This seminar has no fixed time. Here we discuss not necessary new, but fundamental results in mathematics.

2015, Thursday, 12 November, 16.15, Villa Battelle.

• *Les polyèdres ont-ils des jacobiennes ? le cas des graphes. Pierre de la Harpe (University of Geneva) L'application d'Abel-Jacobi plonge une surface de Riemann compacte dans sa jacobienne, qui est un tore complexe. On peut chercher des analogues (nécessairement partiels) de ce résultat fondamental dans des contextes moins riches, par exemple celui des polyèdres, et en particulier celui des graphes. C'est un exercice qui a tenté Roland Bacher, Tatiana Nagnibeda et l'orateur, il y a une vingtaine d'années (article paru en 1997). L'exposé en proposera une nouvelle visite. Parmi les mots clés : laplaciens et valeurs propres, formes harmoniques, —- 2015, 20 October, Tuesday, 15.15, Villa Battelle, (together with Séminaire “Groupes de Lie et espaces des modules”) Algebro-geometric proof of the Atiyah-Bott formula for the cohomology

of the moduli space of bundles on the curve Dennis Gaitsgory (Harvard) Abstract: Let $G$ be a semi-simple and simply connected group and $X$ an algebraic curve. We consider $Bun_G(X)$, the moduli space of $G$-bundles on $X$. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of $Bun_G$, namely $H^*(Bun_G)=Sym(H_*(X)\otimes V)$, where V is the space of generators for $H^*_G(pt)$. Their proof uses the interpretation of $Bun_G$ as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (and can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field (in particular, leading to the Tamagawa number formula for function fields). As its main geometric ingredient, it uses non-abelian Poincaré duality. In the course of the proof we will show how tools from higher category theory can be used to prove some rather concrete results. —- 2015, 19 October, Monday, 16.15, Villa Battelle, An introduction to spectral graph theory Anders Karlsson (UniGe) Like numbers, graphs belong to the most basic of mathematical structures. Every finite graph can be encoded by a Laplacian (as is the case for Riemannian manifolds). The Laplacian is a semi-positive symmetric matrix and as such it has semi-positive real eigenvalues. To what extent do the eigenvalues determine the graph? The first non-trivial eigenvalue measures the connectivity of the graph. This leads to the definition of expanders. These are sequences of highly connected sparse graphs which by now have many and varied applications, in communication network theory, computer science, and number theory, for topics such as error-correcting codes and random number generators. Like in algebra, one considers spectral invariants in terms of symmetric functions in the eigenvalues, such as the coefficients of the characteristic polynomial or zeta functions. These invariants often enumerate things. One such example is the number of spanning trees that appears in electrical network theory, theoretical chemistry, mathematical physics, and knot theory. Via zeta functions, this invariant is then an analog of the determinant of the Laplacian of manifolds, which appears in topology starting with Ray-Singer and in physics beginning with Hawking. I will give the basic definitions and facts, and explain a few examples, theorems, applications, and open questions. —- 2015, 24 September, Thursday, 16.15, Villa Battelle. On plane real algebraic curves of degree 8**

Stepan Orevkov.

We discuss the problem of classification of 8 degree curves up to isotopy (survey of result and methods).

Old web-page is http://www.unige.ch/math/folks/langl/battelle/