The Hodge index theorem of Faltings and Hriljac asserts that the Neron-Tate height pairing on a projective curve over a number field is equal to a certain intersection pairing in the setting of Arakelov geometry. In the talk, I will present an extension of this result to adelic line bundles on higher dimensional varieties over finitely generated fields. Then I will talk about its relation to the non-archimedean Calabi-Yau theorem and its application to algebraic dynamics. This is a joint work with Shou-Wu Zhang.

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Compactifications of M5-branes on non-trivial 3-manifolds lead to a N=2 supersymmetric theories in 3 dimensions. If the 3-manifold in question is a knot complement, various Chern-Simons amplitudes – or corresponding knot invariants – for this knot determine the properties of the resulting 3d, N=2 theory. This relation between N=2 theories and Chern-Simons theory is referred to as the 3d-3d correspondence. M-theory realization of this correspondence implies that N=2 theories obtained in this way possess one special flavor symmetry, which is related to certain deformation of Chern-Simons theory and homological knot invariants. In this talk I will discuss properties of these refined/homological invariants, and their role in the 3d-3d correspondence.

Dans cet exposé, j’expliquerai le comportement de la conjecture de Birch et Swinnerton-Dyer dans la Zp-extension cyclotomique : si p est un nombre premier ordinaire, il existe une fonction L p-adique classique, avec des invariants canoniques qui fournissent une description adéquate de ce comportement. Si p est un nombre premier supersingulier, on peut le décrire par une paire de fonctions L p-adiques (avec une paire d’invariants), auquel cas il y a des phénomènes mystérieux.

We define the characteristic cycle of a locally constant étale sheaf on a smooth variety in positive characteristic, ramified along the boundary, as a cycle in the cotangent bundle of the variety, at least on a neighborhood of the generic point of the divisor on the boundary.

We discuss a compatibility with pull-back and local acyclicity in non-characteristic situations. We also give a relation with the characteristic cohomology class and the Euler-Poincaré characteristic.

All necessary biological knowledge for understanding the hot biological problem(s) underlying the modeling will be provided.

We study two classes of n-generated quadratic algebras over a field K. The first is the class Cn of all n-generated PBW algebras with polynomial growth and finite global dimension. We show that a PBW algebra A is in Cn iff its Hilbert series is HA(z) = 1/(1-z)n. Furthermore each class Cn contains a unique (up to isomorphism) monomial algebra A = K ‹ x1, … , xn › / (xj xi | 1 ≤£ i < j £ n). The second is the class of n-generated quantum binomial algebras A, where the defining relations are nondegenerate square-free binomials xy – cxy zt, with nonzero coefficients cxy. Our main result shows that the following conditions are equivalent: (i) A is a Yang-Baxter algebra, that is the set of quadratic relations R defines canonically a solution of the Yang-Baxter equation. (ii) A is an Artin-Schelter regular PBW algebra. (iii) A is a PBW algebra with polynomial growth. (iv) A is a binomial skew polynomial ring. (v) The Koszul dual A! is a quantum Grassmann algebra.

Given an F-crystal C over a scheme S of positive characteristic p, one can associate many reduced locally closed subschemes T of S defined by the property that a suitable invariant of F-crystals is constant on the fibres of C over points of S. A basic problem is to study different properties of such locally closed subschemes T. In this talk, we present a survey of purity results for locally closed subschemes T of S, including some of them which were obtained by us and by the graduate student Jinghao Li.

A technical ingredient in Faltings’ original approach to p-adic comparison theorems involves the construction of K(π, 1)-neighborhoods for a smooth scheme X over a mixed characteristic dvr with a perfect residue field: every point x ∈ X has an open neighborhood U whose general fiber is a K(π, 1) scheme (a notion analogous to having a contractible universal cover). I will show how to extend this result to the logarithmically smooth case, which might help to simplify some proofs in p-adic Hodge theory.