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.