Nonlinear Partial Differential Equations in Freiburg
Mittwoch, 2.10.24, 09:00-10:00, Hörsaal II, Albertstr. 23b
Homepage von Analysis-Gruppe:\nhttp://home.mathematik.uni-freiburg.de/analysis/
A Journey with Michael Ruzicka
Mittwoch, 2.10.24, 09:45-10:45, Hörsaal II, Albertstr. 23b
In 1996, Professors Rajagopal and Ruzicka introduced a new model for electrorheological fluids. In this talk, I will reflect on how this model sparked the beginning of a rewarding scientific journey shared between Michael Ruzicka and myself. I will also present some of the subsequent advancements of those results.
Codimension two mean curvature flow of entire graphs
Mittwoch, 2.10.24, 11:00-12:00, Hörsaal II, Albertstr. 23b
We consider the graphical mean curvature flow of maps \(f:\bmathbb{R}^m\bto\bmathbb{R}^n\), \(m\bge 2\), and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken. In the case of uniformly area decreasing maps \(f:\bmathbb{R}^m\bto\bmathbb{R}^2\), we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander. This is joint work with Andreas Savas-Halilaj.
On the stability of bifurcating periodic patterns of compressible viscous fluid equations
Mittwoch, 2.10.24, 11:45-12:45, Hörsaal II, Albertstr. 23b
In systems of equations describing the motion of viscous fluids, solutions exhibit various interesting spatiotemporal pattern dynamics. In this talk, we will consider the bifurcation and stability problem of spatially periodic vortex patterns in a rotating fluid system. We will present results on the bifurcation of stationary periodic patterns and the stability and instability of bifurcating periodic patterns when the Mach number is small.
Conformal fill in by Poincaré-Einstein metrics in dimension 4
Mittwoch, 2.10.24, 14:00-15:00, Hörsaal II, Albertstr. 23b
Given a closed riemannian manfiold of dimension 3 \((M^3,[h])\), when will we fill in an asymptotically hyperbolic Einstein manifold of dimension 4 \((X^4,g_+)\) such that \(r^2 g_+|_M =h\) on the boundary \(M=\bpartial X\) for some defining function \(r\) on \(X^4\)? This problem is motivated by the correspondance AdS/CFT in quantum gravity proposed by Maldacena in 1998 et comes also from the study of the structure of asymptotically hyperbolic Einstein manifolds.\nIn this talk, I discuss some existence and uniqueness issue of asymptotically hyperbolic Einstein manifolds in dimension 4. It is based on the recent works with Alice Chang.
Geometrical criteria and "regular" solutions to the Navier-Stokes equations
Mittwoch, 2.10.24, 14:45-15:45, Hörsaal II, Albertstr. 23b
In this talk, I will give an overview of results on linking the regularity of the solutions to the 3D Navier-Stokes equations and the geometry of the velocity/vorticity fields. In particular, starting from the celebrated criterion of Constantin and Fefferman (Indiana Univ. Math. J., 1993) about the vorticity direction, I will present some variations and improvement along various paths.
Stability for anisotropic curvature functionals
Mittwoch, 2.10.24, 16:00-17:00, Hörsaal II, Albertstr. 23b
Anisotropic integrands are used to model surface energies of inhomogeneous materials, where the energy density depends on the direction of the surface. Under suitable assumptions, minimisers of the associated functionals are known to be so-called Wulff shapes, those being certain convex bodies which induce a Minkowski norm on Euclidean space. The first variation being the anisotropic mean curvature, the anisotropic Alexandrov-type theorem says that a surface with constant anisotropic mean curvature must be the corresponding Wulff shape. There are various related rigidity results for other anisotropic curvature functionals and in this talk we discuss their stability: In case the curvature condition is “almost” satisfied, is the surface “close” to the Wulff shape? This is joint work with Xuwen Zhang (University of Freiburg).
Regularity of functions in fractional Orlicz--Sobolev spaces
Mittwoch, 2.10.24, 16:45-17:45, Hörsaal II, Albertstr. 23b
Homogeneous fractional Orlicz--Sobolev spaces extend classical fractional Sobolev spaces governed by the Gagliardo--Slobodetskii seminorm, which were introduced in late 1950's and which are defined in terms of a non-integer smoothness parameter. They provide a natural framework for solutions to nonlocal elliptic problems associated with non-polynomial nonlinearities. This is achieved by replacing the power type integrability with an integrability condition expressed in terms of a Young function. Fractional Orlicz--Sobolev spaces were introduced in 2019 by Fernandez-Bonder and Salort, and their functional properties have been intensively investigated ever since, either out of the pure mathematical curiosity, or in connection with one or more of its many applications.\b\b\nLike for any type of Sobolev spaces, relations to other function spaces constitute a fundamental issue in the theory of fractional Orlicz--Sobolev spaces, as they provide a crucial tool for transferring regularity from the data to solution in related differential equations. Properties of functions in these spaces are governed by the smoothness parameter and the Young function.\b\b\nWe shall focus on various types of regularity of functions belonging to fractional Orlicz--Sobolev spaces in both the so-called subcritical and supercritical regimes. In the former regime we shall mainly concentrate on sharp embeddings into spaces defined in terms of global integrability properties of functions, called rearrangement-invariant spaces, while, in the latter regime, we shall call for finer properties of functions such as criteria for continuity, optimal moduli of continuity, or a control of mean oscillation expressed by a membership into spaces of generalized Campanato type.\b\b\nThe aptness of the notion of fractional Orlicz--Sobolev spaces which we adopt is supported by the fact that, unlikely in the classical case, setting the smoothness parameter as an integer exactly matches their counterparts for integer-order Orlicz--Sobolev spaces. Interestingly, customary techniques that have proved appropriate for classical fractional Sobolev spaces, such as characterizations of Hölder spaces in terms of Campanato spaces, Littlewood--Paley decompositions and Hardy-type inequalities fail to yield optimal conclusions for fractional Orlicz--Sobolev spaces. This discrepancy forces us to adopt novel approaches.\b\b\nIn the talk we shall give a survey of recent results on fractional Orlicz--Sobolev spaces obtained jointly with Angela Alberico, Andrea Cianchi and Lenka Slavíková.
The Penrose inequality in extrinsic geometry
Mittwoch, 2.10.24, 17:30-18:30, Hörsaal II, Albertstr. 23b
The Riemannian Penrose inequality is a fundamental result in mathematical relativity. It has been a long-standing conjecture of G. Huisken that an analogous result should hold in the context of extrinsic geometry. In this talk, I will present recent work that resolves this conjecture: The exterior mass m of an asymptotically flat support surface \(S\bsubset \bmathbb{R}\) with nonnegative mean curvature and outermost free boundary minimal surface D is bounded in terms of\n\[\nm\bge \bsqrt{\bfrac{|D|}{\bpi}}.\n\]\nIf equality holds, then the unbounded component of \(S\bsetminus\bpartial D\) is a half-catenoid. To prove this result, we develop the theory of a weak foliation of the region above \(S\) by minimal capillary surfaces supported on \(S\) that emerges from \(D\) and admits a nondecreasing quantity associated with its leaves.
Homogeneous rearrangement-invariant function spaces
Donnerstag, 3.10.24, 09:00-10:00, Hörsaal II, Albertstr. 23b
Let us consider the dilation operator, defined in the usual way by\n\[\nD_r f(t)=f(rt),\bquad t>0,\]\nwhere \(r>0\) is the dilation parameter and \(f:[0,\binfty)\bto \bmathbb{R}\) is a Lebesgue-measurable function. With \(p\bin [1,\binfty]\), a space \((X,\b|\bcdot\b|_X)\), consisting of functions defined on \([0,\binfty)\), is called \(p\)-homogeneous if the function norm \(\b|\bcdot\b|_X\) satisfies\n\[\n\b|D_r f \b|_X=r^{−\bfrac1p}\b|f\b|_X\]\nfor every \(f\bin X\) and \(r>0\). We will focus on the homogeneity property within the class of rearrangement-invariant spaces, i.e., function spaces \(X\) where the norm \(\b|f\b|_X\) depends only on the measure of the level sets of \(f\). More precisely, given a measurable \(f:[0,\binfty)\bto R\), its nonincreasing rearrangement is given by\n\[\nf^∗(t)=inf{s>0; \blambda({x\bin [0,\binfty):|f(x)|>s})\ble t},\bquad t>0,\]\nwhere \(\blambda\) stands for the Lebesgue measure. A Banach function space \((X,\b|\bcdot\b|_X)\) is then called rearrangement-invariant (r.i.) if \(\b|f\b|_X=\b|g\b|_X\) whenever \(f^∗=g^∗\). \n\nIn the talk, various properties of homogeneous r.i. spaces will be discussed. We will see that a typical example of a such space is the Lorentz \(L^{p,q}\) space. However, the homogeneity property is not restricted only to this class of spaces, which will be shown by constructing other examples, for instance by using certain interpolation and extrapolation techniques.
Ricci flow with \(L^p\) bounded scalar curvature
Donnerstag, 3.10.24, 09:45-10:45, Hörsaal II, Albertstr. 23b
In this talk, we show that localised, weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed \(n\)-dimensional Kähler Ricci flow always hold. These integral estimates improve and extend the integral curvature estimates shown in an earlier paper by the speaker. If \(M^4\) is closed and four dimensional, and the spatial \(L^p\) norm of the scalar curvature is uniformly bounded for some \(p>2\), for \(t\bin [0,T)\), \(T<\binfty\), then we show:\n\na) a uniform bound on the spatial \(L^2\) norm of the Riemannian curvature tensor for \(t\bin [0,T)\),\n\nb) uniform non-expanding and non-inflating estimates for \(t\bin [0,T)\),\n\nc) convergence to an orbifold as \(t\bto T\),\n\nd) existence of an extension of the flow to times \(t\bin [0,T+\bsigma)\) for some \(\bsigma >0\) using the orbifold Ricci flow.\n\nThis is joint work with Jiawei Liu.
A new convergent method to solve the Fokker-Planck equation in higher dimensions
Donnerstag, 3.10.24, 11:00-12:00, Hörsaal II, Albertstr. 23b
A recent topic in the area of numerics of PDEs is their approximation in higher space dimensions. While different strategies are available by now that are supported by motivating heuristics, it is that most of them lack a rigorous convergence theory. In the talk, I propose a new discretization approach for the Fokker-Planck equation, and prove its convergence. The method combines tools from stochastics and mathematical statistics, as does the related convergence theory. This is joint work with Max Jensen (UC London) and Fabian Merle (U Tübingen).
Non-degeneracy of the bubble in a fractional and singular 1D Liouville equation
Donnerstag, 3.10.24, 11:45-12:45, Hörsaal II, Albertstr. 23b
In this talk, we will focus on a stationary fractional nonlinear equation with exponential non-linearity defined on the whole real line in presence of a singular term. We will show the non-degeneracy of its solutions. This particular equation appears as a limit problem to physical models for the description of galvanic corrosion phenomena for simple electrochemical systems. We use conformal transformations to rewrite the linearized equation as a Steklov eigenvalue problem posed in a Lipschitz bounded domain. We conclude by proving the simplicity of the corresponding eigenvalue. The argument used to prove our main result can also be applied to prove that the second eigenvalue of Steklov’s problem on the ellipse is simple, as long as the ellipse is not a circle. The talk is based on a work done in collaboration with G. Mancini and A. Pistoia.
About the Onset of the Hopf Bifurcation for Convective Flows in Horizontal Annuli
Donnerstag, 3.10.24, 14:00-15:00, Hörsaal II, Albertstr. 23b
Experimental and numerical results can not yet settle whether, between horizontal coaxial cylinders, if the curvature is large, the first transition for convection is an exchange of stability or rather an Hopf bifurcation. We directly show that if the curvature tends to infinity, no periodic linear perturbation exists when the Rayleigh number is equal to the critical one for non-linear stability.
Rigidity on capillary constant mean curvature hypersurfaces
Donnerstag, 3.10.24, 14:45-15:45, Hörsaal II, Albertstr. 23b
In this talk, we introduce our recent developments, joint with Professor Guofang Wang, on capillary constant mean curvature (CMC) hypersurfaces, including a resolution to a question of Ros and Sternberg-Zumbrun on classification of stable capillary CMC hypersurfaces in a Euclidean ball via Minkowski-type formula, and a new proof of Alexandrov-Wente’s theorem on embedded capillary CMC hypersurfaces in a half-space via Heintze-Karcher-type inequality.
Fully-discrete finite element approximation of an unsteady electro-rheological fluid flow model: convergence and error analysis
Donnerstag, 3.10.24, 16:00-17:00, Hörsaal II, Albertstr. 23b
In this talk, a fully-discrete approximation of an unsteady electro-rheological fluid flow model employing an implicit Euler step in time and a discretely inf-sup-stable finite element approximation in space is examined for its well-posedness, stability, and (weak) convergence under minimal regularity assumptions on the data. Furthermore, numerical experiments are presented to complement the theoretical findings.
Gromov-Hausdorff stability of tori under Ricci and integral scalar curvature bounds
Donnerstag, 3.10.24, 16:45-17:45, Hörsaal II, Albertstr. 23b
In this talk, we provide characterizations on the Gromov-Hausdorff stability to a flat torus from a closed Riemannian manifold with Ricci and integral scalar curvature bounds, in terms of harmonic maps and harmonic forms. Applications include a new topological stability result to a flat torus. This is a joint work with C. Ketterer, I. Mondello, R. Perales and C. Rigoni.
Magma and aircraft hulls: Convection is fascinating
Donnerstag, 3.10.24, 17:30-18:30, Hörsaal II, Albertstr. 23b
Since 1992 I have been thinking about Fluid flow. My first publication was about the Boussinesq approximation of power-law materials. My co-authors chose it in order to apply well established tools to a slightly nonstandard problem - the flow of magma. One of them was Michael Růžička. A few years later we started to build convection models together with the right frame for their validation with Yoshiyuki Kagei and Michael Růžička. When collecting ideas for my habilitation I decided to let power-law materials convect again and presented model derivation and solvability theory in one volume. After that from 2006 on Arianna Passerini invited Michael Růžička and myself to study flow in annuli. Which seemed simple enough but keeps us interested and busy up to today. I invite the audience to revisit the Boussinesq approximation and the study of convection in different domains.
Short closed geodesics and the Willmore energy
Freitag, 4.10.24, 09:00-10:00, Hörsaal II, Albertstr. 23b
In this talk we study the relation between two geometric quantities for smooth closed \(2d\)-surfaces \(\bSigma\) -- the Willmore bending energy \(W(\bSigma)\) and the minimal length of a closed geodesic \(\bell(\bSigma)\). It turns out that for surfaces of Willmore energy less than \(6\bpi\) (with normalized area), \(\bell(\bSigma)\) is bounded below in terms of \(W(\bSigma)\). The threshold of \(6\bpi\) is optimal for such a result -- we will see that surfaces above this threshold can indeed have geodesic bottlenecks. Our inequality can be proved very easily if one assumes that the shortest closed geodesic has no self-intersections. The discussion of this assumption leads to intriguing insights. This is joint work with Fabian Rupp (Vienna) and Christian Scharrer (Bonn).
Endpoint maximal regularity and free boundary problem for the Navier-Stokes system in scaling invariant case
Freitag, 4.10.24, 09:45-10:45, Hörsaal II, Albertstr. 23b
We consider the free boundary problem of the incompressible Navier-Stokes system in the scaling critical Besov space. By translation into the Lagrange coordinate, the system can be considered on the fixed region without the convection term, while all the spatial derivatives are changed into the covariant derivatives involving higher order nonlinearity and the system becomes a quasi-linear system. To control such terms, we introduce endpoint maximal regularity of the solution and show the global existence of the free boundary problem for the Naiver-Stokes system near the half Euclidean space. This talk is based on a joint work with Senjo Shimizu (Kyoto University).
Gauss curvature flow: it's variations and applications
Freitag, 4.10.24, 11:00-12:00, Hörsaal II, Albertstr. 23b
Firey introduced the Gauss curvature flow in 1974 to model evolution of tumbling stones. Andrews proved the convergence to a round point in dimension two in 1999. The same result holds for in high dimensions, the flow converges to a soliton (Guan-Ni) and the soliton is sphere (Brendle-Choi-Daskalopoulos). The convergence relies on the estimates of entropy type quantity and related entropy points. This approach can be adapted to deal with variations of Gauss curvature type flows: inhomogeneous and anisotropy Gauss curvature flows, and application to the Lp-Minkowski type problems. We will also discuss a type of anisotropic flow arising from the \(L^p\) Christoffel-Minkowski problem.
The Frobenius relation in string topology
Montag, 21.10.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
String topology is the study of algebraic operations on the homology of the free loop space of a closed manifold. Two prominent operations are the Chas-Sullivan product and the Goresky-Hingston coproduct. It is an important question what structure these two operations form together. We show that under a transversality condition a Frobenius-type relation for the product and the coproduct holds. As an application this yields the behaviour of the coproduct on product manifolds. This talk is based on joint work with Nathalie Wahl.\n
Mathematics and Arts
Dienstag, 22.10.24, 16:00-17:00, Raum 125, Ernst-Zermelo-Str. 1
Our civilization has been developed based on modern knowledge and tradition, or oriental and occidental, but there is an important third source which has often been neglected: interdisciplinary and intercultural inference and exchange. In this talk, Professor Cai will focus on the interplay between science and humanity and, in particular, the similarities between mathematics, arts, and daily life. Prof. Cai will demonstrate it through classical and modern paintings together with photographs that he has taken all around the world. \n\nDr. Cai Tianxin is a mathematician, poet and essayist, an outstanding professor of Zhejiang University, China. He is the author of over 30 books of literature, academic and popular science works. His work has been translated and published into more than 20 languages, including 7 in English. He is a winner of the 2013 Naji Naaman Poetry Award (Beirut) and the 2019 Kadark Literature Award (Darka), the 2017 National Award of Science and Technology (Beijing) for his book Mathematical Legends and the 2024 National Award for Science Popularization, he is the only mathematician who won the prize awarded every 4 years.\n
Verbindung zwischen schulischer und akademischer Mathematik – Lehramtsaufgaben als Mittel zur Adressierung der doppelten Diskontinuität?
Dienstag, 22.10.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Lehramtsstudierende zeigen häufig Schwierigkeiten, Verbindungen zwischen der akademischen\nMathematik, die sie an der Hochschule lernen, und der Schulmathematik, die sie später unterrichten\nsollen, herzustellen. Viele Hochschulen versuchen diesem bereits von Felix Klein (1908)\nbeschriebenen Problem der doppelten Diskontinuität durch den Einsatz professionsspezifischer\nÜbungsaufgaben zu begegnen. Diese „Lehramtsaufgaben“ adressieren explizit Verbindungen\nzwischen Schul- und Hochschulmathematik und sollen dazu beitragen, dass Studierende\nHochschulmathematik als relevant für die Schulmathematik wahrnehmen. In dem Vortrag wird\nanhand einer Fragebogenstudie erörtert, inwiefern Lehramtsaufgaben einer wahrgenommenen\ndoppelten Diskontinuität entgegenwirken können. Ergänzend werden Ergebnisse einer\nInterviewstudie vorgestellt zu der Frage, was genau Lehramtsaufgaben für Studierende relevant\nmacht, sodass zusammenfassend diskutiert werden kann, welche Ansatzpunkte sich zur\nWeiterentwicklung professionsspezifischer Lerngelegenheiten zur Adressierung der doppelten\nDiskontinuität ableiten lassen.
Semi-algebraic differential forms
Freitag, 25.10.24, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Modifying a construction by Hardt, Lambrechts, Turchin and Volić, we will present (\(\bmathbb{Q}\)-)semi-algebraic differential forms and explain connections to period numbers.
A comparison principle based on couplings of partial integro-differential operators
Freitag, 25.10.24, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
In this talk, we present a new perspective on the comparison principle for viscosity solutions of Hamilton-Jacobi (HJ), HJ-Bellman, and HJ-Isaacs equations. Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test-function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with Lévy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to both continuous and discrete operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology. We apply our theory to derive well-posedness results for partial integro-differential operators. In the context of spatially dependent Lévy operators, we show that the comparison principle is implied by a Wasserstein-contractivity property on the Lévy jump measures.\n\nJoint work with Serena Della Corte (TU Delft), Richard Kraaij (TU Delft) and Max Nendel (University of Bielefeld)
Self-adjoint codimension 2 boundary conditions for Dirac operators
Montag, 28.10.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
Joint work with Nadine Große.\n\nLet \(N\) be an oriented compact submanifold in an oriented complete Riemannian manifold \(M\). We assume that \(M\bsetminus N\) is spin and carries a unitary line bundle \(L\). We study the associated twisted Dirac operator, a priori defined on smooth section with compact support in the interior of \(M\bsetminus N\). We are interested in self-adjoint extensions of this operator.\n\nIf \(N\) has codimension~\(1\), then this is the well-studied subject of classical\nboundary values for Dirac operators. If \(N\) has codimension at least \(3\), or if \(N\) has codimension \(2\) and if \(L\) has trivial monodromy around \(N\), then we obtain a unique self-adjoint extension which coincides with the classical self-adjoint Dirac operator on \(M\). The submanifold \(N\) is thus ``invisible''.\n\nThe main topic of this talk is thus the case of codimension~\(2\) with non-trivial monodromy. We will classify all selfadjoint extensions.\n\nThis work is motivated by work of Portman, Sok and Solovej, who treated the special case of \(M=S^3\) with a link, a case important in mathematical physics.\nWe thank Boris Botvinnik and Nikolai Saveliev for stimulating discussions about this topic.
On a Complete Riemannian Metric on the Space of Closed Embedded Curves
Dienstag, 29.10.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
On a Complete Riemannian Metric on the Space of Closed Embedded Curves\njoint work with Elias Döhrer and Henrik Schumacher (Chemnitz University\nof Technology / Univ. of Georgia)\n\nIn pursuit of choosing optimal paths in the manifold of closed embedded\nspace curves we introduce a Riemannian metric which is inspired by a\nself-contact avoiding functional, namely the tangent-point potential.\nThe latter blows up if an embedding degenerates which yields infinite\nbarriers between different isotopy classes.\n\nFor finite-dimensional Riemannian manifolds the Hopf—Rinow theorem\nstates that the Heine—Borel property (bounded sets are relatively\ncompact), geodesic completeness (long-time existence of geodesic\nshooting), and metric completeness of the geodesic distance are\nequivalent. Moreover, it states that existence of length-minimizing\ngeodesics follows from each of these statements. Albeit the Hopf—Rinow\ntheorem does not hold true in this generality for infinite-dimensional\nRiemannian manifolds, we can prove all its four assertions for a\nsuitably chosen Riemannian metric on the space of closed embedded\ncurves.\n
Biased random walk on dynamical percolation
Donnerstag, 7.11.24, 15:00-16:00, Hörsaal II, Albertstr. 23b
As an example for a random walk in random environment, we study biased random walk for dynamical percolation on the d-dimensional lattice. We establish a law\nof large numbers and an invariance principle for this random walk using regeneration times.\nMoreover, we verify that the Einstein relation holds, and we investigate the speed of the walk\nas a function of the bias. While for d = 1 the speed is increasing, we show that in general this\nfails in dimension d ≥ 2. As our main result, we establish two regimes of parameters, separated\nby a critical curve, such that the speed is either eventually strictly increasing or eventually\nstrictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical\npercolation cluster, where the speed is known to be eventually zero.\n\nBased on joint work with Sebastian Andres, Dominik Schmid and Perla Sousi.
Weyl formulae for some singular metrics with application to acoustic modes in gas giants
Montag, 11.11.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
We prove eigenvalue asymptotics of the Laplace-Beltrami operator for certain singular Riemannian metrics. This is motivated by the study of propagation of soundwaves in gas planets. This is joint work with Yves Colin de Verdière, Maarten de Hoop and Emmanuel Trélat.
Rigorous justification of kinetic equations: Recent progress and finite size corrections
Dienstag, 12.11.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
The justification of kinetic equations for long times is a longstanding mathematical challenge; in fact, Hilbert's 6th problem refers specifically to the Boltzmann equation. In my talk I will discuss the case of hard spheres and the very recent progress by Deng & Hani. Finally, I will present results on finite size corrections.
On (non-)elimination of imaginaries
Dienstag, 12.11.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Imaginaries are a way of introducing canonical representatives of equivalence classes. Given a theory, an important question is whether equivalence classes are already coded in the models of the theory so that one can avoid working with imaginaries (i.e. we say that the theory then eliminates imaginaries).\n\nIn this talk, we want to present a criterion that yields the failure of elimination of imaginaries due to equivalence classes that arise as cosets of subgroups. We will illustrate the main ideas by considering the example of the theory of beautiful pairs of algebraically closed fields. Pillay and Vassiliev proved that this theory does not have elimination of imaginaries. In this talk, we will present a different proof that generalizes to more theories of fields.
Central limit theorems under non-stationarity via relative weak convergence
Freitag, 15.11.24, 13:00-14:00, Raum 232, Ernst-Zermelo-Str. 1
Traditional central limit theorems (CLTs) have limited applicability to non-stationary sequences, restricting their use in real-world, dynamic data contexts. We introduce relative weak convergence, a generalisation of classical weak convergence that provides a natural framework for CLTs in non-stationary settings. Within this framework we can prove multivariate, uniform, and sequential relative CLTs for dynamic sequences under general assumptions. These results bridge a critical gap in asymptotic statistics, enabling researchers to rigorously study the asymptotic behaviour of dynamic sequences that do not satisfy traditional assumptions of stationarity.
Symplectic topology and rectangular peg problem
Montag, 18.11.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
The rectangular peg problem, an extension of the square peg problem which has a long history, is easy to outline but challenging to prove through elementary methods. I will report the recent progress on the existence and multiplicity results, utilizing advanced concepts from symplectic topology, e.g. J-holomorphic curves and Floer theory.
Numbers on barcodes and torsion theory
Montag, 25.11.24, 16:00-17:00, Raum 404, Ernst-Zermelo-Str. 1
Morse function on a manifold M is called strong if all its critical points have different critical values. Given a strong Morse function f and a field F we construct a bunch of elements of F, which we call Bruhat numbers (they're defined up to sign). More concretely, Bruhat number is written on each bar in the barcode of f. It turns out that if homology of M over F is that of a sphere, then the product of all the numbers is independent of f. We then construct the barcode and Bruhat numbers with twisted (a.k.a. local) coefficients and prove that mentioned product equals the Reidemeister torsion of M. In particular, it's again independent of f. This way we link Morse theory to the Reidemeister torsion via barcodes. Based on a joint work with Petya Pushkar. \n
Das Rigidity Phänomen einer Artificial Venus Flytrap
Dienstag, 26.11.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Wir werden ein Modell zur Beschreibung einer Artificial Venus Flytrap formulieren, bei dem das Material als rigid angenommen wird. Während dieses rigide Modell numerisch das Phänomen der Curvature Inversion erfasst, werden wir sehen, dass die Annahme der Rigidity dazu führt, dass die planare Lösung die einzige exakte Lösung ist. Darauf aufbauend werden wir nicht-planare Lösungen betrachten, sobald wir die Rigidity-Annahme fallenlassen. Schließlich werden wir besprechen, wie sich eine Beschreibung einer Limit-Theorie angehen lässt.
Omega-kategorische Ringe
Dienstag, 26.11.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
A spectral result for massive electromagnetic Dirac Hamiltonians
Montag, 2.12.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
From the physical perspective of relativistic wave mechanics, Dirac operators on Riemannian manifolds can be interpreted as infinitesimal generators of time translation, i.e. as Hamiltonians. Most mathematical research on Dirac operators focuses on what amounts to the study of free, massless fields. In this talk however, we will consider a Dirac Hamiltonian that is coupled to an electromagnetic field and a spatially non-constant mass term. After motivating and setting up the necessary notions, we will proceed to show that, given a suitably behaved "potential well", the spectrum of such a Dirac Hamiltonian must be discrete. The result being presented is a generalization of a theorem by N. Charalambous and N. Große in 2023.
Model theory of difference fields with an additive character on the fixed field
Dienstag, 3.12.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Motivated by work of Hrushovski on pseudofinite fields with an additive character we investigate the theory ACFA+ which is the model companion of the theory of difference fields with an additive character on the fixed field working in (a mild version of) continuous logic. Building on results by Hrushovski we can recover it as the characteristic 0 asymptotic theory of the algebraic closure of finite fields with the Frobenius-automorphism and the standard character on the fixed field. We characterise 3-amalgamation in ACFA+ and obtain that ACFA+ is simple as well as a description of the connected component of the Kim-Pillay group. If time permits we present some results on higher amalgamation.\n
Apps, Projekte und KI-Tools für den digitalen Mathematikunterricht
Dienstag, 3.12.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Im Zeitalter der Künstlichen Intelligenz sollten digitale Medien im Mathematikunterricht nicht nur zur Reproduktion von Wissen, sondern vor allem zur Förderung und Stärkung von Kompetenzen eingesetzt werden. Dazu ist die Verknüpfung von mobilen Endgeräten mit individuellen, forschenden, kreativen und projektartigen Arbeitsaufträgen unerlässlich. Im Vortrag werden zunächst praktische Hinweise zum effektiven Einsatz digitaler Medien im Mathematikunterricht gegeben und einzelne Apps vorgestellt, die den Unterricht bereichern können. Anschließend wird auf die Bedeutung einer projektorientierten Lernkultur und den gezielten Einsatz von künstlicher Intelligenz eingegangen. Langfristiges Ziel eines digital angereicherten Mathematikunterrichts ist die\nEtablierung einer neuen Lern- und Prüfungskultur, die es den Schülerinnen und Schülern ermöglicht, ihre Fähigkeiten in einer dynamischen und vernetzten Welt optimal zu entfalten.
TBA
Donnerstag, 5.12.24, 10:15-11:15, Raum 414, Ernst-Zermelo-Str. 1
On a generalization of indiscernible sequences
Dienstag, 10.12.24, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Guingona and Hill introduced and studied a new hierarchy of dividing lines for first-order structures, denoted by (NCK)K , where K ranges in the theorie of ultrahomogeneous omega-categorical Ramsey structures. In a subsequent paper, Guigonna, Hill and Scow give a characterisation in terms of (generalised) K-indiscernible sequences.\nIn this talk, I will present a joint work with Nadav Meir and Aris Papadopoulos, in which we develop around these notions of K-indiscernibility. In particular, we will answer (negatively) a question posed by Guingona and Hill about the linearity of the NC_K hierarchy. As an application, we will also see that the ordered random graph admits a unique proper Ramsey reduct, namely the linear order.
Sequential topological complexities and sectional categories of subgroup inclusions
Donnerstag, 12.12.24, 10:00-11:00, Raum 414, Ernst-Zermelo-Str. 1
The sequential topological complexities (TCs) of a space are integer-valued homotopy invariants that are motivated by the motion planning problem from robotics and express the complexity of motion planning if the robots are supposed to make predetermined intermediate stops along their ways. After outlining their definitions, I will discuss the sequential TCs of aspherical spaces in the first part of my talk and describe how they can be investigated by purely algebraic means. We will also take a look at a generalization of this algebraic setting, namely sectional categories of subgroup inclusions. In the second part of my talk, I will present a general lower bound on their values and derive consequences for sequential TCs and parametrized topological complexities of epimorphisms. We will investigate the methodology of the proof of this lower bound, in which all key steps are carried out using elementary homological algebra. This is joint work with Arturo Espinosa Baro, Michael Farber and John Oprea.
Moduli spaces of 3-manifolds with boundary are finite
Montag, 16.12.24, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In joint work with Rachael Boyd and Corey Bregman we study the classifying\nspace \(B\bmathrm{Diff}(M)\) of the diffeomorphism group of a connected, compact, orientable\n3-manifold \(M\). I will recall the construction of this \(B \bmathrm{Diff}(M)\), also known as\nthe "moduli space of \(M\)", and explain how it parametrises smooth families of\nmanifolds diffeomorphic to \(M\).\n\nMilnor's prime decomposition and Thurston's geometrisation conjecture allow us\nto cut \(M\) into "geometric pieces", which each admit complete, locally\nhomogeneous Riemannian metric. For such geometric manifolds \((N,g)\) recent work\nusing Ricci flow shows that a certain space of metrics is contractible and thus\nthat the generalised Smale conjecture (often) holds: the diffeomorphism group\n\(\bmathrm{Diff}(N)\) is homotopy equivalent to the isometry group \(\bmathrm{Isom}(N,g)\).\n\nThe purpose of this talk is to explain a technique for computing the moduli\nspace \(B\bmathrm{Diff}(M)\) in terms of the moduli spaces of the pieces. We use this to\nprove that if \(M\) has non-empty boundary, then \(B \bmathrm{Diff}(M\btext{ rel boundary})\) has the\nhomotopy type of a finite CW complex, as was conjectured by Kontsevich.\n\n\n
The 1d inelastic Boltzmann equation for moderately hard potentials
Dienstag, 17.12.24, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Inelastic interaction of granular matter is a common phenomenon in natural processes. A mathematical description of such behaviour is given by a modification of the Boltzmann equation, where the dissipation of kinetic energy during collisions characterises the inelasticity at the particle level.\n\nIn this talk, we consider the occurrence of self-similar behaviour in the long-time limit for the one-dimensional inelastic Boltzmann equation. More precisely, we prove that self-similar profiles are unique in the regime of moderately hard potentials. The proof relies on a perturbation argument from the Maxwell model, together with a spectral gap for the corresponding linearised operator.
Ein roter Faden durch die Stochastik
Dienstag, 17.12.24, 18:30-19:30, Hörsaal II, Albertstr. 23b
Wenn man darauf verzichtet, Wahrscheinlichkeiten als objektiv - unabhängig vom Menschen - existierende Größen zu begreifen, sie stattdessen konsequent als unsichere und revidierbare Festlegungen deutet, die als Modelle dadurch entstehen, dass aus Erfahrungen Erwartungen werden, verschwinden seit Jahrzehnten ungelöste didaktische Probleme.\n\nUnter dem Hut eines neu entwickelten Wahrscheinlichkeits-begriffs wachsen der klassische (LAPLACE), der frequentistische (V. MISES), der subjektivistische (BAYES) und der axiomatische (KOLMOGOROFF) Wahrscheinlichkeitsbegriff zu einer Einheit zusammen. Mit seiner Hilfe entspinnt sich ein roter Faden durch die Stochastik, der Wahrscheinlichkeitsrechnung mit beschreibender und beurteilender Statistik von der Grundschule bis zum Abitur … und auch danach … zu einer Einheit verschmelzen lässt.\n\nMöglicherweise wird der in diesem Experimentalvortrag, vorgenommene Perspektivwechsel Ihr stochastisches Weltbild erweitern, vielleicht sogar konstruktiv erschüttern, wenigstens ein bisschen!
Pattern retrieval in the Hopfield model
Freitag, 20.12.24, 12:00-13:00, Raum 404, Ernst-Zermelo-Str. 1
Hopfield proposed in 1982 as a simple model capable to store and\nsuccessively retrieve a number of high dimensional patterns, which\nrepresents nowadays a cornerstone of artificially intelligence (he was\nawarded the Nobel prize in Physics 2024 for this work). I will present some\nold and new mathematical results about pattern retrieval in the Hopfield\nmodel along with some open problems.\n
Distinguishing Variants of Friedman's Property
Dienstag, 7.1.25, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
We introduce parametrized variants of Friedman's property. \(F(\blambda,\bkappa)\) states that any function from \(\bkappa\) into \(\blambda\) is constant on a closed set of order type \(\bomega_1\). The principle \(F^+((D_i: i\bin\bomega_1),\bkappa)\) (for \((D_i : i\bin\bomega_1)\) a partition of \(\bomega_1\)) states that for any sequence \((A_i: i\bin\bomega_1)\) of stationary subsets of \(E_{\bomega}^{\bkappa}\) there is a normal function \(f\bcolon\bomega_1\bto\bkappa\) such that \(f[D_i]\bsubseteq A_i\). We will prove all possible implications between instances of both properties and show the optimality of our results by obtaining suitable independence results.
Sharp functional inequalities and their stability
Donnerstag, 9.1.25, 15:00-16:00, Hörsaal II, Albertstr. 23b
The Sobolev inequality is a paradigmatic example of a functional inequality with many applications in the Calculus of Variations, Geometric Analysis and PDEs. In some of these applications the optimal value of the constant is of importance, as is a characterization of the set of optimizers. The stability question is whether functions whose Sobolev quotient is almost minimal are close to minimizers of the inequality and, if so, in which sense. We give a gentle introduction to this question and review some recent results on the Sobolev inequality and other functional inequalities of a similar nature.
The pro-etale homotopy type
Freitag, 10.1.25, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
After reviewing the classical construction of the etale homotopy type by Artin-Mazur we define a pro-etale analogue for the pro-etale site of a scheme. An important difference between the etale and pro-etale site of a scheme is that the latter has enough weakly contractible objects. Using this fact we prove that the pro-etale homotopy type of a qcqs scheme is determined by a single split affine weakly contractible hypercovering. Lastly and if time permits, we discuss the pro-etale homotopy type of a field.
On the L^p spectrum
Montag, 13.1.25, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In this talk we will consider the \(L^p\) spectrum of the Laplacian on differential forms. In particular, we will show that the resolvent set of the Laplacian on \(L^p\) integrable \(k\)-forms lies outside a parabola whenever the volume of the manifold has an exponential volume growth rate, removing the requirement on the manifold to be of bounded geometry. Moreover, we find sufficient conditions on an open Riemannian manifold so that a Weyl criterion holds for the \(L^p\)-spectrum of the Laplacian on \(k\)-forms, and we provide a detailed description of the \(L^p\) spectrum of the Laplacian on \(k\)-forms over hyperbolic space. The above results are joint work with Zhiqin Lu.
Random Generation: from Groups to Algebras
Freitag, 17.1.25, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, with emphasis on finite simple groups. In this talk, based on joint work with Aner Shalev, we study similar notions for finite and profinite associative algebras.\n\nLet \(k\) be a finite field. Let \(A\) be a finite associative algebra over \(k\), and let \(P(A)\) be the probability that two random elements of \(A\) will generate it. It is known that, if \(A\) is simple, then \(P(A) \bto 1\) as \(|A| \bto \binfty\). We extend this result for larger classes of finite associative algebras. For \(A\) simple, we estimate the growth rate of \(P(A)\) and find the best possible lower bound for it. We also study the random generation of \(A\) by two special elements.\n\nNext, let \(A\) be a profinite algebra over \(k\). We show that \(A\) is positively finitely generated if and only if \(A\) has polynomial maximal subalgebra growth. Related quantitative results are also\nobtained.
Problemlösen in Prüfungssituationen
Dienstag, 21.1.25, 18:30-19:30, Hörsaal II, Albertstr. 23b
Problemlösefähigkeit gilt als eine der Kernkompetenzen für das Lernen und Arbeiten im 21.\nJahrhundert und ist eine der zentralen Begründungen für den Mathematik-Unterricht. Im Unterricht\nbieten „Problemlöseaufgaben“ (auch Modellierungs- und Begründungsaufgaben) ein großes\nPotenzial zur kognitiven Aktivierung. Daher bilden seit 2021 „Problemlösen“ und „Modellieren“\neigene Bildungsplaneinheiten im beruflichen Gymnasium in Baden-Württemberg. Seit dem Abitur\n2024 findet sich in der schriftlichen Prüfung des beruflichen Gymnasiums eine eigene,\nausgewiesene „Problemlöseaufgabe“, über die speziell diese Prozesskompetenz beurteilt werden\nsoll. Im Vortrag wird die Entwicklung und bisherige Erfahrung am beruflichen Gymnasium zum\nThema „Problemlösen in Prüfungssituationen“ vorgestellt.
Bericht über aktuelle Projekte, praktische Fragen
Freitag, 24.1.25, 09:00-10:00, Raum 232, Ernst-Zermelo-Straße 1
A sharp isoperimetric gap theorem in non-positive curvature
Montag, 27.1.25, 16:15-17:15, Raum 404, Ernst-Zermelo-Str. 1
In joint work with Cornelia Drutu, Panos Papasoglu, and Stephan Stadler, we\nstudy isoperimetric\ninequalities for null-homotopies of Lipschitz 2-spheres in Hadamard manifolds\nor, more generally,\nproper CAT(0) spaces. In one dimension less, for fillings of circles by discs,\nit is known that a \nquadratic inequality with a constant smaller than the sharp threshold\n\(1/(4\bpi)\) implies that the \nunderlying space is Gromov hyperbolic and satisfies a linear inequality. Our\nmain result is a first \nanalogous gap theorem in higher dimensions, yielding exponents arbitrarily\nclose to 1. Towards \nthis we prove a Euclidean isoperimetric inequality for null-homotopies of\n2-spheres, apparently \nmissing in the literature, and introduce so-called minimal tetrahedra, which we\ndemonstrate satisfy \na linear inequality.
The tangent-point energy for surfaces and its symmetric critical points
Dienstag, 28.1.25, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We will prove the existence of several distinct surfaces of the same given genus that are critical points of the tangent-point energy.\nThe first step of this proof is to pull the tangent-point energy into our comfort zone. The key idea of this step is to describe the surfaces by embeddings of a \(2\)D manifold \(M\) into \(\bmathbb{R}^3.\) We will define the tangent-point energy on the set of \(W^{s,q}\)-embeddings, which is an open subset of the Banach space \(W^{s,q}(M,\bmathbb{R}^3).\) We will discuss this space and characterize the energy space in terms of this regularity. We will see that the tangent-point energy of each \(W^{s,q}\)-embedding is finite, and each surface with finite energy can be described by a \(W^{s,q}\)-embedding. Furthermore, we will show that the tangent-point energy is continuously Fréchet differentiable on this domain.\nOnce we have reached this comfortable situation, we will study the energy landscape. By an application of Palais' principle of symmetric criticality and a symmetry argument, we will establish the claimed result.
Derivation-like theories and neostability
Dienstag, 28.1.25, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Motivated by structural properties of differential field extensions, Omar Leon Sanchez and I introduced the notion of a theory T being derivation-like with respect to another model complete theory \(T_0\). We proved that when T admits a model companion \(T^+\), then several model-theoretic properties are transferred from \(T_0\) to \(T^+\). These properties include completeness, quantifier elimination, stability, simplicity, and NSOP\(_1\). Examples of derivation-like theories are plentiful but are typically obtained by adding extra structure to theories of fields. In this talk I will introduce the central notions, detail how the proofs work by lifting independence relations from \(T_0\) to \(T^+\), and give examples.
B-Spline Discretization of Inextensible Curves
Dienstag, 4.2.25, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
Two models for the numerical approximation of the elastic movement of inextensible curves are investigated. The configuration with the least possible elastic energy can be approached by employing a gradient flow of the corresponding energy functional. This gradient flow is then discretized using cubic spline functions. First, we examine a scheme that is based on functions that are once globally differentiable, and then we try to recreate that scheme using the twice globally differentiable B-splines. We show the convergence of the discretizations to the continuous problem and compare the performance of the two discretization schemes.\n
CoHAs of Weighted Projective Lines
Freitag, 7.2.25, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The Cohomological Hall Algebra (CoHA), introduced by Kontsevich--Soibelman, is a cohomological analogue of the Ringel--Hall algebra. While for a symmetric quiver this algebra is free (super) symmetric, there were essentially no other examples that were known explicitly in terms of generators and relations, until Franzen--Reineke computed the CoHA of (regular representations) of the 2-Kronecker quiver. The 2-Kronecker quiver is derived equivalent to the projective line and regular representations correspond to torsion sheaves; thus Franzen--Reineke's algebra can also be seen as the CoHA of torsion sheaves on \(\bmathbb{P}^1\). We extend this computation to CoHAs of torsion sheaves on the so-called weighted projective lines. As a special case, we also obtain the CoHAs of regular representations of all other extended Dynkin quivers (satisfying a certain natural condition on the Euler form).
Normale Spannbäume
Dienstag, 11.3.25, 14:30-16:00, Seminarraum 404
Das Aufwärts Löwenheim-Skolem-Tarski-Theorem in der infinitären Logik Lω1,ω
Dienstag, 18.3.25, 14:30-16:00, Seminarraum 404