Informalizing formalized mathematics using the Lean theorem prover
Freitag, 21.4.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
One of the applications of interactive theorem provers in\npure mathematics is being able to produce machine-verified formal proofs. I will talk about a less-obvious application, which is using formalized mathematics to author interactive informal expositions. I will demonstrate a prototype of an "auto-informalization" system written in Lean that presents the reader with an interface to view proofs at a desired level of detail. I will also discuss thoughts on the impact of such tools in mathematics. This is joint work with Patrick Massot.\n
Syzygies of the cotangent complex
Freitag, 28.4.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
The cotangent complex is an important but difficult to understand object associated to a map of commutative rings (or schemes). It is connected with some easier to compute invariants: the module of differential forms, the conormal module, and Koszul homology can all be seen as syzygies inside the cotangent complex. Quillen conjectured that, for maps of finite flat dimension, the cotangent complex can only be bounded for locally complete intersection homomorphisms. This was proven by Avramov in 1999. I will explain how to get a new proof by paying attention to these syzygies, and how to simultaneously prove a conjecture of Vasconcelos on the conormal module.
Combinatorics of toric bundles
Freitag, 28.4.23, 14:00-15:00, SR 119
Toric bundles are fibre bundles which have a toric variety as a fiber. One particularly studied class of toric bundles are horospherical varieties which are toric bundles over generalized flag varieties. Similar to toric varieties, toric bundles admit a combinatorial description via polyhedral geometry. In my talk, I will explain such a combinatorial description, and describe a couple of results which rely on it. In particular, I will present a generalization of the BKK theorem and the Fano criterion for toric bundles.
Heights on commutative algebraic groups
Freitag, 5.5.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
In diophantine geometry a height function is measure of the complexity of an algebraic number. This talk is a presentation of my master thesis and will look at height estimates on connected commutative algebraic groups for taking integral multiples of points. Such estimates are used in the Analytic Subgroup Theorem by Gisbert Wüstholz.
A categorical perspective on scattering amplitudes
Freitag, 12.5.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
Scattering amplitudes are physical observables which play a central role in interpreting scattering experiments at particle colliders. In recent years, a new perspective on scattering amplitudes, known as amplituhedron programme, has revealed a fascinating link to various mathematical structures of a combinatorial nature, such as positive Grassmannians and cluster algebras. In this talk I will explain this connection from the point of view of derived and cluster categories of type A quivers, from which the formulae for scattering amplitudes can be obtained from projectives of hearts of intermediate t-structures. This talk is based on arXiv:2101.02884 joint with K. Ray and on arXiv:2112.14288 joint with P. Oak, A. Pal, K. Ray and H. Treffinger.
Hodge numbers of moduli spaces of principal bundles on curves
Freitag, 16.6.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
The Poincaré series of moduli stacks of semistable G-bundles on curves has been computed by Laumon and Rapoport. In this joint work with Melissa Liu, we show that the Hodge-Poincaré series of these moduli stacks can be computed in a similar way. As an application, we obtain a new proof of a joint result of the speaker with Erwan Brugallé, on the maximality on moduli spaces of vector bundles over real algebraic curves.
Elliptic Lie Theory : state of art
Freitag, 23.6.23, 10:30-11:30, Hörsaal II, Albertstr. 23b
After introducing the so-called elliptic root system, \nI will explain some motivations. The aim of this talk \nis to present the state of art on this class of \nroot systems.
On the Six Functor Formalism for Nori Motivic Sheaves
Freitag, 14.7.23, 10:30-11:30, Hörsaal II, Albertstr. 23b