Classification of principal bundles via motivic homotopy theory
Freitag, 20.10.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will talk about recent joint work with Aravind Asok and Marc Hoyois on classification results for principal bundles over smooth affine varieties. The main point will be to explain how the general homotopical results boil down to very concrete examples of interesting octonion algebras.
K-theory of locally compact modules
Freitag, 27.10.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
We generalize a recent result of Clausen: For a number field with integers O, we compute the K-theory of locally compact O-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite different: Instead of a homotopy coherent cone construction in infinity categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact which might be of independent interest. As in Clausen's work, our computation works for all localizing invariants, not just K-theory.
Walks, Difference Equations and Elliptic Curves
Freitag, 3.11.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
In the recent years, the nature of the generating series of walks in\nthe quarter plane has attracted the attention of many authors in\ncombinatorics and probability. The main questions are: are they\nalgebraic, holonomic (solutions of linear differential equations) or at\nleast hyperalgebraic (solutions of algebraic differential equations)? In\nthis talk, we will show how the nature of the generating function can\nbe approached via the study of a discrete functional equation over a\ncurve E, of genus zero or one. In the first case, the functional\nequation corresponds to a so called q-difference equation and all the\nrelated generating series are differentially transcendental. For the\ngenus one case, the dynamic of the functional equation corresponds to\nthe addition by a given point P of the elliptic curve E. In that\nsituation, one can relate the nature of the generating series to the\nfact that the point P is of torsion or not.\n This is a collaboration with T. Dreyfus (Irma, Strasbourg), J. Roques\n(Institut Fourier, Grenoble) and M.F. Singer (NCSU, Raleigh).
The algebraic topology of p-adic Lie groups
Donnerstag, 9.11.17, 10:15-11:15, Raum 403, Eckerstr. 1
Lazard proved that the cohomology of compact p-adic Lie groups satisfies a version of Poincare duality. This indicates an analogy between p-adic Lie groups and real manifolds. I will explain some results which develop this analogy further.
From elementary number theory to string theory and back again
Freitag, 10.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
I will describe some surprising interactions between number theory, algebraic geometry and mirror symmetry that have appeared in my recent work with Mircea Mustata and Chenyang Xu and that have led to a solution of Veys' 1999 conjecture on poles of maximal order of Igusa zeta functions. The talk will be aimed at a general audience and will emphasize some key ideas from each of the fields involved rather than the technical aspects of the proof.
Modular representations of sl_n with two-row nilpotent p-character
Mittwoch, 15.11.17, 10:45-11:45, Hörsaal FRIAS
We will study the category of modular representations of the\nspecial linear Lie algebra with a central p-character given by a\nnilpotent whose Jordan type is a two-row partition. Building on work of\nCautis and Kamnitzer, we construct a categorification of the affine\ntangle calculus using these categories; the main technical tool is a\ngeometric localization-type result of Bezrukavnikov, Mirkovic and\nRumynin. Using this, we give combinatorial dimension formulae for the\nirreducible modules, composition multiplicities of the simples in the\nbaby Vermas, and a description of the Ext spaces. This Ext algebra is an\n"annular" analogues of Khovanov's arc algebra, and can be used to give\nan extension of Khovanov homology to links in the annulus. This is joint\nwith Rina Anno and David Yang.
Functorial test modules
Freitag, 17.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
In my talk I will report on joint work with Manuel Blickle. I will explain how one can generalize the definition of test ideals \btau to so-called Cartier modules in a functorial way. We obtain several transformation rules with respect to f^! and f_* for various classes of morphisms f: X \bto Y, e.g. for f smooth one has an isomorphism f^! \btau = \btau f^!. Part of the reason for working in this generality is that one has an equivalence with constructible etale p-torsion sheaves up to nilpotence of Cartier modules and these results further support the idea that the test module construction relates to etale nearby cycles similarly to the complex situation where multiplier ideals relate to complex nearby cycles.
Local systems in motivic homotopy theory
Freitag, 24.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
Local systems in motivic homotopy theory
Freitag, 24.11.17, 10:15-11:15, Hörsaal FRIAS, Albertstr. 19
This talk is first devoted to explain how Voevodsky's theory of motivic complexes is built on, and provides, a natural notion of local system. Two key players involved here are homotopy sheaves and cycle modules, which are respectively the theoretical and concrete side of the same notion. The first part of the talk will recall these two notions and their fundamental link.\n The second part of the talk will move on relative motivic complexes. I will explain the construction of a natural t-structure, analogous to the perverse t-structure (but not realized to it!), obtained in collaboration with Bondarko along ideas of Ayoub. Besides describing some of its good properties, I will explain a program to compute its heart, in terms of cycle modules.\n If time allows, I will wander a little bit on the aimed tool motivating these technicalities: the delta-homotopy Leray spectral sequence.\n
Hodge theory and formality
Freitag, 1.12.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
Given a differential graded algebra or any algebraic structure\nin chain complexes, one may ask if it is quasi-isomorphic to its\nhomology equipped with the zero differential. This property is called\nformality and has important consequences in algebraic topology. For\ninstance, if the de Rham algebra of a manifold is formal, then certain\nhigher operations in cohomology, called Massey products, are known to\nvanish. In this talk, I will first discuss the notion of formality and\nits consequences in different algebraic and topological contexts. Then,\nI will explain how mixed Hodge theory and Galois actions can be used to\nprove formality, for algebraic structures arising from the category of\ncomplex algebraic varieties.
Recent results and open problems about Oeljeklaus-Toma manifolds
Freitag, 8.12.17, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
Oeljeklaus - Toma manifolds are compact complex manifolds associated to number fields with at least one real and and at least one complex place. The construction is similar to tori, but it involves not only the lattice of integers but also a suitable group of units. We investigate how the number-theoretic properties influence their geometric properties: existence of special metrics, existence of closed subvarieties, etc. The talk is based mainly on joint work with L.Ornea and M. Verbitsky.
1-Motives
Donnerstag, 14.12.17, 11:15-12:15, Hörsaal FRIAS, Albertstr. 19
In this expository talk we want to present Deligne's category of 1-motives and its realisations. This ties up\nwith the talk of Wüstholz on Friday, but the two talks\nwill be independent of each other.
Period domains
Freitag, 15.12.17, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Kac-Moody algebras and derived algebraic geometry
Dienstag, 9.1.18, 09:15-10:15, SR 119, Eckerstr. 1
Kac-Moody algebras are usually used in physics to describe\nsome 2-dimensional conformal field theories. In this talk, we will introduce a new version of Kac-Moody algebras supposed to describe some physical phenomenons in higher dimensions.\n\nThose new algebras are in fact Lie algebras up to homotopy and we will study them using tools from algebraic topology and derived algebraic geometry.
Adelic formal neighborhoods and chiral modules with support
Donnerstag, 11.1.18, 09:15-10:15, SR 119, Eckerstr. 1
In this talk I will explain how one can use Ind-Pro objects\nto consider the restriction to formal and punctured formal\nneighborhoods of a subvariety Z in a variety X. This restriction functor can then be applied to give a description of modules for a chiral algebra over X supported at Z in terms of modules for an associative algebra object over Z. This is work in progress.\n\n\n(no worries, the speaker will explain what a "chiral algebra" is...)
Unlikely intersections between isogeny orbits and curves
Freitag, 19.1.18, 10:15-11:15, Hörsaal FRIAS, Albertsstr. 19
In the spirit of the Mordell-Lang conjecture, we consider the intersection of a curve in a family of abelian varieties with the images of a finite-rank subgroup of a fixed abelian variety A0 under all isogenies between A0 and some member of the family. After excluding certain degenerate cases, we can prove that this intersection is finite. This proves the so-called André-Pink-Zannier conjecture in the case of curves. We can even allow translates of the finite-rank subgroup by abelian subvarieties of controlled dimension if we strengthen the degeneracy hypotheses suitably. In my talk, I will try to explain the motivation for this problem and give an outline of the proof, which follows a strategy due to Pila-Zannier.
On the motivic Tamagawa number of number fields
Donnerstag, 25.1.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
The Tamagawa number of a linear algebraic group is a classical arithmetic invariant.\nIt was computed and related to special values of L-functions in the second half of the 20th century.\nIn 1990 Bloch and Kato proposed a Tamagawa number of a (certain kind of) motive, related it to special L-values of motives and stated a conjecture on their value in the spirit of the classical case. We will discuss motivic Tamagawa numbers for the (seemingly easiest) unknown case, namely the twisted motive of a number field. I will present precise formulas relating these two notions of Tamagawa numbers to one another and to Borel regulators.
The Koszul duality between D-modules and Omega-modules
Freitag, 26.1.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Koszul duality between D-modules and Omega-modules (dg modules over the algebraic de Rham complex) has first been studied by Kapranov, and subsequently by various authors including Beilinson-Drinfeld and Positselski. In this talk, I will report on work in progress (joint with Dmitri Pavlov) on a refinement of Koszul duality. An application of this refinement is a presentation of the derived pushforward and pullback functors for D-modules which avoids the usual mixing of right and left adjoints.
Cohomological properties of OT manifolds
Dienstag, 30.1.18, 11:00-12:00, SR 119
Resolution of singularities of the cotangent sheaf of a singular variety
Freitag, 9.2.18, 10:30-11:30, Hörsaal FRIAS, Albertsstr. 19
The subject of the talk is resolution of singularities of differential forms on an algebraic or analytic variety. We address the problem of finding a resolution of singularities \(\bsigma : X \bto X_0 \) of a singular algebraic or analytic variety \(X_0\) such that the pulled back cotangent sheaf of \(X_0\) (i.e., the pull-back of the Kahler differential forms defined in \(X_0\)) is given, locally in \(X\), by monomial differential forms (with respect to a suitable coordinate system). This problem is related with monomialization of maps, the \(L^2\) cohomology of singular varieties and reduction of singularities of vector-fields. In a work in collaboration with Bierstone, Grandjean and Milman, we give a positive answer to the problem when \(dim\b, X_0 \bleq 3\).
On the topology of smooth hypersurfaces
Freitag, 9.2.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
To what extent the Chern class of a divisor (in singular\ncohomology) determines its topology?\nDiscussion of a conjecture by Totaro concerning the topology\nof smooth hypersurfaces on projective manifolds.
Algebraic curves and modular forms of low degree
Freitag, 23.2.18, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
For genus 2 and 3 modular forms are intimately connected with\nthe moduli of curves of genus 2 and 3. We give an explicit way to\ndescribe such modular forms for genus 2 and 3\nusing invariant theory and give some applications.\nThis is based on joint work with Fabien Clery and Carel Faber.\n
Beilinson conjectures for curves
Donnerstag, 1.3.18, 15:00-16:00, Raum 404, Eckerstr. 1