program discussion
Montag, 15.10.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Shape Analysis and Non-Linear PDEs
Dienstag, 16.10.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Donnerstag, 18.10.18, 17:00-18:00, Hörsaal II, Albertstr. 23b
Projectivity of Rigid Group Actions on Complex Tori
Freitag, 19.10.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we shall discuss a result already obtained by Torsten Ekedahl around 1999, stating that every complex torus \(T\) admitting a rigid group action of a finite group \(G\) is in fact projective, i.e., an Abelian variety. Firstly, we shall explain the notion of “deformations of the pair \((T,G)\)“; afterwards the proof of Ekedahl’s Theorem will be outlined and the projectivity of \(T\) will be shown explicitly. If time allows, applications of Ekedahl’s result will be explained towards the end of the seminar talk. This is (partly) joint work with Fabrizio Catanese.\n\n
The eta invariant under conic degeneration
Montag, 22.10.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
I will report on the progress of my phd thesis, presenting a result on the behaviour of the eta invariant of the Dirac and Hodge operator under conic degeneration: under favorable assumptions, realized when e.g. the link of a cone is a space form, the eta invariant tends to the invariant of the degenerated space plus a constant term. I´ll discuss part of the proof and, if time allows, ongoing work for simple edge spaces.
Asymptotic rigidity of layered structures and its applications in homogenization theory
Dienstag, 23.10.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Rigidity results in elasticity are powerful statements that allow to derive global properties of a deformation from local ones. The classical Liouville theorem states that every local isometry of a domain corresponds to a rigid body motion. If connectedness of the set fails, clearly, global rigidity can no longer be true. \nIn this talk, I will present a new type of asymptotic rigidity lemma, which shows that if an elastic body contains sufficiently stiff connected components arranged into fine parallel layers, then macroscopic rigidity up to horizontal shearing prevails in the limit of vanishing layer thickness. The optimal scaling between layer thickness and stiffness can be identified using suitable bending constructions. This result constitutes a useful tool for proving homogenization results of variational problems modeling high-contrast bilayered composites. We will finally utilize it to characterize the homogenized Gamma-limits of two models inspired by nonlinear elasticity and finite crystal plasticity. \n\nThis is joint work with Fabian Christowiak (Universität Regensburg).\n
Blow up criteria for geometric flows on surfaces
Dienstag, 23.10.18, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
I will present a scheme to prove blow up criteria for various (intrinsic) geometric flows on closed surfaces. \nThe uniformization theorem allow us to split a curve of Riemannian metrics into a curve of constant curvature metrics and conformal factors. A further refinement, due to Buzano, Rupflin and Topping, allows us to view the evolution of the curve of constant curvature metrics as a finite dimensional dynamical system.\n\nCombining this splitting with a compactness theorem adapted to the situation allows us to apply standard PDE techniques to rule out blow up under certain conditions, depending on the flow. I will discuss the approach for the specific example of the harmonic Ricci flow.\n
Curve flows with a global forcing term
Dienstag, 23.10.18, 17:30-18:30, Raum 404, Ernst-Zermelo-Str. 1
We study curve shortening flow with global forcing terms for embedded, closed, smooth curves in the plane. We derive an analogue to Huisken's distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below \(-\bpi\) and show that this condition is sharp. For bounded forcing terms, this excludes singularities in finite time. For immortal flows whose forcing terms provide non-vanishing enclosed area and bounded length, we prove convexity in finite time and smooth and exponential convergence to a circle. In particular, the above holds for the area preserving curve shortening flow.
On a question of Babai and Sós, a nonstandard approach
Mittwoch, 24.10.18, 16:30-17:30, Raum 404, Ernst-Zermelo-Str. 1
Abstract: In 1985, Babai and Sós asked whether there exists a constant c>0 such that every finite group of order n has a product-free set of size at least cn, where a product-free set of a group is a subset that does not contain three elements x,y and z satisfying xy=z. Gowers showed that the answer is no in the early 2000s, by linking the existence of product-free sets of large density to the existence of low\ndimensional unitary representations.\n\nIn this talk, I will provide an answer to the aforementioned question by model theoretic means. Furthermore, I will relate some of Gowers' results to the existence of nontrivial definable compactifications of nonstandard finite groups.
Finiteness of perfect torsion points of an abelian variety
Freitag, 26.10.18, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
I will report on a joint work with Emiliano Ambrosi. Let k be a field\nthat\nis finitely generated over the algebraic closure of a finite field. As\na\nconsequence of the theorem of Lang-Néron, for every abelian variety\nover k\nwhich does not contain any isotrivial abelian variety, the group of\nk-rational torsion points is finite. We show that if k^perf is a\nperfect\nclosure of k, the group of k^perf-rational torsion points is finite as\nwell. This gives a positive answer to a question asked by Hélène\nEsnault in\n2011. To prove the theorem we translate the problem to a certain\nquestion\non morphisms of F-isocrystals. Subsequently, we handle the question\nstudying the monodromy groups of the F-isocrystals involved. We can\nprove\nthat a certain monodromy group is "big" via an argument with Frobenius\ntori. Then class field theory and some considerations on the slopes\nconclude the proof. As an additional outcome of our work we prove a\nweak\n(weak) semi-simplicity statement for p-adic representations coming\nfrom\npure overconvergent F-isocrystals.\n
The Casson Invariant and Feynman diagrams
Montag, 29.10.18, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
I will review the definition of the Casson-Walker invariant of rational homology spheres and its connection to Feynman graphs. Then I will discuss some recent computations involving the cutting and gluing of these diagrams, and some conjectures that result from these computations.
wird noch bekanntgegeben
Dienstag, 30.10.18, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10