Hwang–Mok rigidity of cominuscule homogeneous varieties in positive characteristic
Friday, 18.10.13, 10:00-11:00, Raum 404, Eckerstr. 1
Jun-Muk Hwang and Ngaiming Mok have proved the rigidity of\nirreducible Hermitian symmetric spaces of compact type under Kaehler degeneration. I adapt their argument to the algebraic setting in positive characteristic, where cominuscule homogeneous varieties serve as an analogue of Hermitian symmetric spaces. The main result gives an\nexplicit (computable in terms of Schubert calculus) lower bound on the characteristic of the base field, guaranteeing that a smooth projective family with cominuscule homogeneous generic fibre is isotrivial. The bound depends only on the type of the generic fibre, and on the degree of an invertible sheaf whose extension to the special fibre is very ample. An important part of the proof is a characteristic-free analogue of Hwang and Mok’s extension theorem for maps of Fano varieties of Picard number 1, a result I believe to be interesting in its own right.\n
Exterior power operations on Witt rings
Friday, 25.10.13, 10:00-11:00, Raum 404, Eckerstr. 1
The Witt ring of a field has a natural filtration by powers of the so-called fundamental ideal. We exhibit a possible generalization of this filtration to Grothendieck-Witt rings of vector bundles over a scheme or of representations of an affine algebraic group via exterior power operations. Our main technical result is that the resulting λ-structures are special. The talk will close with a few example calculations and many open question.
Fibered Derivators, (Co)homological Descent and the Six-Functor Formalism
Friday, 8.11.13, 10:15-11:15, Raum 404, Eckerstr. 1
Derivators have been introduced by Grothendieck, as one of his last mathematical contributions, to simplify, extend, and conceptually clarify the\nnotions of derived and triangulated categories. Extended to the context of fibered (multi-)categories (e.g. any kind of sheaves of abelian groups on spaces, schemes, stacks etc.)\nthe notion allows for a neat solution to problems of cohomological descent as well as homological descent. This gives an elegant way\nof extending the six-functor formalism of Grothendieck (which encodes, among other things, dualities like e.g. Serre duality, Poincar'e-Verdier duality)\nto stacks.
Nichthomogene affine Flächen mit riesiger Automorphismengruppe
Friday, 15.11.13, 10:15-11:15, Raum 404, Eckerstr. 1
A diagram algebra for categorifying gl(1|1)
Friday, 22.11.13, 10:15-11:15, Raum 404, Eckerstr. 1
Friday, 29.11.13, 10:15-11:15, Raum 404, Eckerstr. 1
The geometry of singularities in the Minimal Model Program and applications to singular spaces with trivial canonical class
Friday, 29.11.13, 10:15-11:15, Raum 404, Eckerstr. 1
This talk surveys recent results on the singularities of\nthe Minimal Model Program and discusses applications to the study of\nvarieties with trivial canonical class. Comparing the étale fundamental\ngroup of a klt variety with that of its smooth locus, we show that any\nflat holomorphic bundle, defined on the smooth part of a projective\nklt variety is algebraic and extends across the singularities. This\nallows to generalise a famous theorem of Yau, which states that any\nRicci-flat Kähler manifold with vanishing second Chern class is an\nétale quotient of a torus.\n\nThis is joint work with Daniel Greb and Thomas Peternell\n
Differential forms in the h-topology
Friday, 6.12.13, 10:15-11:15, Raum 404, Eckerstr. 1
The h-topology on the category of separated schemes of\nfinite type over a field of characteristic zero was\nintroduced by Voevodsky to study the homology of schemes. I\nwill discuss in my talk the sheaves of h-differential\nforms, i.e., the appropriate notion of differential forms\nin the h-topology. Subsequently I will study their\nbehaviour on rationally chain connected spaces and the\nconnection to algebraic de Rham cohomology. This is joint\nwork with Annette Huber-Klawitter.\n
Graph algebras
Friday, 13.12.13, 10:15-11:15, Raum 404, Eckerstr. 1
From a graph (e.g., cities and flights between them) one can generate an algebra which captures the movements along the graph.\n\nThis talk is about one type of such correspondences, i.e., Leavitt path algebras.\n\nDespite being introduced only 8 years ago, Leavitt path algebras have arisen in a variety of different contexts as diverse as analysis, symbolic dynamics, noncommutative geometry and representation theory. In fact, Leavitt path algebras are algebraic counterpart to graph C*-algebras, which has become an area of intensive research. There are strikingly parallel similarities between these two theories. Even more surprisingly, one cannot (yet) obtain the results in one theory as a consequence of the other; the statements look the same, however the techniques to prove them are quite different (as the names suggest, one uses Algebra and other Analysis). These all suggest that there might be a bridge between Algebra and Analysis yet to be uncovered.\n\nIn this talk, we introduce Leavitt path algebras and then try to understand the behaviour and to classify them by means of (graded) K-theory. We will ask nice questions!\n
Friday, 20.12.13, 10:15-11:15, Raum 404, Eckerstr. 1
Friday, 10.1.14, 10:15-11:15, Raum 404, Eckerstr. 1
Lie subalgebras of vector fields and the Jacobian conjecture
Friday, 24.1.14, 10:15-11:15, Raum 404, Eckerstr. 1
Higher Tannaka duality and realizations of mixed motives
Thursday, 30.1.14, 11:00-12:00, Raum 218, Eckerstr. 1
Tilting and Parity
Friday, 31.1.14, 10:15-11:15, Raum 404, Eckerstr. 1
In the representation theory of reductive groups in positive\ncharacteristic, there is a very special class of objects called tilting\nmodules. Two important theorems state that the class of tilting modules\nis closed under tensor product and restriction to Levi subgroups.\n\nSimilarly, on (generalized) flag varieties, there is a special class of\ngeometric objects called parity sheaves. I will explain two newer\ntheorems that are analogues of the tensor product and restriction\ntheorems mentioned above.\n\nThe main goal of the talk will be to explain how these two pictures fit\ntogether.\n\nThis is based on joint work with Daniel Juteau and Geordie Williamson.
Weightless cohomology of algebraic varieties
Thursday, 6.2.14, 11:00-12:00, Raum 218, Eckerstr. 1
Using Morel's weight truncations in categories of mixed sheaves, we attach to any variety defined over complex numbers, over finite fields or even over a number field, a series of groups called the weightless cohomology groups . These lie between the usual cohomology and the intersection cohomology, have a natural ring structure, satisfy Kunneth, and are functorial for certain morphisms.\n The construction is motivic and naturally arises in the context of Shimura Varieties where they capture the cohomology of Reductive Borel Serre compactification. The construction also yields invariants of singularities associated with the combinatorics of the boundary divisors in any resolution.
Motives of rigid analyic varieties over perfectoid fields
Friday, 7.2.14, 10:15-11:15, Raum 404, Eckerstr. 1
The aim of the talk is to outline the proof of the equivalence between the category of motives of rigid analytic varieties (defined by Ayoub adapting Voevodsky's construction) over a perfectoid field of mixed characteristic and over the associated (tilted) perfectoid field of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger regarding the isomorphism of the two absolute Galois groups. A main tool for constructing the equivalence is Scholze's theory of perfectoid spaces, which will also be briefly discussed.\n
Friday, 14.2.14, 10:15-11:15, Raum 404, Eckerstr. 1
Soulé-elements, p-adic periods & an explicit reciprocity law
Wednesday, 19.2.14, 14:15-15:15, Raum 404, Eckerstr. 1
Intersection theory on singular varieties
Tuesday, 25.2.14, 10:15-11:15, Hörsaal II, Albertstr. 23b
We introduce some ideas from motivic cohomology into the study of singular varieties. Our approach is modeled on the intersection homology of Goresky-MacPherson; our goal is to intersect cycles on a stratified singular variety provided the cycles do not meet the strata too badly. We define "perverse" analogues of Chow groups and motivic cohomology. Properties include homotopy invariance, a localization theorem, and a splitting theorem. As a consequence we obtain pairings between certain "perverse" cycle groups on a singular variety.\n\n
Multiple polylogarithm cycles and mixed Tate motives over the projective line minus three points
Monday, 17.3.14, 10:15-11:15, Raum 404, Eckerstr. 1
After introducing the multiple polylogarithms functions and the mixed Tate motive context,\nwe will present Bloch and Kritz description of motivic polylogarithms using algebraic cycles. More generally,\nwe will show how this approach can be generalized to multiple polylogarithms. We will insist on the geometric\nsituation and on some low weight examples.
Tates Residuo, transposed to the K-theory spektrum
Thursday, 27.3.14, 10:15-11:15, Raum 127, Eckerstr. 1
Ich werde eine hübsche Idee von John Tate zum klassischen Residuum auf Kurven aufgreifen. Arbarello, deConcini und Kac haben darauf aufbauend einen klassischen Reziprozitätssatz von Weil für K_2 Gruppen auf Kurven bewiesen. Wir dehnen diese Idee auf das volle K-Theoriespektrum sowie höhere Dimension aus, und noch weiter. Insbesondere erhält man diese klassischen Resultate als Spezialfälle wieder".\n(gem. mit Michael Gröchenig, Imperial UK und Jesse Wolfson, Northwestern US)