Steenrod-Squares in differential cohomology
Dienstag, 2.6.20, 16:15-17:15, virtuellen SR 125
In my master's thesis I tried to construct the Steenrod squaring operations in a\ndifferential cohomology theory.\nOne construction method for singular cohomology is based on the graded\ncommutativity of the cup product. For odd Squares this method can be imitated in\nthe Cheeger-Simons-Model of differential cohomology and produces\nthe same differential operations which were already given by D. Grady and H.\nSati. It also gives another explanation\nwhy the even differential Squares cannot exist in general.\nIf time permits, I can discuss an approach to some even Squares that makes use of\nWu classes.\n\n
Twists of quaternionic Kähler manifolds
Montag, 8.6.20, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces
Freitag, 12.6.20, 10:30-11:30, virtueller Raum 404
K3 surfaces have been extensively studied over the past decades for\nseveral reasons. For once, they have a rich and yet tractable geometry\nand they are the playground for several open arithmetic questions.\nMoreover, they form the only class which might admit more than one\nelliptic fibration with section. A natural question is to ask if one\ncan classify such fibrations, and indeed that has been done by several\nauthors, among them Nishiyama, Garbagnati and Salgado.\nIn this joint work with A. Garbagnati, C. Salgado, A. Trbović and R.\nWinter we study K3 surfaces defined over a number field k which are\ndouble covers of extremal rational elliptic surfaces. We provide a list\nof all elliptic fibrations on certain K3 surfaces together with the\ndegree of the field extension over which each genus one fibration is\ndefined and admits a section. We show that the latter depends, in\ngeneral, on the action of the cover involution on the fibers of the\ngenus one fibration.
ODE methods in non-local equations
Dienstag, 16.6.20, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk, I will introduce some ODE-type methods to find radially symmetric solutions for non-local PDEs. I will provide an overview and some improvements of the main ideas behind these methods. The main underlying idea is understanding a non-local PDE problem as an infinite dimensional ODE system. These methods were first developed in the non-local gluing scheme to find solutions for the fractional Yamabe problem, which are singular along a non-zero dimensional submanifold. I will also show some new applications.\n\nIn particular, I will present a variation of constants formula for fractional Hardy operators, a suitable extension in the spirit of Caffarelli–Silvestre and an equivalent formulation as an infinite system of second order constant coefficient ODEs. Moreover, I will show that, like in classical ODEs, quantities such as the Hamiltonian and Wronskian may then be used. As an example of applications, we obtain a Frobenius theorem and a new proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane-Emden equation. We also establish new Pohozaev identities.\n\nThis is a joint work with Weiwei Ao, Hardy Chan, Marco Fontelos, María del Mar González and Juncheng Wei.
TBA
Montag, 22.6.20, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Donnerstag, 25.6.20, 17:00-18:00, Hörsaal II, Albertstr. 23b
"(Ir)rationality of L-values"
Freitag, 26.6.20, 10:30-11:30, virtueller Raum 404
Euler’s beautiful formula\nζ(2n) = −\n(2πi) 2n\nB 2n .\n2(2n)!\ncan be seen as the starting point of the investigation of special values of L-\nfunctions. In particular, Euler’s result shows that all critical zeta values are ra-\ntional up to multiplication with a particular period, here the period is a power of\n(2πi). Conjecturally this is expected to hold for all critical L-values of motives.\nIn this talk, we will focus on L-functions of number fields. In the first part of the\ntalk, we will discuss the ’critical’ and ’non-critical’ L-values exemplary for the\nRiemann zeta function. Afterwards, we will head on to more general number\nfields and explain our recent joint result with Guido Kings on the algebraicity\nof critical Hecke L-values for totally imaginary fields up to explicit periods.
"(Ir)rationality of L-values"
Freitag, 26.6.20, 10:30-11:30, v404
Euler’s beautiful formula\nζ(2n) = −\n(2πi) 2n\nB 2n .\n2(2n)!\ncan be seen as the starting point of the investigation of special values of L-\nfunctions. In particular, Euler’s result shows that all critical zeta values are ra-\ntional up to multiplication with a particular period, here the period is a power of\n(2πi). Conjecturally this is expected to hold for all critical L-values of motives.\nIn this talk, we will focus on L-functions of number fields. In the first part of the\ntalk, we will discuss the ’critical’ and ’non-critical’ L-values exemplary for the\nRiemann zeta function. Afterwards, we will head on to more general number\nfields and explain our recent joint result with Guido Kings on the algebraicity\nof critical Hecke L-values for totally imaginary fields up to explicit periods.
Shape optimization of convex, rotational symmetric domains for an eigenvalue problem arising in optimal insulation
Dienstag, 30.6.20, 14:15-15:15, virtueller SR 226, HH-10
I will present the main results from my master thesis about shape optimization for an eigenvalue problem arising in optimal insulation. Under the assumption of convexity, the existence of an optimal domain can be proven, but due to the difficulties of approximating convex domains in \(\bmathbb{R}^3\), the constraint to rotational symmetric domains is used to reduce the problem to a two-dimensional setting, allowing the numerical approximation of optimal domains.