Birational geometry of foliations
Freitag, 2.7.21, 10:30-11:30, virtueller Raum Lasker
I will try to explain, by means of some examples and recent results, how the classical framework of the Minimal Model Program has been extended to the case of foliation, in particular in low dimension, as well as, how it has been used to initiate a systematic study and classification of foliations from a birational view point. \nThe talk will feature joint work with C. Spicer.
Singularity categories and singular Hochschild cohomology
Montag, 5.7.21, 16:15-17:15, Anderssen (BBB)
The singularity category was introduced by Buchweitz and then rediscovered by Orlov motivated by the homological mirror symmetry conjecture. Following Buchweitz, in analogy with Hochschild cohomology, one defines the singular Hochschild cohomology of an algebra as the Yoneda algebra of the diagonal bimodule in the singularity category of bimodules. \n\nThe first half of the talk is an introduction to singularity categories and Hochschild cohomology. The second half will show that singular Hochschild cohomology is endowed with the same rich algebraic structure as classical Hochschild cohomology, namely a Gerstenhaber bracket in cohomology and a B-infinity structure at the cochain complex level. We will also talk about its relationship with the deformation theory of singularities.
Regularitätsbedingungen für den verallgemeinerten Cantorraum
Dienstag, 6.7.21, 14:30-15:30, BBB-Raum Philidor
Jedes Baumforcing P definiert eine Regularitätsbedingung, die P-Messbarkeit genannt wird. So korrespondiert z.B. die\nCohen-Messbarkeit mit der Baire-Eigenschaft. Es wurde gezeigt, dass für eine große Familie von Teilmengen der reellen Zahlen sowohl Cohen- als auch Mathias-messbar jeweils Silver-messbar implizieren.\n\nWir zeigen, dass sich diese Resultate auf den verallgemeinerten Cantorraum übertragen lassen.\nEs werden alle grundlegenden Begriffe eingeführt und gezeigt, wie sich bekannte Baumforcings für überabzählbare Kappa verallgemeinern lassen.\n\n
Frobenius kernels for automorphism group schemes
Freitag, 9.7.21, 10:30-11:30, virtueller Raum Lasker
We establish structure results for Frobenius kernels of automorphism group schemes for surfaces of general type in positive characteristics. It turns out that there are surprisingly few possibilities. This relies on properties of the famous Witt algebra, a simple Lie algebra without finite-dimensional counterpart over the complex numbers, together with is twisted forms. The result actually holds true for rather general schemes, under the assumption that the Frobenius kernel has large isotropy group at the generic point. This is joint work with Nikolaos Tziolas.\n
Deep Learning for Brain Signals
Freitag, 9.7.21, 13:00-14:00, online: Zoom
Relativistische Modelle des Universums um einen zentralen Stern
Montag, 12.7.21, 16:15-17:15, bbb Raum Anderssen
Wir betrachten in diesem Vortrag eine zentrale Masse, die wir als statisch und kugelsymmetrisch annehmen. Ziel wird es sein, die diese Masse umgebende Raumzeit differentialgeometrisch zu beschreiben. Wir werden hierzu zwei Modelle entwickeln und untersuchen: Die intuitivere Schwarzschild-Raumzeit, sowie die Kruskal-Raumzeit. Dabei werden wir ein besonderes Augenmerk auf die auftretenden Singularitäten legen, wobei wir zwischen Koordinatensingularitäten und physikalischen Singularitäten unterscheiden.
Trees, Stationary Reflexion, and Mahlo Cardinals
Dienstag, 13.7.21, 13:30-14:30, BBB Philidor
A major thread of set-theoretic research focuses on realizing the compactness properties of large cardinals at accessible cardinals like \(\baleph_2\) or \(\baleph_{\bomega+1}\), thus answering questions that one could naturally pose without realizing that large cardinals are relevant. We discuss recent work with Thomas Gilton and Sarka Stejskalova in which we realized an array of consistency results concerning variants of the tree property and the stationary reflection for double successors of regular cardinals like \(\baleph_2\).
Finite Element Approximation of Hamilton-Jacobi-Bellman equations with nonlinear mixed boundary conditions
Dienstag, 13.7.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
\n\nWe show uniform convergence of monotone P1 finite element methods to the viscosity solution of isotropic parabolic Hamilton-Jacobi-Bellman equations with mixed boundary conditions on unstructured meshes and for possibly degenerate diffusions. Boundary operators can generally be discontinuous across face-boundaries and type changes. Robin-type boundary conditions are discretised via a lower Dini derivative. In time the Bellman equation is approximated through IMEX schemes. Existence and uniqueness of numerical solutions follows through Howard’s algorithm. We show how equations of this type naturally appear in models of mathematical finance.
Cohen-Lenstra-Martinet heuristics on class groups of number fields
Freitag, 16.7.21, 10:30-11:30, virtueller Raum Lasker
In the 1980s Cohen and Lenstra proposed a probabilistic model\nfor the behaviour of class groups of quadratic number fields. A few\nyears later, it was generalised by Cohen and Martinet to class groups\nof more general families of number fields. Recently, in joint work with\nLenstra we disproved these conjectures -- in two completely different\nways, and in joint work with Lenstra and Johnston we have offered a\ncorrected version. In my talk I will give an overview of this work.
Freitag, 16.7.21, 13:00-14:00, online: Zoom
Determinants, group cocycles and multiplicative Chern character
Montag, 19.7.21, 16:15-17:15, Anderssen (BBB)
The well known central extension of loop groups is an example of a group two-cocycle naturally constructed from action of the restricted linear group on a certain non-linear category of idempotents in a polarised Hilbert space. We will explain the concepts involved in this construction, its generalisation to a construction of higher cocycles and give some examples of non-trivial three-cocycles for the double loop group, both formal and smooth. On the other hand, these group cocycles lead to functionals on algebraic K-theory, the so called regulators. We will sketch this relation and, in particular, the relation to the Tate tame symbol in algebraic geometry and multiplicative Chern of Connes and Karoubi associated to universal finitely summable Fredholm modules.
Variable exponent Bochner-Lebesgue spaces with symmetric gradient structure
Dienstag, 20.7.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
We introduce function spaces for the treatment of non-linear parabolic equations with variable log–Hölder continuous exponents, which only incorporate information of the symmetric part of a gradient. As an analog of Korn’s inequality for these functions spaces is not available, the construction of an appropriate smoothing method proves itself to be difficult. Using a pointwise Poincaré inequality near the boundary of a bounded Lipschitz domain\ninvolving only the symmetric gradient, we construct a smoothing operator with convenient properties. In particular, this smoothing operator leads to several density results, and therefore to a generalized formula of\nintegration by parts with respect to time. Using this formula and the theory of maximal monotone operators, we prove an abstract existence result."
The effective model structure and infinity-groupoid objects
Freitag, 23.7.21, 10:30-11:30, virtueller Raum Lasker
I will discuss a construction of a new model structure on\nsimplicial objects in a countably lextensive category (i.e., a category\nwith well behaved finite limits and countable coproducts). This builds\non previous work on a constructive model structure on simplicial sets,\noriginally motivated by modelling Homotopy Type Theory, but now\napplicable in a much wider context. This is joint work with Nicola\nGambino, Simon Henry and Christian Sattler.\n
Federated analysis using different cohorts - are we comparing apples and oranges?
Freitag, 23.7.21, 13:00-14:00, online: Zoom