Positive scalar curvature and stable homotopy theory
Montag, 2.11.15, 16:15-17:15, Raum 404, Eckerstr. 1
Optimal rigidity estimates for nearly umbilical surfaces (2)
Dienstag, 3.11.15, 16:00-17:00, Raum 125, Eckerstr. 1
Sonderkolloquium Stochastik
Donnerstag, 5.11.15, 12:30-13:30, Raum 404, Eckerstr. 1
Alle Vorträge finden im Seminarraum 404 in der Eckerstraße 1 statt.\n\n12:30 Uhr Dr. Philipp Harms \nHypoelliptic diffusions in mathematical finance\n\n17:00 Uhr: Dr. Martin Herdegen \nGleichgewichtsmodelle mit kleinen Transaktionskosten
Donnerstag, 5.11.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
Sonderkolloquium Stochastik
Freitag, 6.11.15, 08:00-09:00, Raum 404, Eckerstr. 1
Alle Vorträge finden im Seminarraum 404 in der Eckerstraße 1 statt.\n\n08:00 Uhr: Dr. Jakob Söhl \nStatistik für Levy-Prozesse und Diffusionen\n\n10:00 Uhr: Dr. Joscha Diehl \nMethoden für stochastische partielle Differentialgleichungen
Foam categories from categorified quantum groups
Freitag, 6.11.15, 10:15-11:15, Hörsaal II, Albertstr. 23b
About 15 years ago, Khovanov introduced an homological invariant of knots categoryfying the Jones polynomial. Though this polynomial can be viewed both from a representation-theoretic and a diagrammatic point of view, for long only the latter version had been categorified.\nI will explain how, inspired by the concept of skew-Howe duality developed by Cautis, Kamnitzer, Morrison and Licata, one can describe the cobordism categories used in Khovanov homology from categorified quantum groups. In turn, this method allowed us to precisely redefine the sln generalizations of these categories, yielding a combinatorial and integral description of Khovanov-Rozansky homologies.
Foam categories from categorified quantum groups
Freitag, 6.11.15, 10:15-11:15, Hörsaal II, Albertstr. 23b
About 15 years ago, Khovanov introduced an homological invariant of knots categoryfying the Jones polynomial. Though this polynomial can be viewed both from a representation-theoretic and a diagrammatic point of view, for long only the latter version had been categorified.\nI will explain how, inspired by the concept of skew-Howe duality developed by Cautis, Kamnitzer, Morrison and Licata, one can describe the cobordism categories used in Khovanov homology from categorified quantum groups. In turn, this method allowed us to precisely redefine the sln generalizations of these categories, yielding a combinatorial and integral description of Khovanov-Rozansky homologies.
t.b.a., part I
Montag, 9.11.15, 16:15-17:15, Raum 404, Eckerstr. 1
Mass in Kähler geometry
Dienstag, 10.11.15, 16:00-17:00, Raum 125, Eckerstr. 1
We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.
Super-Ricci flows of metric measure spaces
Donnerstag, 12.11.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
A time-dependent family of Riemannian manifolds is a\nsuper-Ricci flow if \(2 Ric + \bpartial_t g \bge 0\).\nThis includes all static manifolds of nonnegative Ricci\ncurvature as well as all solutions to the Ricci flow\nequation.\nWe extend this concept of super-Ricci flows to\ntime-dependent metric measure spaces. In particular, we\npresent characterizations in terms of dynamical convexity\nof the Boltzmann entropy on the Wasserstein space as well\nin terms of Wasserstein contraction bounds and gradient\nestimates. And we prove stability and compactness of\nsuper-Ricci flows under mGH-limits.
New counterexamples to Quillen's conjecture
Freitag, 13.11.15, 10:15-11:15, Raum 404, Eckerstr. 1
In the talk I will explain the computation of cohomology of \(GL_3\) over function rings of affine elliptic curves. The computation is based on the study of the action of the group on its associated Bruhat-Tits building. It turns out that the equivariant cell structure can be described in terms of a graph of moduli spaces of low-rank vector bundles on the corresponding complete curve. The resulting spectral sequence computation of group cohomology provides very explicit counterexamples to Quillen's conjecture. I will also discuss a possible reformulation of the conjecture using a suitable rank filtration.
t.b.a., part II
Montag, 16.11.15, 16:15-17:15, Raum 404, Eckerstr. 1
Duality, regularity and uniqueness for \(BV\)-minimizers
Dienstag, 17.11.15, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
In my talk I will discuss similar convex variational integrals under a\nlinear growth condition. After a short introduction to the dual problem\nin the sense of convex analysis I will explain the duality relations\nbetween generalized minimizers and the dual solution. The duality\nrelations can be interpreted as mutual representation formulas, and in\nparticular they allow to deduce statements on uniqueness and regularity\nfor generalized minimizers. The results presented in this talk are based\non a joined project with Thomas Schmidt (Erlangen).
Quantitative rigidity results for conformal immersions
Dienstag, 17.11.15, 16:00-17:00, Raum 125, Eckerstr. 1
Bemerkungen über einige Redukte
Mittwoch, 18.11.15, 16:30-17:30, Raum 404, Eckerstr. 1
Aspherical manifolds, what we know and what we do not know
Donnerstag, 19.11.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
Der Vortrag wird kurzfristig abgesagt!
Six operations on dg enhancements of derived categories of sheaves and applications
Freitag, 20.11.15, 10:15-11:15, Raum 404, Eckerstr. 1
We lift Grothendieck’s six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. If time permits we give applications concerning homological smoothness of derived categories of schemes.
Rayleigh-Benard convection at finite Prandtl number: bounds on the Nusselt number.
Dienstag, 24.11.15, 14:00-15:00, Raum 226, Hermann-Herder-Str. 10
We consider Rayleigh-Benard convection at finite Prandtl number\nas modeled by the Boussinesq equation. We are interested in the scaling\nof the average upward heat transport, the Nusselt number, in \nterms of\nthe Rayleigh number, and the Prandtl number.\n\nIn this talk I present a rigorous upper bound for the Nusselt number reproducing \nboth physical\n scalings in some parameter regimes up to logarithms. \n\nThis is a joint work with Felix Otto and Antoine Choffrut.\n
A phase field model for Willmore´s energy with topological constraint
Dienstag, 24.11.15, 16:00-17:00, Raum 125, Eckerstr. 1
We consider the problem of minimizing Willmore’s energy on confined and connected surfaces with prescribed surface area. To this end, we approximate the surface by a level set function u admitting the value +1 on the inside of the surface and -1 on its outside. The confinement of the surface is now simply given by the domain of definition of u. A diffuse interface approximation for the area functional, as well as for Willmore’s energy are well known. We address the main difficulty, namely the topological constraint of connectedness by a penalization of a geodesic distance which is chosen to be sensitive to connected components of the phase field level sets and provide a proof of Gamma-convergence of our model to the sharp interface limit in case of a two-dimensional ambient space. Furthermore, we show some numerical results. This is joint work with Stephan Wojtowytsch (Durham University) and Antoine Lemenant (Universit Paris 7).
The Nash problem for arc spaces
Donnerstag, 26.11.15, 17:00-18:00, Hörsaal II, Albertstr. 23b
Algebraic varieties (zeroes of polynomial equations) often present singularities: points around which the variety fails to be a manifold, and where the usual techniques of calculus encounter difficulties. The problem of understanding singularities can be traced to the very\nbeginning of algebraic geometry, and we now have at our disposal many tools for their study. Among these, one of the most successful is what is known as resolution of singularities, a process that transforms (often in an algorithmic way) any variety into a smooth one, using a\nsequence of simple modifications.\n\nIn the 60's Nash proposed a novel approach to the study of\nsingularities: the arc space. These spaces are natural higher-order analogs of tangent spaces; they parametrize germs of curves mapping into the variety. Just as for tangent spaces, arc spaces are easy to understand in the smooth case, but Nash pointed out that their geometric structure becomes very rich in the presence of\nsingularities.\n\nRoughly speaking, the Nash problem explores the connection between the topology of the arc space and the process of resolution of singularities. The mere existence of such a connection has sparked in recent years a high volume of activity in singularity theory, with connections to many other areas, most notably birational geometry and\nthe minimal model program.\n\nThe objective of this talk is to give an overview of the Nash problem. I will give a precise description of the problem, and discuss the most recent developments, including the proof of Fernandez de Bobadilla and\nPe Pereira of the Nash conjecture in dimension two, and our extension of their result to arbitrary dimension.
The arc space of Grassmannians
Freitag, 27.11.15, 10:00-11:00, Raum 404, Eckerstr. 1
Arc spaces can be used as an effective tool to compute invariants of\nsingularities of algebraic varieties. In this talk, I will explain how this can\nbe achieved for a classical example: the singularities of Schubert varieties\ninside the Grassmannian. This involves a delicate study of the combinatorics\ninside of the arc space of the Grassmannian. The main tool I will discuss is a\nstratification of the arc space which plays the role of a Schubert cell\ndecomposition for lattices. Analyzing the geometric structure of the resulting\nstrata leads to the computation of invariants, mainly the log canonical\nthreshold of pairs invoving Shubert varieties.
Symplectic topology of classical field theories via polysaturated models
Montag, 30.11.15, 16:15-17:15, Raum 404, Eckerstr. 1
Hamiltonian PDE, arising e.g. in classical field theories and quantum mechanics, can be viewed as infinite-dimensional Hamiltonian systems. In this talk I show that analogues of the classical rigidity results from symplectic topology, such as Gromov's nonsqueezing theorem and the Arnold conjecture, also hold for these Hamiltonian PDE. In order to establish the existence of the relevant holomorphic curves, I use the surprising fact from logic that each separable symplectic Hilbert space is contained in a symplectic vector space which behaves as if it were finite-dimensional. As a concrete result I show (without experiment !) that every Bose-Einstein condensate, which is constrained to a circle and annoyed by a time-periodic exterior potential, has infinitely many time-periodic quantum states.