Bloch's formula
Friday, 2.11.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we discuss Bloch's formula for smooth and singular schemes. The formula relates Chow group of cycles on a scheme with the cohomology of K-sheaves or K^M-sheaves, where K and K^M stand for K-theory and Milnor K-theory, respectively. In smooth case, the formula is a corollary to the Gersten resolution. As Gersten resolution for these sheaves is not available on singular schemes, in a joint work with Prof. Amalendu Krishna, we use Cousin complex to study the Bloch's map. \n\nWe begin the talk by recalling the definition of Chow groups and Milnor K-groups and briefly discuss the formula for smooth schemes. In the case of singular schemes, we use Cousin complex to define Bloch's map. We then prove the formula for affine schemes over algebraically closed fields and for regular in codimension one projective schemes over algebraically closed fields. At last, Bloch's formula with modulus will be discussed. \n\n
Global hyperbolic propagators: a microlocal-analytic approach
Monday, 5.11.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
n my talk I will discuss how one can, in the spirit of some classical results due to Laptev, Safarov and Vassiliev, write the propagator of a class of hyperbolic operators on manifolds as one single oscillatory integral with complex-valued phase function, global both in space and in time. In particular, a refined, geometric version of the method will be presented, in the Riemannian setting: the adoption of a distinguished complex-valued phase function, naturally dictated by the geometric framework, will allow us to visualise the process of circumventing topological obstructions. The microlocal method is explicit and constructive; the calculation of the subprincipal symbol of the propagator enables us to recover asymptotic spectral properties of the operators at hand. I will discuss explicit formulae and recent results for the wave operator. Time permitting, the extension of the method to Lorentzian spacetimes will be briefly analysed.\nThis is joint work with D. Vassiliev (UCL) and M. Levitin (Reading).
Global hyperbolic propagators: a microlocal-analytic approach
Monday, 5.11.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In my talk I will discuss how one can, in the spirit of some classical results due to Laptev, Safarov and Vassiliev, write the propagator of a class of hyperbolic operators on manifolds as one single oscillatory integral with complex-valued phase function, global both in space and in time. In particular, a refined, geometric version of the method will be presented, in the Riemannian setting: the adoption of a distinguished complex-valued phase function, naturally dictated by the geometric framework, will allow us to visualise the process of circumventing topological obstructions. The microlocal method is explicit and constructive; the calculation of the subprincipal symbol of the propagator enables us to recover asymptotic spectral properties of the operators at hand. I will discuss explicit formulae and recent results for the wave operator. Time permitting, the extension of the method to Lorentzian spacetimes will be briefly analysed.\nThis is joint work with D. Vassiliev (UCL) and M. Levitin (Reading).
Phasenfeldmethoden zur Querschnittsoptimierung eines Pflanzenstiels
Tuesday, 6.11.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Mathematik hat viele Gesichter
Tuesday, 6.11.18, 19:30-20:30, Hörsaal II, Albertstr. 23b
Mathematik hat viele Gesichter - angewandt, abgewandt und zugewandt\n\n...angewandt: Mathematik lernen - wozu soll das gut sein? Eine Antwort darauf ist ein anwendungs- und realitätsorientierter Mathematikunterricht. Er zeigt: Mathematik ist nützlich.\n\n...abgewandt: Doch Mathematik kann auch einfach nur "schön" sein. Für nichts gut. Einfach nur schön. In einen allgemeinbildenden Mathematikunterricht gehört auch diese Seite.\n\nDazu stelle ich eine Reihe überraschend einfacher, anschaulich-begreifbarer Beispiele vor. Und neben angewandt und abgewandt wird etwas Drittes deutlich, nämlich zugewandt: Um den Schülerinnen und Schülern "meine" Mathematik näherbringen zu können, muss ich mich ihnen zuwenden - ehrlich, transparent, klar, verlässlich.
Thursday, 8.11.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Geometry of intersections of some secant varieties to algebraic curves
Friday, 9.11.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. In this talk we consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We will describe some interesting cases with unexpected transversality properties and, if time permits, explain a general method to verify when this intersection is empty.
Statistical estimation under differential privacy constraints
Friday, 9.11.18, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
One of the many new challenges for statistical inference in the information age is the increasing concern of data privacy protection. \nA particularly fruitful approach, that offers strong protection against privacy breaches, is the concept of `differential privacy' (Dwork et al. (2006)). The idea is that instead of the original database, only a randomly perturbed version is released for further analysis. Such a randomization mechanism is said to provide differential privacy if the conditional distribution of the released database given the original data does not depend too much on any individual entry of the true database. \n\nThis talk attempts to provide a general introduction to the notion of differential privacy from the point of few of mathematical statistics, but is directed at a broad audience. After discussing the main ideas of differential privacy, we will briefly recall the concept of minimax optimal estimation and survey some of the few existing results from the statistics literature on estimation under differential privacy. In this setup, the objective is not only to come up with an optimal estimation procedure that efficiently recovers information from the randomized observations, but also to devise a randomization mechanism that best facilitates subsequent estimation while respecting the required privacy provisions. \n\nIn the second part of the talk, we will present some of our own results on minimax optimal locally private estimation of linear functionals. Our analysis allows for a quantification of the price, in terms of statistical accuracy, that has to be paid for achieving differential privacy. This price appears to be highly problem dependent.\n
Discrete Gaussians, theta functions and abelian varieties
Monday, 12.11.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The Gaussian distribution is a central object in mathematics and it can be characterised as the unique probability on the real numbers that maximises entropy, for fixed mean and variance. It turns out that the same property can be used to define a discrete Gaussian distribution on the integers. Moreover, the discrete Gaussian is parametrised naturally by the Riemann theta function, and, as such, it has a natural connection to the geometric theory of complex tori, or, more precisely, abelian varieties. The aim of the talk is to present this connection and to show how question in probability give rise to natural problems in geometry and viceversa. This is joint work with Carlos Amendola (TU Munich)
Splitting trees
Wednesday, 14.11.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
We investigate some types of non-ccc tree forcings for adding splitting reals. In particular we focus on some questions concerning the ideals and the regularity properties associated with such splitting trees.\nWe aim to provide a proof for a positive answer to the following question posed by Spinas: Can one prove in ZFC that the additivity of the splitting tree ideal is less than the bounding number?
How to optimally stir your coffee: Challenges in differential equations
Thursday, 15.11.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Mixing of different fluids is an ubiquitous phenomenon in sciences, technology, and everyday life. Nevertheless it is fair to say that we are quite far from a clear mathematical understanding of its analytical properties. In this talk I will present my perspective on this problem by describing a suitable mathematical framework for mixing phenomena and by proving a "toy theorem" in a simplified setting. The role of measure theory in the analysis of irregular partial differential equations will be emphasised.
The André–Oort conjecture: statement and motivation
Friday, 16.11.18, 09:30-10:30, Raum 404, Ernst-Zermelo-Str. 1
Introduction to o-minimal structures
Friday, 16.11.18, 11:00-12:00, Raum 404, Ernst-Zermelo-Str. 1
The Manin-Mumford conjecture for algebraic tori
Friday, 16.11.18, 13:45-14:45, Raum 404, Ernst-Zermelo-Str. 1
The proof of the André–Oort conjecture
Friday, 16.11.18, 15:15-16:15, Raum 404, Ernst-Zermelo-Str. 1
Hyperkähler manifolds from higher sections
Monday, 19.11.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Almost by definition, every Hyperkähler manifold M comes with a 2-sphere of complex structures. These all combine to a family of Kähler manifolds over the complex projective line CP1. Its total space, the twistor space Z of M, has many interesting properties. For example, it carries a real involution covering the antipodal map of CP1. A section of the projection from Z to CP1 is called a real section if it is invariant under the real involution of Z. \nIn case M is the moduli space of Hitchin’s self-duality equations, every solution of these equations determines a real section. Simpson called these preferred sections and posed the question if every real section is a preferred section. A positive answer would imply a complex-analytic way to determine solutions of the self-duality equations. This would be surprising because these are non-linear PDEs. However, a negative answer has recently been given by Heller-Heller by constructing so-called higher sections. \nIn this talk, we show that a symmetric subspace of higher sections is a Hyperkähler manifold which is unexpected because they are a priori unrelated to points of the initial Hyperkähler manifold M. Moreover, we explain the implications to a complex-analytic approach to solutions of the self-duality equations as envisioned by Simpson.
Null-Eins-Gesetze und Logik der Zufallsgraphen
Tuesday, 20.11.18, 19:30-20:30, Hörsaal II, Albertstr. 23b
In der Schule ist die Ergebnismenge eines Wahrscheinlichkeitsraumes meist endlich. Wie können wir nun die Wahrscheinlichkeit berechnen, dass ein zufällig gegebener endlicher Graph eine konkrete Eigenschaft besitzt, zum Beispiel, dass er eine gerade Anzahl von Knoten oder von Kanten besitzt? Weil es unendlich viele endliche Graphen gibt, ist die Wahrscheinlichkeit zunächst einmal unbestimmt. Wir müssen zunächst klären, was mit "zufällig gegeben" gemeint sein könnte. Überraschenderweise kann uns die Mathematische Logik helfen, solche Fragen aus der Wahrscheinlichkeitstheorie und der Kombinatorik zu beantworten.\nDieser Vortrag richtet sich an ein allgemeines mathematisches Publikum und verlangt keine Vorkenntnisse in Mathematischer Logik.
From collapsing functions to admissible sets
Wednesday, 21.11.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
I will present a notion of "almost" order-preserving collapsing function, mapping large ordinals (uncountable/non-recursive) to smaller ones (countable/recursive). While this notion is inspired by impredicative ordinal analysis it does, I believe, lead to natural and elegant objects of set theory. I will show that the existence of collapsing functions is equivalent to the existence of admissible sets, and hence to Pi^1_1-comprehension. This result can also be read as a combinatorial characterization of the Church-Kleene ordinal. A preprint is available as arXiv:1809.06759\n
Two polarized K3 surfaces associated to the same cubic fourfold
Friday, 23.11.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
For infinitely many d, Hassett showed that special cubic fourfolds of\ndiscriminant d are related to polarized K3 surfaces of degree d via\ntheir Hodge structures. For half of the d, a generic special cubic has\nnot one but two different associated K3 surfaces. This induces an\ninvolution on the moduli space of polarized K3 surfaces of degree d. We\ngive a geometric description of this involution. As an application, we\nobtain examples of Hilbert schemes of two points on K3 surfaces that are\nderived equivalent but not birational.
\(G_2\)-orbifolds with ADE-singularities
Monday, 26.11.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
We study \(G_2\)-orbifolds whose singularities are modelled on \(\bmathbb R^3\btimes\bmathbb R^4/G\),\nwhere \(G\) is a finite subgroup\nof \(SU(2)\). Orbifolds of this kind have applications in M-theory and they may\ndefine boundary components of the \nmoduli space of parallel \(G_2\)-structures. We show how the existing construction\nmethods for \(G_2\)-manifolds can be \nmodified such that they produce \(G_2\)-orbifolds. A recent result of D. Joyce and\nS. Karigiannis allows us to resolve \nthe singularities under certain circumstances. Therefore, we are able to\nconstruct smooth \(G_2\)-manifolds with new \nvalues of the second and third Betti number.
Strain-gradient plasticity with cross-hardening
Tuesday, 27.11.18, 15:00-16:00, Raum 226, Hermann-Herder-Str. 10
Nichtäquationale Theorien
Wednesday, 28.11.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Quantorenelimination für reine Erweiterungen von abelschen Gruppen
Wednesday, 28.11.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Für dreisortige Strukturen (A,B,C), wo B eine reine Erweiterung von\nA ist, und C der Quotient B/A, geben wir eine Quantorenlimination an, die\nFormeln phi(x,..) in Formeln psi(r(x,..),..) übersetzt, die nur noch über die\nSorten A und C sprechen. Dabei sind die Terme r,.. einfache definierbare\nFunktionen von B nach A^eq und C.\n\nDas ist Teil einer gemeinsamen Arbeit mit Aschenbrenner, Chernikov und Gehret.\nAls Folgerung ergibt sich dort zum Beispiel\n\n Sei (K,O) ein henselscher Körper, dessen\n Restklassenkörper k die Charakteristik 0 hat. Dann ist\n (K,O) genau dann distal, wenn k und die Wertegruppe distal\n sind.\n
Quantorenelimination für reine Erweiterungen von abelschen Gruppen
Wednesday, 28.11.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Für dreisortige Strukturen (A,B,C), wo B eine reine Erweiterung von\nA ist, und C der Quotient B/A, geben wir eine Quantorenlimination an, die\nFormeln phi(x,..) in Formeln psi(r(x,..),..) übersetzt, die nur noch über die\nSorten A und C sprechen. Dabei sind die Terme r,.. einfache definierbare\nFunktionen von B nach A^eq und C.\n\nDas ist Teil einer gemeinsamen Arbeit mit Aschenbrenner, Chernikov und Gehret.\nAls Folgerung ergibt sich dort zum Beispiel\n\n Sei (K,O) ein henselscher Körper, dessen\n Restklassenkörper k die Charakteristik 0 hat. Dann ist\n (K,O) genau dann distal, wenn k und die Wertegruppe distal\n sind.\n
Quantorenelimination für reine Erweiterungen von abelschen Gruppen
Wednesday, 28.11.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Für dreisortige Strukturen (A,B,C), wo B eine reine Erweiterung von\nA ist, und C der Quotient B/A, geben wir eine Quantorenlimination an, die\nFormeln phi(x,..) in Formeln psi(r(x,..),..) übersetzt, die nur noch über die\nSorten A und C sprechen. Dabei sind die Terme r,.. einfache definierbare\nFunktionen von B nach A^eq und C.\n\nDas ist Teil einer gemeinsamen Arbeit mit Aschenbrenner, Chernikov und Gehret.\nAls Folgerung ergibt sich dort zum Beispiel\n\n Sei (K,O) ein henselscher Körper, dessen\n Restklassenkörper k die Charakteristik 0 hat. Dann ist\n (K,O) genau dann distal, wenn k und die Wertegruppe distal\n sind. \n\n
Logarithms, constructible functions and integration on non-archimedean models of the theory of the real field with restricted analyticfunctions with value groups of finite archimedean rank
Thursday, 29.11.18, 16:15-17:15, Raum 414, Ernst-Zermelo-Str. 1
We work in a model of the theory of the real field with restricted analytic functions such\nthat its value group has finite archimedean rank. An example is given by the field of Puiseux series over\nthe reals. We show how one can extend the restricted logarithm to a global logarithm with values in the\npolynomial ring over the model with dimension the archimedean rank. The logarithms are determined\nby algebraic data from the model, namely by a section of the model and by an embedding of the value\ngroup into its Hahn group. If the archimedean rank of the value group coincides with the rational rank the\nlogarithms are equivalent. We illustrate how one can embed such a logarithm into a model of the real field\nwith restricted analytic functions and exponentiation. This allows us to define constructible functions with\ngood lifting properties. As an application we establish a full Lebesgue measure and integration theory\nwith values in the polynomial ring.\n
Chow schemes in mixed characteristic
Friday, 30.11.18, 10:30-11:30, Hörsaal, Otto-Krayer-Haus
Spaces parametrizing positive algebraic cycles have been in use in algebraic geometry for a long time.\nHowever in positive and mixed characteristic we do not know to which extent these spaces can be understood in terms of moduli problems. Some progress has been made however:\nin '96 Suslin and Voevodsky introduced a presheaf of effective relative zero cycles on the category of normal varieties and proved that it is isomorphic to the presheaf represented by infinite symmetric powers (after localization by the characteristic of the field when it is positive). The aim of this talk is to explain how Suslin and Voevodsky's theorem\ncan be generalized to schemes of mixed characteristic and also to higher dimensional cycles. We intend the talk to be understandable for algebraic geometers of various backgrounds thus we start by recalling the definition of a relative cycle and give an insightful example as well as introduce other useful notions such as Voevodsky's h-topology.\nAfter stating our theorem we briefly explain the strategy behind its proof and give a relatively detailed proof of one of the key components.
siehe link in der Zusammenfassung
Friday, 30.11.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1