Classes in Zakharevich K-groups constructed from Quillen K-theory
Freitag, 22.10.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
(joint work with M. Groechenig)\n\nThe Grothendieck ring of varieties K0(Var) is defined a lot like K0\nin Quillen K-theory. However, vector bundles get replaced by varieties\nand instead of quotienting out exact sequences, we quotient out Z -> X\n<- U relations, where Z is a closed subvariety and U its open\ncomplement.\nThis ring plays a crucial role in motivic integration, as in the proof\nthat K-equivalent [that's yet another meaning of K...] varieties have\nthe same Hodge numbers.\n\n Even though things look very analogous, K0(Var) is not the K0 group\nof some abelian category (or Waldhausen etc). Usual K-theory\nfoundations do not apply. Zakharevich and Campbell established that\nthere is an analogous theoretical formalism nonetheless, so there are\nalso higher K-groups corresponding to K(Var). However, until recently,\nit was not known whether any such Kn(Var) for n>0 contains any\nnon-zero element beyond torsion classes. Some months ago, we managed to\ngive the first construction of such, indeed showing that for all odd\nn>=3 the group Kn(Var) is infinite-dimensional. To do this, we develop\ntwo new tools. Joint with M. Groechenig and A. Nanavaty, motivic\nrealizations give rise to maps out of K(Var), and (joint just with M.\nGroechenig) there is a kind of exponential map from Quillen K-theory to\nK(Var), allowing us to import Quillen K-theory classes to give rise to\nclasses living on abelian varieties in K(Var).
Algebraic independence and special functions
Freitag, 29.10.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk we are going to see that the solutions of many linear functional equations do not satisfy algebraic differential equations. As an application, we will see how it yields to the proof of the algebraic independence of some special functions.\n\nThis is a joint work with B. Adamczewski, C. Hardouin and M. Wibmer
Strict minimality and algebraic relations between solutions in Poizat’s family of equations.
Freitag, 5.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In the last twenty years, techniques from model theory and differential algebra have been successfully applied to study integrability and algebraic dependence/independence of solutions in certain families of algebraic differential equations. I will discuss some of these techniques on a concrete example: the family of (non-linear) differential equations of the form y’’/y’ = f(y) where f(y) is a rational function. From a model-theoretic perspective, the study of this family was initiated by Poizat in the 80’s. \n\nThis is joint work with James Freitag, David Marker and Ronnie Nagloo.
Rigidity of Hyperelliptic Manifolds
Freitag, 12.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
A rigid compact complex manifold is one which admits no non-trivial deformations in a neighborhood of the base point. I will start by discussing the notions of rigidity and hyperelliptic manifolds (these are torus quotients with certain properties). After that, I will explain how to classify rigid hyperelliptic manifolds in complex dimension four (which is the minimal dimension in which rigid hyperelliptic manifolds exist). The classification has differential and complex geometric as well as algebraic flavors. This is joint work with Christian Gleißner (Universität Bayreuth).\n\n\n
A geometric approach to the charge statistic
Freitag, 19.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The charge is a statistic defined on the set of Young tableaux by Lascoux and Schützenberger in 1978. They showed that the generating functions for this statistic are the q-analogue of the weight multiplicities in type A.\nIn this talk, I will explain a different and geometrically motivated approach to the charge statistic. The q-weight multiplicities can in fact be obtained as Kazhdan-Lusztig polynomials for the affine Grassmannian. In this setting, we will recover the charge statistic by studying hyperbolic localization for a family of cocharacters.
Schinzel's Hypothesis with probability 1 and rational points on varieties in families
Freitag, 26.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In a joint work with Efthymios Sofos we prove that Schinzel's Hypothesis (H) holds for 100% of polynomials of any fixed degree. I will pass in silence the proof of this analytic result, but will explain how to deduce from this that among varieties in specific families over Q, a positive proportion have rational points. The main examples are varieties given by generalised Châtelet equations (a norm form equals a polynomial) and diagonal conic bundles of any fixed degree over the projective line.\n
Proofs by example and numerical Nullstellensätze
Freitag, 3.12.21, 10:30-11:30, BBB room
We study the proof method "proof by example" in which a general statement can be proved by verifying it for a single example. This strategy can indeed work if the statement in question is an algebraic identity and the example is "generic". This talk addresses the problem of construction a practical example, which is sufficiently generic, for which the statement can be verified efficiently, and which even allows for a numerical margin of error.\nOur answer comes in the form of a numerical Nullstellensatz, which is based on Diophantine geomery, in particular an arithemetic Nullstellensatz and a new effective Liouville-Lojasiewicz type inequality.\n\nIf time permits we moreover consider "proofs by several examples", which in addition requires a conceptual notion of sufficient genericity of a set of points. Besides theoretical and algorithmic criteria for sufficient genericity, we obtain several new types of Nullstellensätze in the spirit of the combinatorial Nullstellensatz and the Schwartz-Zippel lemma, also for varieties.
C^1-triangulations of semi-algebraic sets
Freitag, 17.12.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
If one wants to treat integration of differential forms over semi-algebraic sets analogous to the case of smooth manifolds, it is desirable to have triangulations of semi-algebraic sets that are globally of class C^1.\nWe will present a proof of the existence of such triangulations using the 'panel beating' method introduced by Ohmoto-Shiota (2017) and discuss possible generalizations.
Does hyperbolic 3-geometry provide an infinite family of fields with class number one?
Freitag, 21.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The class number one problem dating back to Gauss' work on quadratic\nforms asks whether infinitely many number fields of ideal class number\none exist. In an exciting 2017 paper, Ulf Rehmann and Ernest Vinberg\nhave described a candidate family of fields defined in hyperbolic\n3-geometry which, in all computed examples, turned out to have class\nnumber one. In the talk, we will introduce this family of fields and\nexamine its class number phenomenon from different perspectives: by an\nanalogy to a known class number formula in hyperbolic 3-geometry, by\nempirical computations and by estimates with known class number\nstatistics.
The standard conjecture of Hodge type for abelian fourfolds
Freitag, 28.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The standard conjecture of Hodge type for abelian fourfolds
Freitag, 28.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Let S be a surface, V be the Q-vector space of divisors on S modulo numerical equivalence and d be the dimension of V. The intersection product defines a non degenerate quadratic form on V. The Hodge index theorem says that it is of signature (1,d-1).\nIn the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is a consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable, thanks to p-adic Hodge theory. Moreover, using classical product formulas on quadratic forms, the p-adic result will give non-trivial information on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.
Higgs bundles twisted by a vector bundle
Freitag, 4.2.22, 10:30-11:30, Talk on BBB