Proofs by example and numerical Nullstellensätze
Friday, 3.12.21, 10:30-11:30, BBB room
We study the proof method "proof by example" in which a general statement can be proved by verifying it for a single example. This strategy can indeed work if the statement in question is an algebraic identity and the example is "generic". This talk addresses the problem of construction a practical example, which is sufficiently generic, for which the statement can be verified efficiently, and which even allows for a numerical margin of error.\nOur answer comes in the form of a numerical Nullstellensatz, which is based on Diophantine geomery, in particular an arithemetic Nullstellensatz and a new effective Liouville-Lojasiewicz type inequality.\n\nIf time permits we moreover consider "proofs by several examples", which in addition requires a conceptual notion of sufficient genericity of a set of points. Besides theoretical and algorithmic criteria for sufficient genericity, we obtain several new types of Nullstellensätze in the spirit of the combinatorial Nullstellensatz and the Schwartz-Zippel lemma, also for varieties.
Knightian uncertainty in Financial Markets
Friday, 3.12.21, 12:00-13:00, online: Zoom
In this talk we revisit uncertainty in probability when the underlying probability measure can not be estimated in a reliable way which is often the case in financial markets. We will see some applications where upper and lower bounds are of interest which lead to non-linear expectation operators in contrast to the very familiar and well-known linear expectation.
Recently on arXiv: 'Systole and small eigenvalues of hyperbolic surfaces' / 'Classical KMS Functionals and Phase Transitions in Poisson Geometry'
Monday, 6.12.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
(i) Let S be a closed orientable hyperbolic surface with Euler characteristic \(\bchi\), and let \(\blambda_k(S)\) be the \(k\)-th positive eigenvalue for the Laplacian on \(S\). According to famous result of Otal and Rosas, \(\blambda_−\bchi >0,25\). In this article, we prove that if the systole of S is greater than \(3,46\), then \(\blambda_{−\bchi−1}>0,25\). This inequality is also true for geometrically finite orientable hyperbolic surfaces without cusps with the same assumption on the systole.\n\n(ii)The authors study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. They discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinstein's seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, they focus on the case of \(\bflat\)-Poisson manifolds, where they provide a complete characterization of the convex cone of KMS measures.
On the canonical base property and transfer of internality
Tuesday, 7.12.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Baldwin and Lachlan proved that an uncountably categorical structure is largely controlled by a strongly minimal set D and the Canonical Base Property (CBP) states that over a realization of a stationary type, its canonical base is always almost internal to the strongly minimal set D.\nChatzidakis, Moosa and Pillay showed under the assumption of the CBP that every almost D-internal type transfers internality on intersections and more generally on quotients. Both properties do not hold in the uncountably categorical structure without the CBP, produced by Hrushovski, Palacín and Pillay.\n\nIn this talk, I will show that transfer of internality of quotients already implies the CBP and present the counter-example to the CBP as an additive cover of the complex numbers. In order to show that this structure does not transfer internality, we must consider imaginary elements (definable equivalence classes) and obtain a connection between elimination of finite imaginaries and the failure of the CBP.
Verständnisorientierter Stochastikunterricht am Gymnasium: Anforderungen an die Lehrerbildung
Tuesday, 7.12.21, 19:30-20:30, Hörsaal II, Albertstr. 23b
Der gymnasiale Stochastikunterricht ist momentan von einer starken Rezeptorientierung geprägt, der auch geltende Bildungspläne Vorschub leisten. Zudem bringt ein Großteil der Lehrkräfte entweder keine Stochastik-Kenntnisse aus dem Studium mit, oder diese Kenntnisse sind in erster Linie durch eine rein mathematische Stochastik mit Elementen der Maßtheorie geprägt. Stochastik gilt gemeinhin als schwierig, weil sie im Spannungsfeld zwischen Mathematik, Modellbildung und persönlichen Erfahrungen mit stochastischen Vorgängen steht. Im Vortrag stelle ich mein Konzept für eine grundständige, im Wesentlichen auf die Tafel verzichtende Stochastik-Vorlesung vor, die auf den Kenntnissen des ersten Studienjahres aufbaut und obigem Spannungsfeld Rechnung trägt.
The Index theorem on end-periodic manifolds
Monday, 13.12.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
Atiyah and Singer published 1963 a formula for the index of an elliptic operator over a closed oriented Riemannian manifold just containing topological terms, known as the Atiyah-Singer index theorem. Forty-one years later they were awarded with the Abel Prize, among other things, for this deep result connecting topology, geometry and analysis.\nIn this talk the Atiyah-Singer index theorem will be formulated and a proof via the heat equation and it's asymptotic expansion will be sketched. Further, a modification of this proof leads to an index theorem for end-periodic Dirac operators discovered by Mrowka, Ruberman and Saveliev in 2014. This end-periodic index theorem and how it is related to the classical Atiyah-Patodi-Singer index theorem for manifolds with boundary are also treated in the talk.
C^1-triangulations of semi-algebraic sets
Friday, 17.12.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
If one wants to treat integration of differential forms over semi-algebraic sets analogous to the case of smooth manifolds, it is desirable to have triangulations of semi-algebraic sets that are globally of class C^1.\nWe will present a proof of the existence of such triangulations using the 'panel beating' method introduced by Ohmoto-Shiota (2017) and discuss possible generalizations.
Sharp adaptive similarity testing with pathwise stability for ergodic diffusions
Friday, 17.12.21, 12:00-13:00, online: Zoom
Suppose we observe an ergodic diffusion with unknown drift. We develop a fully data-driven nonparametric test for the null hypothesis that the drift is similar to a reference drift under supremum loss. Our procedure turns out to be asymptotically optimal in both rate and constant. Moreover, we investigate its behavior if the true process was driven by a fractional Brownian motion with Hurst index close to 1/2.
Weak Dual Pairs in Dirac-Jacobi Geometry
Monday, 20.12.21, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
Adopting the omni-Lie algebroid approach to Dirac-Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac-Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. \n\nIn my talk I will give a short introduction to Dirac-Jacobi geometry, introduce the notion of weak dual pairs, explain some cases where they exist and apply this to prove a normal form theorem, which locally in special cases gives the \nlocal structure theorems by Dazord, Lichnerowicz and Marle for Jacobi structures on the one hand, and the Weinstein splitting theorem on the other hand, which are generalizations of the Darboux theorem for contact (resp. symplectic) manifolds.