I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

/pdf/deformation-quantization-of-poisson-manifolds-4tp4kn8ari.pdf

Deformation Quantization of Poisson Manifolds

Sphere Packings, Lattices and Groups

Advances in mathematics

We show that the Kervaire invariant one elements θj ∈ π2j+1−2S exist only for j ≤ 6. By Browder’s Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology.

https://math.ucla.edu/~mikehill/Talks/ICM.pdf

On the nonexistence of elements of Kervaire invariant one

Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The $qf$-model structure on $\mathcal{K}_B$ Equivariant $qf$-type model structures Ex-fibrations and ex-quasifibrations The equivalence between Ho$G\mathcal{K}_B$ and $hG\mathcal{W}_B$ Parametrized equivariant stable homotopy theory: Introduction to Part III Enriched categories and $G$-categories The category of orthogonal $G$-spectra over $B$ Model structures for parametrized $G$-spectra Adjunctions and compatibility relations Module categories, change of universe, and change of groups Parametrized duality theory: Introduction to Part IV Fiberwise duality and transfer maps Closed symmetric bicategories The closed symmetric bicategory of parametrized spectra Costenoble-Waner duality Fiberwise Costenoble-Waner duality Homology and cohomology, Thom spectra, and addenda: Introduction to Part V Parametrized homology and cohomology theories Equivariant parametrized homology and cohomology Twisted theories and spectral sequences Parametrized FSP's and generalized Thom spectra Epilogue: Cellular philosophy and alternative approaches Bibliography Index Index of notation.

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Parametrized homotopy theory

Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence

We prove convergence of the motivic Adams spectral sequence to completions at p and η under suitable conditions. We also discuss further conditions under which η can be removed from the statement.

/pdf/convergence-of-the-motivic-adams-spectral-sequence-3px63j5uoc.pdf

Convergence of the Motivic Adams Spectral Sequence

The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism

On the Picard group of the stable A1-homotopy category

We discuss certain calculations in the 2-complete motivic stable homotopy category over an algebraically closed field of characteristic 0. Specifically, we prove the convergence of motivic analogues of the Adams and Adams-Novikov spectral sequences, and as one application, discuss the 2-complete version of the complex motivic J -homomorphism.

/pdf/remarks-on-motivic-homotopy-theory-over-algebraically-closed-2f2z8wu5lq.pdf

Po Hu

Papers

Real-oriented homotopy theory and an analogue of the Adams–Novikov spectral sequence

Convergence of the Motivic Adams Spectral Sequence

The homotopy limit problem for Hermitian K-theory, equivariant motivic homotopy theory and motivic Real cobordism

On the Picard group of the stable A1-homotopy category

Remarks on motivic homotopy theory over algebraically closed fields