Strict minimality and algebraic relations between solutions in Poizat’s family of equations.
Freitag, 5.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In the last twenty years, techniques from model theory and differential algebra have been successfully applied to study integrability and algebraic dependence/independence of solutions in certain families of algebraic differential equations. I will discuss some of these techniques on a concrete example: the family of (non-linear) differential equations of the form y’’/y’ = f(y) where f(y) is a rational function. From a model-theoretic perspective, the study of this family was initiated by Poizat in the 80’s. \n\nThis is joint work with James Freitag, David Marker and Ronnie Nagloo.
Multiplication of differential operators in terms of connections using Lie-Rinehart algebras
Montag, 8.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
The multiplication of two differential operators in an open set of \(\bmathbb{R}^n\)\nis explicitly known in terms of their standard symbols: this is a motivating point\nfor the theory of deformation quantization. On a differentiable manifold equipped\nwith a connection ∇ in the tangent bundle the same formula --where partial\nderivatives are replaced with symmetrized covariant derivatives-- will be wrong in\ngeneral, and one has to correct it by terms containing torsion and curvature and\ntheir covariant derivatives. In our work with my PhD student Hamilton Menezes de Araujo\nwe shall give an 'explicit formula' of the corrected formula in the more algebraic framework\nof Lie-Rinehart algebras L (G.S.Rinehart, 1963, which are now being used in the study\nof singular foliations) over a commutative unital K-algebra A (where the ground ring K should\ncontain the rational numbers as a subring). L generalizes the\nLie algebra of vector fields (more generally Lie algebroids) and A the algebra of\nsmooth functions in differential geometry. The enveloping algebra of L over A introduced\nby Rinehart plays the rôle of the differential operator algebra. The arising combinatorial problems\ncan conveniently be treated in terms of the fibrewise shuffle comultiplication in the free algebra\ngenerated by L over A and the associated convolution products. The torsion and curvature\nterms arise in a morphism of Lie-Rinehart algebras Z from the free Lie algebra generated\nby L over A equipped with a Lie-Rinehart bracket isomorphic to the one on M.Kapranov's\n'free path Lie algebroid' (2007) to L which are related to the (infinitesimal)\nholonomy of the connection. Z is obtained by a simple explicit linear recursion.\nThe framework allows to discuss `family theorems' by replacing the ground ring K\nbut the smooth function algebra of the base of a fibered manifold.
Asymptotic Stability in a free boundary model of cell motion
Dienstag, 9.11.21, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
We introduce a free boundary model of the onset of motion of a living cell (e.g. a keratocyte) driven by myosin contraction, with focus on a transition from unstable radial stationary states to stable asymmetric moving states. This model generalizes a previous 1D model (Truskinovsky et al.) by combining a Keller-Segel model, a Hele-Shaw boundary condition, and the Young-Laplace law with a nonlocal regularizing term, which precludes blow-up or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We show that this model has a family of asymmetric traveling wave solutions bifurcating from a family of stationary states. Our goal is to establish observable steady cell motion with constant velocity. Mathematically, this amounts to proving stability of the traveling wave solutions, which requires generalization of the standard notion of stability. Our main result is establishing nonlinear asymptotic stability of traveling solutions. To this end, we derive an explicit asymptotic formula for the stability-determining eigenvalue from asymptotic expansions in small speed. This formula greatly simplifies the numerical computation of the sign of this eigenvalue and reveals the physics underlying onset of the cell motion and stability of moving states. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading.\n\nThis is joint work with V. Rybalko (Verkin Institute, Ukraine) and C. A. Safsten (Penn State, PhD student).
Construction and Representation of Generalised Equitable Preference Relations (Based on a Joint Paper with Ram Sewak Dubey)
Dienstag, 9.11.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In recent years a strict connection between set theory and social welfare relations has been studied in theoretical economics. In this talk I present some generalised versions of redistributional equity principles for infinite populations. More specifically we focus on the representation and construction of social welfare relations satisfying these generalised principles and we also combine them with other known efficiency and intergenerational equity principles in economic theory; in particular, important roles from set theory are played by ultrafilters and non-Ramsey sets.\n\n
Rigidity of Hyperelliptic Manifolds
Freitag, 12.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
A rigid compact complex manifold is one which admits no non-trivial deformations in a neighborhood of the base point. I will start by discussing the notions of rigidity and hyperelliptic manifolds (these are torus quotients with certain properties). After that, I will explain how to classify rigid hyperelliptic manifolds in complex dimension four (which is the minimal dimension in which rigid hyperelliptic manifolds exist). The classification has differential and complex geometric as well as algebraic flavors. This is joint work with Christian Gleißner (Universität Bayreuth).\n\n\n
Nu invariants of extra twisted connected sums
Montag, 15.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
The \(\bnu\) invariant is an invariant of \(G_2\)-structures on closed 7-manifolds. It can be computed in examples and has been used to show that for some closed spin 7-manifolds, the moduli space of \(G_2\)-metrics is not connected.\n\nIn this talk, we will present the computation for extra twisted connected sums and show how to obtain a tractable formula in the end.
An obstacle problem for the p−elastic energy
Dienstag, 16.11.21, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
In this talk we seek to minimize the p-elastic curvature energy\nE(u) := \bintgraph(u) |κ|^p ds\namong all graphs u ∈ W^2,p (0, 1)∩ W0^1,p(0, 1) that satisfy the obstacle constraint u(x) ≥ ψ(x) for all x ∈ [0, 1]. Here ψ ∈ C^0([0, 1]) is an obstacle function. The energy functional imposes three major challenges that we need to overcome:\n\n1. Lack of coercivity.\n\n2. Loss of regularity on the coincidence set {u = ψ}.\n\n3. (For p > 2:) Degeneracy of the Euler-Lagrange equation.\n\nWe will develop methods to examine all three phenomena. A key ingredient for\nthis analysis goes back to L. Euler: One can find a substitution that makes the\nEuler-Lagrange equation elliptic.\n\nFinally, we are able to show sharp existence (and non-existence) results and\ndiscuss the optimal regularity of minimizers.
The number of models of the theory of existentially closed differential fields revisited.
Dienstag, 16.11.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
In 1973, Shelah showed that the theory of existentially closed differential fields of characteristic 0 although complete and totally transcendental admits the maximal number of models in any uncountable cardinality. This was extended by Hrushovski-Sokolovic and independently by Pillay to countable models in the 90’s. \n\nIn my talk, I will discuss how to recover (local and global versions of) Shelah’s result from the study of a specific family of differential equations: the differential equations of the form y’’/y’ = f(y) where f(y) is an arbitrary rational function introduced by Poizat in the 90's. \n\nThis is joint work with J. Freitag, D. Marker and R. Nagloo.
Mathematikunterricht in einer durch Digitalisierung geprägten Welt
Dienstag, 16.11.21, 19:30-20:30, Hörsaal II, Albertstr. 23b
Der erfolgreiche Einsatz digitaler Medien stellt eine der wesentlichen Herausforderung des heutigen Mathematikunterrichts dar, was sich nicht erst durch die aktuelle Situation um die COVID19-Pandemie gezeigt hat. Im Vortrag wird am Beispiel einer Studie zur Bruchrechnung aufgezeigt, wie eine Implementation digitaler Tools in den Regelunterricht aussehen kann und welche Vorteile für das Lehren und Lernen von Mathematik erwartet werden können. Weiter werden auf der Basis eine Forschungssynthese Gelingensfaktoren für den Einsatz digitaler Medien aufgezeigt und dargestellt, welche aktuellen Herausforderungen die Forschung zur Digitalisierung des Mathematikunterrichts mit Blick auf die Unterrichtspraxis beschäftigen.
Tag der offenen Tür
Mittwoch, 17.11.21, 10:30-11:30, BigBlueButton. Es gibt synchrone und asynchrone Angebote, siehe Webseite (dazu auf Titel klicken)
A geometric approach to the charge statistic
Freitag, 19.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The charge is a statistic defined on the set of Young tableaux by Lascoux and Schützenberger in 1978. They showed that the generating functions for this statistic are the q-analogue of the weight multiplicities in type A.\nIn this talk, I will explain a different and geometrically motivated approach to the charge statistic. The q-weight multiplicities can in fact be obtained as Kazhdan-Lusztig polynomials for the affine Grassmannian. In this setting, we will recover the charge statistic by studying hyperbolic localization for a family of cocharacters.
A convergent algorithm for the interaction of mean curvature flow and diffusion
Montag, 22.11.21, 12:15-13:15, Raum 226, Hermann-Herder-Str. 10
\nIn this talk we will present an evolving surface finite elment algorithm for the interaction of forced mean curvature flow and a diffusion process on the surface.\nThe evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by the above coupled geometric PDE system. The coupled system is inspired by the gradient flow of a coupled energy, we will use this model for introductury purposes.\nWe will present two algorithms, based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. In some sense, these algorithms return home, since they were heavily inspired by the works of Professor Dziuk.\nFor one of the numerical methods we will give some insights into the stability estimates which are used to prove optimal-order \(H1\)-norm error estimates for finite elements of degree at least two.\nWe will present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.\nThe talk is based on joint work with C.~M.~Elliott (Warwick) and H.~Garcke (Regensburg).
TBA
Dienstag, 23.11.21, 14:15-15:15, Raum 226, Hermann-Herder-Str. 10
TBA
TBA
Dienstag, 23.11.21, 14:15-15:15, https://uni-freiburg.zoom.us/j/61095147552?pwd=VDNSdnRnMVFCbVgxTVJ0QWNmeU0yQT09#success
TBA
On abelian corners and squares
Dienstag, 23.11.21, 14:45-15:45, Raum 404, Ernst-Zermelo-Str. 1
Given an abelian group G, a corner is a a subset of pairs of the form \(\b{(x,y), (x+g, y), (x, y+g)\b}\) with \(g\) non trivial. Ajtai and Szemerédi proved that, asymptotically for finite abelian groups, every dense subset \(S\) of \(G\btimes G\) contains an corner. Shkredov gave a quantitative lower bound on the density of the subset \(S\). In this talk, we will explain how model-theoretic conditions on the subset \(S\), such as local stability, will imply the existence of corners and of cubes for (pseudo-)finite abelian groups. This is joint work with D. Palacin (Madrid/Freiburg) and J. Wolf (Cambridge).\n
Mathematisches Kolloquium:
Johan Commelin "Liquid Tensor Experiment"
Donnerstag, 25.11.21, 17:00-18:00, Hörsaal Pharmazie (Hermann-Herder-Str. 7)
In December 2020, Peter Scholze posed a challenge to formally verify the main theorem on liquid \(\bmathbb{R}\)-vector spaces, which is part of his joint work with Dustin Clausen on condensed mathematics. I took up this challenge with a team of mathematicians to verify the theorem in the Lean proof assistant. Half a year later, we reached a major milestone, and our expectation is that in a couple of months we will have completed the full challenge, In this talk I will give a brief motivation for condensed/liquid mathematics, a demonstration of the Lean proof assistant, and discuss our experiences formalizing state-of-the-art research in mathematics.
Liquid Tensor Experiment
Donnerstag, 25.11.21, 17:00-18:00, Hörsaal Pharmazie, Hermann-Herder-Str. 7
In December 2020, Peter Scholze posed a challenge to formally verify\nthe main theorem on liquid \(\bmathbb{R}\)-vector spaces,\nwhich is part of his joint work with Dustin Clausen on condensed\nmathematics.\nI took up this challenge with a team of mathematicians\nto verify the theorem in the Lean proof assistant.\nHalf a year later, we reached a major milestone,\nand our expectation is that in a couple of months\nwe will have completed the full challenge.\n\nIn this talk I will give a brief motivation for condensed/liquid\nmathematics,\na demonstration of the Lean proof assistant,\nand discuss our experiences formalizing state-of-the-art research in\nmathematics.
Schinzel's Hypothesis with probability 1 and rational points on varieties in families
Freitag, 26.11.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
In a joint work with Efthymios Sofos we prove that Schinzel's Hypothesis (H) holds for 100% of polynomials of any fixed degree. I will pass in silence the proof of this analytic result, but will explain how to deduce from this that among varieties in specific families over Q, a positive proportion have rational points. The main examples are varieties given by generalised Châtelet equations (a norm form equals a polynomial) and diagonal conic bundles of any fixed degree over the projective line.\n
Freitag, 26.11.21, 12:00-13:00, online: Zoom
https://conferencekuwert.github.io/directions/
Montag, 29.11.21, 00:00-01:00, KG1 : Platz der Universität 3
Geometric PDEs in Freiburg: A conference in honor of the 60th Birthday of Ernst Kuwert. \n\n
Elliptic Genera and \(G_2\)-manifolds
Montag, 29.11.21, 16:15-17:15, Hörsaal II, Albertstr. 23b
In 1988 Witten showed that the universal elliptic genus of a manifold \(M\) can be interpreted as the index of a twisted Dirac operator on the the loop space of \(M\). Furthermore he discovered, that the index of this Dirac operator has similar modular properties, if one restricts to string manifolds. The resulting modular form is now called the Witten genus.\n\nIn my talk I will give an introduction to modular forms and I will formaly derive the Witten genus from the index theorem.\n\nIf we compare the Witten genus with the elliptic genus in dimension \(8\), there occur characteristic classes, which are connected with the Nu-invariant of \(G_2\)-manifolds.
On the Distributivity of Perfect Tree Forcings for Singulars of Uncountable Cofinality
Dienstag, 30.11.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Forcing with perfect trees is a major topic of research in set theory. One example is Namba forcing, which was originally developed as an example of a forcing that is \((\baleph_0,\baleph_1)\)-distributive but not \((\baleph_0,\baleph_2)\)-distributive. A recent paper of Dobrinen, Hathaway, and Prikry shows that a classical singular Namba forcing \(P_\bkappa\) is \((\bomega,\bnu)\)-distributive for \(\bnu<\bkappa\) if \(\bkappa\) is a singular strong limit cardinal of countable cofinality. The authors then ask whether this result generalizes, i.e\(.\) if \(P_\bkappa\) is (cf\((\bkappa),\bnu)\)-distributive for \(\bnu<\bkappa\) if \(\bkappa\) has uncountable cofinality. In joint work with Heike Mildenberger, we answer this question in the negative by showing that in this case \({\bmathbb P}_\bkappa\) is not (cf\((\bkappa),2)\)-distributive.
On the Distributivity of Perfect Tree Forcings for Singulars of Uncountable Cofinality
Dienstag, 30.11.21, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Forcing with perfect trees is a major topic of research in set theory. One example is Namba forcing, which was originally developed as an example of a forcing that is \((\baleph_0,\baleph_1)\)-distributive but not \n\((\baleph_0,\baleph_2)\)-distributive. A recent paper of Dobrinen, Hathaway, and Prikry shows that a classical singular Namba forcing \n\(P_\bkappa\) is \((\bomega,\bnu)\)-distributive for \(\bnu<\bkappa\) if \(\bkappa\) is a singular strong limit cardinal of countable cofinality. The authors then ask whether this result generalizes, i.e. if \n\(P_\bkappa\) is (cf\((\bkappa),\bnu)\)-distributive for \(\bnu<\bkappa\) if \(\bkappa\) has uncountable cofinality. In joint work with Heike Mildenberger, we answer this question in the negative by showing that in this case \n\(P_\bkappa\) is not (cf\((\bkappa),2)\)-distributive.