Refined Weyl Law for the Perturbed Harmonic Oscillator
Monday, 3.5.21, 16:15-17:15, Anderssen (BBB)
We consider the quantum harmonic oscillator \(H_0=(1/2)(-\bDelta+|x|^2)\). The underlying classical flow is periodic with period \(2\bpi\). By an explicit calculation one can see that the solution operator to the dynamical Schrödinger equation of \(H_0\) is the identity (modulo a sign) at \(2\bpi\bmathbb{Z}\) and locally smoothing otherwise. This periodicity is related to a sharp remainder estimate for the\ncounting function of the eigenvalues of \(H_0\). If we perturb the operator by a pseudodifferential operator of lower order, then we break the symmetry and could hope for an improved remainder estimate. We will present results on recurrence of singularities for these operators as well as an improved remainder estimate.\n\nThis is based on joint work with Oran Gannot, Jared Wunsch, and Steve\nZelditch.
Variational models for line-defects in materials.
Tuesday, 4.5.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
The purpose of the seminar is to describe a fully 3d model for dislocations derived by the asymptotic analysis of geometrically nonlinear elastic energy with quadratic growth. Precisely we obtain, through a Γ-convergence result, that the energy stored by a distribution of dislocations in a crystal is the contribution of a volume term representing the elastic energy and a line tension term representing the plastic energy.
Überlagerungen der komplexen Zahlen und die kanonische Basis Eigenschaft
Tuesday, 4.5.21, 14:30-15:30, Philidor
Baldwin und Lachlan zeigten, dass überabzählbar kategorische Theorien durch ihre streng minimalen Mengen bestimmt werden. Beispielsweise ist jede unendliche einfache überabzählbar kategorische Gruppe fast streng minimal, das heißt, algebraisch über einer streng minimalen Menge. Eine natürliche überabzählbar kategorische Konstruktion ist die sogenannte Überlagerung einer streng minimalen Menge. Sie ist im Allgemeinen nicht fast streng minimal, jedoch sind alle Fasern in definierbarer Bijektion mit der streng minimalen Menge, d.h. intern zu dieser. In diesem Vortrag werden wir Überlagerungen der komplexen Zahlen im Hinblick auf die kanonische Basis Eigenschaft (CBP) untersuchen. Die CBP, deren Ursprung in einer Arbeit von Pillay und Ziegler liegt, verallgemeinert den Begriff der Monobasiertheit, indem Algebraizität durch Internalität ersetzt wird. Sie gilt in zahlreichen algebraischen Strukturen und einige Zeit war nicht klar, ob sie in allen stabilen Theorien von endlichem Rang gilt, bis Hrushovski, Palacin und Pillay (2013) ein Gegenbeispiel veröffentlichten. Wir werden dieses Beispiel als additive Überlagerung der komplexen Zahlen präsentieren und eine genauere Untersuchung des Scheiterns der CBP wird unendlich viele neue Überlagerungen ohne die CBP liefern.\n\n
Higher dimensional slope inequalities
Friday, 7.5.21, 10:30-11:30, virtueller Raum 404
onsider a family of varieties f: X-> T, where T is a curve. We prove several inequalities about the slope of f, which are generalisations of Xiao and Cornalba-Harris inequalities in the case where X is a surface. We then apply our results to the KSB moduli space of stable varieties to study the ample cone of such spaces.\nThe talk is based on a joint work with Giulio Codogni and Filippo Viviani.
Maurer-Cartan elements, twisting and homotopy
Monday, 10.5.21, 16:15-17:15, Anderssen (BBB)
The globalisation of Kontsevich's formality to smooth manifolds depends on choices, namely of a torsion-free covariant derivative and some section of a pro-finite dimensional vector bundle. In my talk, I explain that even if the globalised formality changes with different choices, its homotopy class does not. The idea of the proof relies on some basic knowledge of strong homotopy Lie algebras, their morphisms, Maurer-Cartan elements and the so-called twisting procedure, which I recall in an introductory part. This talk is based on arXiv:2102.10645 joint with Andreas Kraft. \n\n
On the minimization of various energies of Riesz-type.
Tuesday, 11.5.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
\n\n\nThe celebrated liquid drop model by Gamow, which dates back to the 1930's, has attracted since then a lot of attention among physicists and mathematicians. In particular, there has been a deep increase in the mathematical interest about this problem and several generalisations in the last decade, with many proven results but also fundamental questions still open. We will give a general overview on these problems, concluding with some very recent results, obtained in some collaborations with Carazzato, Fusco, Novaga.\n\n
tba
Tuesday, 11.5.21, 14:30-15:30, Philidor
Combinatorial characterizations of Canjar filters
Tuesday, 11.5.21, 14:30-15:30, Philidor
An often important property of a forcing notion is whether or not it adds dominating reals, i.e. whether there exists a real in the generic extension of the forcing which eventually dominates all reals from the ground model.\nFamously Mathias forcing does add dominating reals. However, this might not be the case for the Mathias forcing associated with a nonprincipal filter F, MA(F), consisting of conditions in which the infinite set has to be in F. For example if F is the Frechet filter, MA(F) will not add dominating reals. This leads to the following question: For which filters F does MA(F) not add dominating reals? Filters for which this is the case are also called Canjar filters, named after a result by Michael Canjar in 1988.\nIn 2014 Hrusak and Minami showed that these filters share a purely combinatorial property. In this talk we will focus on this characterization and various equivalent properties of filters as well as a topological reformulation by Chodounsky, Repovs and Zdomskyy from 2015. \n \n
Highly connected 7-manifolds and non-negative sectional curvature
Friday, 14.5.21, 10:30-11:30, virtueller Raum Lasker
A six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.
t.b.a
Monday, 17.5.21, 16:15-17:15, Anderssen (BBB)
Taylor Scaling for curvature driven interfaces in random media
Tuesday, 18.5.21, 14:15-15:15, Hörsaal II (virtuell: Lasker)
We present a model for curvature driven interface propagation through a homogeneous medium with random obstacles. The energy is fully nonlinear and the dissipation is mixed, capturing both viscous dissipation as well as dry friction. If the interface passes over an obstacle it incurs additional dry friction. This model is relevant for the study of dislocations. Under an applied force, we investigate the pinning (i.e., a solution becomes stuck) and depinning behavior of the interface. We show that the model obeys Taylor Scaling, i.e., the critical pinning force scales like the square root of the concentration of the obstacles. Joint work with Patrick Dondl (Freiburg) and Michael Ortiz (Pasadena).
Forcing With Canjar Filters and With Generic Ultrafilters
Tuesday, 18.5.21, 14:30-15:30, BBB Philidor
In joint work with Christian Bräuninger we used relatives of Canjar filters in forcings with superperfect trees. It is open whether the same goals could be achieved with relatives of Mathias forcing as well. In this talk, I will focus on\nopen questions about ultrafilters, and the few proofs I plan to sketch are about topology and combinatorics. Technical\naspects of iterated forcing will be skipped.\n
Valuation rings in the context of Algebraic Geometry
Thursday, 20.5.21, 11:15-12:15, online: kasparov
Cox rings of algebraic stacks
Friday, 21.5.21, 10:30-11:30, virtueller Raum Lasker
In this talk, I will discuss the construction of Cox rings on algebraic\nstacks. Recall that the Cox ring consists of all global sections of\ndivisors on a given space. Here the definition of the multiplicative\nstructure is a bit subtle. But it turns out that such a structure\nalways exists, and moreover, its (non-)uniqueness can be measured by an\nExt-group. This talk is based on a joint work with Elena Martinengo and\nFabio Tonini.
Uncertainty under small data
Friday, 21.5.21, 13:00-14:00, online: Zoom
Deep dynamic modeling with a small number of time points
Friday, 28.5.21, 13:00-14:00, online: Zoom
Narrow escape problem on Riemannian manifolds
Monday, 31.5.21, 10:15-11:15, ZOOM (link in the email)
We use geometric microlocal methods to compute an asymptotic expansion of the mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. (Joint work with Justin Tzou and Leo Tzou)