Hodge theory and formality
Friday, 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.
Understanding Biological Processes using Stochastic Modelling
Friday, 1.12.17, 12:00-13:00, Raum 404, Eckerstr. 1
The molecular biology of life seems inaccessibly complex, and gene expression is an essential part of it. It is subject to random variation and not exactly predictable. Still, mathematical models and statistical inference pave the way towards the identification of underlying gene regulatory processes. In contrast to deterministic models, stochastic processes capture the randomness of natural phenomena and result in more reliable predictions of cellular dynamics. Stochastic models and their parameter estimation have to take into account the nature of molecular-biological data, including experimental techniques and measurement error.\n \nThis talk presents according modelling and estimation techniques and their applications: the derivation of mRNA contents in single cells; the identification of differently regulated cells from heterogeneous populations using mixed models; and parameter estimation for stochastic differential equations using computer-intensive Markov chain Monte Carlo techniques.
Generalized Seiberg-Witten equations and almost-Hermitian geometry
Monday, 4.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
I will talk about a generalisation of the Seiberg-Witten equations introduced by Taubes and Pidstrygach, in dimension 3 and 4 respectively, where the spinor representation is replaced by a hyperKahler manifold admitting certain symmetries. I will discuss the 4-dimensional equations and their relation with the almost-Kahler geometry of the underlying 4-manifold. In particular, I will show that the equations can be interpreted in terms of a PDE for an almost-complex structure on 4-manifold. This generalises a result of Donaldson.
Additive manufacturing of scaffolds for bone regeneration
Tuesday, 5.12.17, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Inverse mean curvature flow in complex hyperbolic space
Tuesday, 5.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
Abstract: We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the complex hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub- Riemannian metric on the sphere. Finally we show that there exists a family of examples such that the Webster curvature of this sub-Riemannian limit is not constant.
tba
Wednesday, 6.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
Infinite Populations, Choice and Determinacy
Wednesday, 6.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
This talk criticizes non-constructive uses of set theory in formal economics. The main focus is on results on preference aggregation and Arrow's theorem for infinite electorates, but the present analysis would apply as well, e.g., to analogous results in intergenerational social choice. To separate justified and unjustified uses of infinite populations in social choice, I suggest a principle which may be called the "Hildenbrand criterion" and argue that results based on unrestricted Axiom of Choice (AC) do not meet this criterion. The technically novel part is a proposal to use a set-theoretic principle known as the Axiom of Determinacy (AD). A particularly appealing aspect of AD from the point of view of the research area in question is its game-theoretic flavor.\n
Thursday, 7.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Recent results and open problems about Oeljeklaus-Toma manifolds
Friday, 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.
Borcherds-Kac-Moody Algebras in Conformal Field Theory
Monday, 11.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
Borcherds-Kac-Moody algebras are a generalization of finite dimensional semisimple Lie algebras obtained by weakening the requirements on Cartan matrices. In his proof of the Moonshine Conjectures, Borcherds related them to the vertex operator algebras from conformal field theory. At the same time, there appears to be a connection to automorphic forms via denominator functions. This can hopefully be leveraged, in particular, to investigate so-called Bogomol'nyi-Prasad-Sommerfield states in field theories with supersymmetry.
tba
Wednesday, 13.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
Forcing over ord-transitive models
Wednesday, 13.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
Usually forcing is performed over transitive countable ground models.\nHowever, there are technical means to waive transitivity. In this talk we\nshall focus on the algebraic features of suitable ground models. We explain\nord-transitive models, labelled models, the ord-collapse, and\ntheir relations to the Mostowski collapse.
1-Motives
Thursday, 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.
Thursday, 14.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
Period domains
Friday, 15.12.17, 10:30-11:30, Hörsaal FRIAS, Albertstr. 19
Variational formulas for the Selberg zeta function and applications to curvature asymptotics
Monday, 18.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
In this talk, we will study the Selberg zeta function and its relatives. We will recall the celebrated Selberg trace formula, and the geometric setting of our work, the Teichmüller space of Riemann surfaces of genus, g. As shown by Zograf and Takhtajan, the Selberg trace formula connects the Ricci curvature of the Hodge bundle \(H^0 (K^m)\) over Teichmüller space together with the second variation of the Selberg zeta function at integer points. We will briefly explain this connection and the role of the Selberg trace formula in its derivation. \n \nFurther, we will investigate the behavior of the Selberg zeta function, \(Z(s)\), as a function on Teichmüller space. We will deduce an explicit formula for the second variation of \(\blog( Z(s) )\) via a certain infinite sum involving lengths of closed geodesics of the underlying surface and their variations. We will then utilize this formula to study the asymptotics of the second variation of \(\blog( Z(s) )\) as \(s \bto \binfty\). We shall see that the most prominent role is played by the systole geodesics. Moreover, the dimension of the kernel of the first variation of the latter appears in the signature of the Hessian of \(\blog Z(s)\) for large \(s\). In conclusion, we will show how our variational formula and its asymptotics have interesting implications for the curvature of the Hodge bundle and its relationship to the Quillen curvature. \n\nThis is a joint work with Julie Rowlett and Genkai Zhang.
Tuesday, 19.12.17, 16:15-17:15, Raum 404, Eckerstr. 1
The inverse mean curvature flow and the Riemannian Penrose inequality from Husiken and Imanen.
Tuesday, 19.12.17, 17:00-18:00, Raum 404, Eckerstr. 1
Separabel abgeschlossene Körper sind äquational.
Wednesday, 20.12.17, 16:30-17:30, Raum 404, Eckerstr. 1
Der Imperfektionsgrad eines Körpers \(K\) positiver Charakteristik \(p\) ist im Grunde die linear Dimension von \(K\) als \(K^p\)-Vektorraum. Der Körper \(K\) ist separabel abgeschlossen, falls er keine echte separable algebraische Erweiterung besitzt, wobei ein algebraisches Element \(\balpha\) über \(K\) separabel ist, wenn sein minimal Polynom keine doppelten Nullstellen (im algebraischen Abschluss) hat. \n\nDie Theorie separabel abgeschlossener Körper der Charakteristik \(p>0\) ist axiomatisierbar, und ihre Vervollständigungen werden durch den Imperfektionsgrad bestimmt. Insbesondere ist die Theorie separabel abgeschlossener Körper der Charakteristik \(p>0\) und unendlichen Imperfektionsgrades vollständig und stabil. Diese Theorie hat keine Elimination von Imaginären in der Ringsprache. \n\nIn Zusammenarbeit mit Martin Ziegler werden wir zeigen, dass diese Theorie äquational ist. Äquationalität ist eine Art lokaler Noetherianität und impliziert eine relative Elimination von Imaginären. Wir werden zeigen, dass gewisse Formeln Gleichungen in einem geeigneten Modell sind, nämlich in einem differentiell abgeschlossenen Körper der Charakteristik \(p\), dessen modelltheoretische Eigenschaften von Carol Wood beschrieben wurden.
Complete convex surfaces: intrinsic and extrinsic properties
Thursday, 21.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b
We will examine convex surfaces which divide the ambient Euclidean 3-space into two parts. By the Theorem of Gauss-Bonnet such surfaces need to be of the intrinsic type of a cylinder, plane, or sphere. We will then discuss the extrinsic symmetry property of rotational invariance, and its infinitesimal version at a point of the surface. We outline the proof of a global extrinsic conjecture of Victor Andreevich Toponogov : "Any convex plane admits (at least) one point of infinitesimal symmetry, possibly at infinity". The proof, in collaboration with Brendan Guilfoyle, uses complex analysis and a parabolic curvature flow in the space of lines of Euclidean 3-space.
Thursday, 28.12.17, 17:00-18:00, Hörsaal II, Albertstr. 23b