The Gauss-Green theorem in stratified groups

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the $BV$ fields. They provide the most general setting to establish Gauss-Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss-Green theorem.Comment: 69 page

Formal verification of higher-order probabilistic programs

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show the expressiveness of our logics by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) TT-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs

Computations in formal symplectic geometry and characteristic classes of moduli spaces

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.Comment: 33 pages, final version, to appear in Quantum Topolog

Vertex algebras and the Landau-Ginzburg/Calabi-Yau correspondence

We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a vertex algebra closely related to the Landau-Ginburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi-Yau hypersurfaces.Comment: Latex, 50p. Some typos corrected, the page size may have been fixed. One new result, a theorem on the vertx algebra structure of the Landau-Ginzburg orbifold appears in sect. 5.2.18. This is the final version to appear in the Moscow Mathematical Journa