# Battelle séminaire

This seminar has no fixed time. Here we discuss not necessary new, but fundamental results in mathematics. We ask speakers to do informal talks, accessible for a general audience. The usual time for Battelle seminar is 16.15 on Thursday when there are no colloquium.

2016, Thursday, 17 November, Marcos Mariño, 16.15

**An invitation to Chern-Simons theory**

Chern-Simons theory is a quantum field theory introduced by Witten in 1989 which gives a framework to formulate and calculate knot invariants. I will try to describe some basic aspects of the theory and sketch how it leads to the Jones polynomial of knots.

2016, Monday, 9 May, Ilya Tyomkin (Ben-Gurion University of the Negev), 17.30-18.15

**Nagata’s compactification and factorization of morphisms.**

After constructing first examples of complete non-projective varieties in late 50s, Nagata became interested in the compactification problem in algebraic geometry. In early 60s, he managed to prove that any variety, or more generally, a separated scheme of finite type over a Noetherian base scheme can be embedded as an open subscheme in a complete scheme over the base. This fundamental result is known as Nagata’s compactification theorem.

Reading Nagata’s original papers is not an easy task, since they were written in his own version of pre-Grothendieck language. In 70s, Deligne prepared notes written in modern terms providing a version of Nagata’s approach. He sent them to Nagata, but didn’t make them public for almost 40 years till 2010.

In 2009, Temkin found a new proof of Nagata’s compactification. He proved the following remarkable decomposition theorem: any separated morphism of qcqs schemes can be presented as a composition of an affine morphism followed by a proper morphism. Temkin deduced Nagata’s compactification from the decomposition theorem, since affine morphisms of finite type are easily compactifiable.

Today, several proofs of various versions of Nagata’s theorem are known: In addition to the proofs mentioned above, there is a short proof due to Lütkebohmert (1993) for the Noetherian base, there is a detailed version of Deligne’s proof without the Noetherianity assumption due to Conrad (2007), there are two recent proofs in the category of algebraic spaces due to Conrad, Lieblich, Olsson, and to Temkin and myself. Finally, there is an unfinished work of Rydh in the case of (some) Deligne-Mumford stacks.

In my talk, I’ll explain the problem, its history, and development. I’ll also try to give some hints about existing approaches and main difficulties.

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/