By Hodge symmetry, the Betti numbers of a complex projective smooth variety in odd degrees are even. When the base field has characteristic p, Deligne proved the hard Lefschetz theorem in étale cohomology, and the parity result follows from this. Suh has generalized this to proper smooth varieties in characteristic p, using crystalline cohomology. The purity of intersection cohomology group of proper varieties suggests that the same parity property should hold for these groups in characteristic p. We proved this by investigating the symmetry in the categorical level. In particular, we reproved Suh's result, using merely étale cohomology. Some related results will be discussed. This is joint work with Weizhe Zheng.

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Intuitively, the classical variants of the Lefschetz Section Theorem relate a complex algebraic variety X to the intersection of X with a hyperplane H transversal to X (or, alternatively, to an ample divisor D of X). They are tremendously useful to compute invariants of the variety.

However, Lefschetz Section Theorems also hold for spaces that are constructed combinatorially rather than algebraically. Among other things, I will introduce Lefschetz theorems for — certain real subspace arrangements and their complements, — toric arrangements and their complements and, — matroids and smooth tropical varieties (joint work with Anders Björner). These theorems translate results of Lefschetz, Hamm-Le, Andreotti–Frankel, Bott–Thom, Akizuki–Kodaira–Nakano and Kodaira–Spencer to a combinatorial setting.

I will present some history and histories around the first years of the IHES and answer (briefly) the questions: By whom? Why? How? Where? When? was the IHES created, with documents from the archives – letters, administrative as well as mathematical documents – and people's memories.

Let G be a reductive group over a function field of large enough characteristic. We prove the temperedness at unramified places of automorphic representations of G, subject to a local assumption at one place, stronger than supercuspidality. Such an assumption is necessary, as was first shown by Saito-Kurokawa and Howe-Piatetskii-Shapiro in the 70's. Our method relies on the l-adic geometry of BunG, and on trace formulas. Work with Will Sawin.

Current knowledge of Mercury orbit is mainly brought by the direct radar ranging obtained from the 60s to 1998 and five Mercury flybys made by Mariner 10 in the 70s, and MESSENGER made in 2008 and 2009. On March 18, 2011, MESSENGER became the first spacecraft orbiting Mercury. The radioscience observations acquired during the orbital phase of MESSENGER drastically improved our knowledge of the Mercury orbit. An accurate MESSENGER orbit is obtained by fitting one-and-half years of tracking data using GINS orbit determination software. The systematic error in the Earth-Mercury geometric positions, also called range bias, obtained from GINS are then used to fit the INPOP dynamical modeling of the planet motions. An improved ephemeris of the planets is then obtained, INPOP13a, and used to perform general relativity test of PPN-formalism. Monte Carlo simulations will be introduced for estimating the most significant levels of GR violations in terms of PPN parameters and their correlated parameter (oblateness of the sun) and of time-varying Gravitational constant G.

The notion of Cohomological Hall algebra (COHA) was introduced in our joint paper with Maxim Kontsevich 10 years ago. It can be thought of as a mathematical incarnation of the notion of BPS algebra envisioned by physicists Harvey and Moore in the 90's.

Mathematically, COHA is an associative algebra structure on the cohomology of the moduli stack of objects of a 3-dimensional Calabi-Yau category with coefficients in a certain constructible sheaf. Interesting categories can be of geometric or algebraic origin (sheaves on Calabi-Yau 3-folds, quivers with potential, etc.).

In the talk I plan to discuss actions of COHA on the cohomology of certain instanton moduli spaces (spiked instantons of Nekrasov). This gives a relationship of COHA with affine Yangians and more recent "vertex algebras at the corner" introduced  by Gaiotto and Rapcak.

Parreau compactified the Hitchin component of a closed surface S of genus greater or equal to two in such a way that a boundary point corresponds to the projectivized length spectrum of an action of pi_1(S) on an R-building. We will explain how to use the positivity properties of Hitchin representations introduced by Fock and Goncharov to explicitly describe the geometry of a preferred collection of apartments in the limiting building.

Recent advances in two-loop superstring theory will be discussed, including the structure of supermoduli space, the spontaneous supersymmetry breaking on Calabi-Yau orbifolds, and the matching of the D6 R4 effective low energy corrections to supergravity with predictions from supersymmetry and duality.