in this category. We show that when the action preserves the splitting of the extended tangent bundle, our definition of extended equivariant cohomology agrees with the twisted equivariant de Rham model of Cartan, and for t...

For a variety where a connected linear algebraic group acts with only finitely many orbits, each of which admits an attractive slice, we show that the stratification by orbits is perfect for equivariant intersection cohomology with respect to any equivariant local system. This applies to provide a relationship between the vanishing of the odd dimensional intersection cohomology sheaves and of t...

This article is designed to provide an introduction to some examples of triangulated categories that arise in the study of G-equivariant cohomology theories for a compact Lie group G. We focus on cohomology theories whose values are rational vector spaces since one may often give explicit algebraic constructions of the triangulated category in that case. As general references for equivariant co...

The aim of this paper is to put some recent results of HuangPandz̆ić (conjectured by Vogan) and Kostant on Dirac cohomology in a broader perspective. This is achieved by introducing an induction functor in the noncommutative equivariant cohomology. In this context, the results of Huang-Pandz̆ić and Kostant are interpreted as special cases (corresponding to the manifold being a point) of more gene...