Algebraic cobordism of varieties with G-bundles

Lee and Pandharipande studied a "double point" algebraic cobordism theory of varieties equipped with vector bundles, and speculated that some features of that story might extend to the case of varieties with principal G-bundles. This note shows that this expectation holds rationally, and more generally after inverting the torsion index of the group, for reductive G. We show that (after inverting the torsion index) the full theory for bundles on varieties is an extension of scalars of standard algebraic cobordism, that the theory for a point is dual to Omega^*(BG), and describe how the theory for G compares to that for a maximal torus T.Comment: 20 page

Measurable Dynamics of Maps on Profinite Groups

We study the measurable dynamics of transformations on profinite groups, in particular of those which factor through sufficiently many of the projection maps; these maps generalize the 1-Lipschitz maps on $\mathbb Z_p$.Comment: 18 page

Thom-Sebastiani and duality for matrix factorizations, and results on the higher structures of the Hochschild invariants

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 149-150).The derived category of a hypersurface has an action by "cohomology operations" k[[beta]], deg[beta] = 2, underlying the 2-periodic structure on its category of singularities (as matrix factorizations). We prove a Thom-Sebastiani type Theorem, identifying the k[[beta]]-linear tensor products of these dg categories with coherent complexes on the zero locus of the sum potential on the product (with a support condition), and identify the dg category of colimit-preserving k[[beta]]-linear functors between Ind-completions with Ind-coherent complexes on the zero locus of the difference potential (with a support condition). These results imply the analogous statements for the 2-periodic dg categories of matrix factorizations. We also present a viewpoint on matrix factorizations in terms of (formal) groups actions on categories that is conducive to formulating functorial statements and in particular to the computation of higher algebraic structures on Hochschild invariants. Some applications include: we refine and establish the expected computation of 2-periodic Hochschild invariants of matrix factorizations; we show that the category of matrix factorizations is smooth, and is proper when the critical locus is proper; we show how Calabi-Yau structures on matrix factorizations arise from volume forms on the total space; we establish a version of Knörrer Periodicity for eliminating metabolic quadratic bundles over a base.by Anatoly Preygel.Ph.D

Higher structures on Hochschild invariants of matrix factorizations

Non UBCUnreviewedAuthor affiliation: UC BerkeleyPostdoctora

Dynamics of the p -adic Shift and Applications

There is a natural continuous realization of the one-sided Bernoulli shift on the [p] -adic integers as the map that shifts the coefficients of the [p] -adic expansion to the left. We study this map's Mahler power series expansion. We prove strong results on [p] -adic valuations of the coefficients in this expansion, and show that certain natural maps (including many polynomials) are in a sense small perturbations of the shift. As a result, these polynomials share the shift map's important dynamical properties. This provides a novel approach to an earlier result of the authors.Williams College (Bronfman Science Center)National Science Foundation (U.S.) (REU Grant DMS – 0353634

On measure-preserving c1 transformations of compactopen subsets of non-Archimedean local fields

Abstract. We introduce the notion of a locally scaling transformation defined on a compact-open subset of a non-archimedean local field. We show that this class encompasses the Haar measure-preserving transformations defined by C1 (in particular, polynomial) maps, and prove a structure theorem for locally scaling transformations. We use the theory of polynomial approximation on compact-open subsets of non-archimedean local fields to demonstrate the existence of ergodic Markov, and mixing Markov transformations defined by such polynomial maps. We also give simple sufficient conditions on the Mahler expansion of a continuous map Zp → Zp for it to define a Bernoulli transformation. 1