Koszul duality
Montag, 3.6.24, 15:15-16:15, Raum 218, Ernst-Zermelo-Str. 1
I discuss how to reformulate local Langlands for real groups as an equivalence of categories and check some examples. \nIn the first lecture, I will discuss Koszul selfduality \nfor category O and the general formalism of Koszul duality.
On local boundary conditions for Dirac-type operators
Montag, 3.6.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We give an overview on smooth local boundary conditions for Dirac-type operators, giving existence and non-existence results for local symmetric boundary conditions. We also \n discuss conditions when the boundary conditions are elliptic/regular/Shapiro-Lopatinski (i.e. in particular giving rise to self-adjoint Dirac operators with domain in \(H^1\)). This is joint work with Hanne van den Bosch (Universidad de Chile) and Alejandro Uribe (University of Michigan).
Friedman's and other Reflection Properties
Dienstag, 4.6.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In 1975, Friedman introduced the property \(F(\bkappa)\),\nstating that every subset of \(\bkappa\) either contains or is disjoint\nfrom a closed set of ordertype \(\bomega_1\). Famously, this property\nfollows from the power forcing axiom ``Martin's Maximum''. In this\ntalk, we introduce posets which force the negation of this property and\nother related notions and investigate the patterns in which these\nproperties can fail in connection to large cardinals.
Wann regen Aufgaben Schülerinnen und Schüler zum «Mathematik betreiben» an?
Dienstag, 4.6.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Im Mathematikunterricht ist das Bearbeiten von Aufgaben eine zentrale Unterrichtstätigkeit. Wann regen diese Aufgaben dazu an, dass Mathematik betrieben wird und nicht nur Fertigkeiten trainiert werden? An erprobten Aufgaben aus verschiedenen mathematischen Kontexten der Sekundarstufe1 wird überlegt, wann im Unterrichtsprozess und mit welchen Aufgaben Lernende vielfältig mathematisch tätig sind.
Donnerstag, 6.6.24, 15:00-16:00, Hörsaal II, Albertstr. 23b
Koszul duality methods 2: Equivariant derived category and applications
Montag, 10.6.24, 15:15-16:15, Raum 218, Ernst-Zermelo-Str. 1
Ricci curvature, metric measure spaces and the Riemannian curvature-dimension condition
Montag, 10.6.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
I explain idea of synthetic Ricci curvature bounds for metric measure spaces and one of their applications in Riemannian geometry.
Phase separation on varying surfaces and convergence of diffuse interface approximations
Dienstag, 11.6.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
This talk's topic are phase separations on varying generalized hypersurfaces in\nEuclidian space. We consider a diffuse surface area (line tension) energy of Modica–\nMortola on surfaces and prove a compactness and lower bound estimate in the sharp interface\nlimit. We also consider an application to phase separated biomembranes where a Willmore energy\nfor the membranes is combined with a generalized line tension energy. For a diffuse\ndescription of such energies we give a lower bound estimate in the sharp interface limit. Time permitting I will present recent results about simultaneous phase field approximations of both the biomembrane and the indicator function for one of the two phases defined on the membrane.\n
Developments in Namba Forcing
Dienstag, 11.6.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
One way to study the properties of the infinite cardinals is to examine the extent to which they can be changed by forcing. In 1969 and 1970, Bukovsk{\b'y} and Namba independently showed that \(\baleph_2\) can be forced to be an ordinal of cofinality \(\baleph_0\) without collapsing \(\baleph_1\). The forcings they used and their variants are now known as Namba forcing. Shelah proved that Namba forcing collapses \(\baleph_3\) to an ordinal of cardinality \(\baleph_1\). In a 1990 paper, Bukovsky and Coplakova asked whether there can be an extension that collapses \(\baleph_2\) to an ordinal of cardinality \(\baleph_1\) without collapsing \(\baleph_3\). We will show that a slight strengthening of local precipitousness on \(\baleph_2\) due to Laver allows us to construct such an extension.\n
Analysis of Correlated Many-Body Systems: Bose-Einstein Condensates and Mean Field Spin Glasses
Donnerstag, 13.6.24, 08:30-09:30, Raum 404, Ernst-Zermelo-Str. 1
In this talk, I provide an overview of my recent research on the mathematical analysis of correlated many-body systems. I describe exemplary results concerning dilute quantum systems of interacting bosons and concerning disordered mean field systems of interacting Ising spins. In both cases, I carefully introduce the model, review relevant results and outline current as well as future research directions. The talk is based on joint work with M. Brooks, C. Caraci, J. Oldenburg, A. Schertzer, C. Xu and H.-T. Yau. .
Stabilization by transport noise and enhanced dissipation in the Kraichnan model
Donnerstag, 13.6.24, 10:40-11:40, Raum 404, Ernst-Zermelo-Str. 1
Thanks to the work of Arnold, Crauel, and Wihstutz it is known that for any self-\nadjoint operator T acting on a finite dimensional space with the negative trace the\ncorresponding linear equation dxt = T xt dt can be stabilized by a noise, i.e. there\nexists operator-valued Brownian motion W such that the solution of dxt + dW xt =\nT xt dt vanishes a.s. for any initial value x0 = x. The goal of the talk is to extend this\ntheorem to infinite dimensions. Namely, we prove that the equation dut = ∆ut dt\ncan be noise stabilized and that an arbitrary large exponential rate of decay can\nbe reached. The sufficient conditions on the noise are shown to be satisfied by the\nso-called Kraichnan model for stochastic transport of passive scalars in turbulent\nfluids. This talk is based on joint work with Prof. Benjamin Gess (MPI MiS and\nBielefeld University).
Sub-Riemannian geometries and hypoelliptic diffusion processes
Donnerstag, 13.6.24, 14:00-15:00, Raum 404, Ernst-Zermelo-Str. 1
I will start with an overview on sub-Riemannian geometries, where motion is only possible along certain admissible trajectories, and hypoelliptic diffusion processes, which due to underlying constraints spread in different directions at different orders. Subsequently, I will present two projects where the analysis of stochastic processes on constrained systems has proven to be fruitful. Firstly, I will discuss how a stochastic process introduced jointly with Barilari, Boscain and Cannarsa on surfaces in three-dimensional contact sub-Riemannian manifolds can be used to classify singular points arising in that setting. Secondly, I will show how the study of a standard one-dimensional Brownian motion conditioned to have vanishing iterated time integrals of all orders, which can be rephrased as studying projected hypoelliptic diffusion loops, has led to a novel polynomial approximation for Brownian motion..
Singular PDEs: regularity and homogenization
Donnerstag, 13.6.24, 16:10-17:10, Raum 404, Ernst-Zermelo-Str. 1
I will mostly focus on the regularity theory and homogenization for elliptic equations with degenerate unbounded coefficients, both at the deterministic (mostly done with Mathias Schäffner) as well as stochastic level. While already interesting on its own, I will mention two areas of use for these: study of regularity of critical points for variational integrals as well as invariance principle for random walks in random environments. At the end, I will conclude with a short discussion of few result in quantitative stochastic homogenization..
A semigroup approach for stochastic quasilinear equations driven by rough noise
Freitag, 14.6.24, 08:00-09:00, Raum 404, Ernst-Zermelo-Str. 1
We consider stochastic quasilinear equations perturbed by nonlinear multiplicative noise. Ex-ploring semigroup methods and combining techniques from functional analysis with tools from rough path theory, we establish the pathwise well-posedness of such equations. We apply our results to the stochastic Shigesada-Kawasaki-Teramoto equation describing population segregation by induced cross-diffusion and to the Landau-Lifshitz-Gilbert equation which models the magnetization of a ferromagnetic material. Moreover, we emphasize the advantage of rough path theory in the study of the long-time behavior of such systems. This talk is based on joint works with Antoine Hocquet and Christian Kuehn.
Optimal Transport and Diffusion on varying spaces
Freitag, 14.6.24, 10:10-11:10, Raum 404, Ernst-Zermelo-Str. 1
We discuss contraction estimates of diffusion under optimal transport problems on varying spaces. We further investigate in equivalent formulations and generalizations of these estimates.
From microscopic to macroscopic scales: effective evolution equations of many interacting particles.
Freitag, 14.6.24, 13:30-14:30, Raum 404, Ernst-Zermelo-Str. 1
Systems of interacting particles describing notable physical phenomena, such as time-irreversibility, Bose-Einstein condensation, superconductivity or superfluidity, represent a veritable challenge for mathematicians and physicists. They exhibit a daunting complexity, which renders the exact many-body theory non-approachable, not only from a mathematical viewpoint, but also for computer experiments and simulations. Therefore, an approximate description using effective macroscopic models is highly useful, and the rigorous study of the regime of validity of such approximations is of primary importance in mathematical physics. In this talk, I will present several settings leading to different effective kinetic equations and then I will focus on the mean-field regime for quantum particle systems, highlighting recent significant progress in the mathematical understanding of these systems.
Oscillatory effects in stochastic PDEs
Freitag, 14.6.24, 15:40-16:40, Raum 404, Ernst-Zermelo-Str. 1
I will describe two recent results related to the oscillations of the noise in stochastic PDEs. In the first example, a rougher-than-usual KPZ equation, the fluctuation of the noise is strong enough that on the relevant scale the nonlinearity turns into a new Gaussian noise via a central limit-type theorem. The second example, a 1-dimensional stochastic Allen Cahn equation, is much less singular and a solution theory is fairly unproblematic. Nevertheless, the averaging effects of the noise play a key role in their discretisations: they exhibit four times better temporal pointwise convergence rate than the pointwise regularity of the solution and twice better than the regularity of its single Fourier mode. Based on joint works with Ana Djurdjevac, Helena Kremp, Fabio Toninelli.
A necessary condition for zero modes of the Dirac equation
Montag, 17.6.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
We will state a necessary condition for the existence of a non-trivial solution of the Dirac equation, which is based on a Euclidean-Sobolev-type inequality. First, we will state the theorem in the flat setting and give an overview of the technical issues of the proof. Afterwards, we will consider and point out the main differences in the not necessarily flat setting. This talk is based on a work by R.Frank and M.Loss.
Existence of optimal flat ribbons
Dienstag, 18.6.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We revisit the classical problem of constructing a developable surface along a given Frenet curve \(\bgamma\) in space. First, we generalize a well-known formula, introduced in the literature by Sadowsky in 1930, for the Willmore energy of the rectifying developable of \(\bgamma\) to any (infinitely narrow) flat ribbon along the same curve. Then we apply the direct method of the calculus of variations to show the existence of a flat ribbon along \(\bgamma\) having minimal bending energy. Joint work with Simon Blatt.
Fredholmness of the Laplace operator on singular manifolds with pure Neumann Data
Montag, 24.6.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
Given a smooth Riemannian metric \(h\) on \(M\) one can consider the Laplace problem with pure Neumann data: Let \(\bDelta_h\) be the Laplacian given by \(h\) and \(n_h\) be the outer normal of \(\bpartial M\). Exists an \(u\) such that \((\bDelta_hu,\bpartial_{n_h}u)=(F,G)\) for some given data \(F\) and \(G\). There is no well posedness to this problem on singular mandifolds in regular Sobolev spaces but during the talk I will introduce a scale of weighted Sobolev spaces such that it is Fredholm. In the second part of the talk I will give a formula for the Fredholm Index depending on the chosen weight function.
Unipotent normal subgroups of algebraic groups
Dienstag, 25.6.24, 14:15-15:15, Hörsaal II, Albertstr. 23b
Der angewandte Mathematiker Henry Görtler vor und nach 1945.
Donnerstag, 27.6.24, 15:00-16:00, Hörsaal II, Albertstr. 23b
On finite generation of fundamental groups in algebraic geometry
Freitag, 28.6.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The étale fundamental group of a (quasicompact) variety over complex numbers is (topologically) finitely presented by comparison with the topological case.\nIn characteristic p, the situation is much more subtle, as affine varieties have very large fundamental groups.\nBuilding on a recent breakthrough result by Esnault, Shusterman and Srinivas, I will explain how to extend the finite presentation statement to arbitrary proper varieties (joint work with Srinivas and Stix) and then (at least the finite generation part) to log/tame fundamental groups of schemes and rigid analytic spaces (joint work with Achinger, Hübner and Stix).\nThis requires revisiting the tame topology of rigid spaces and working with a certain class of non-fs log schemes.