Fourier expansions of vector-valued automorphic functions
Monday, 10.1.22, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
In this talk, I provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We will discuss a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.
Adaptive finite elements in convex optimisation
Tuesday, 11.1.22, 02:15-03:15, https://uni-freiburg.zoom.us/j/64858555488?pwd=eGgvUHErS1VZUm43bXl1SXJCMlloUT09
Abstract: The advantages of adaptive discretizations are well-documented, however, many convex optimization algorithms are not able to utilize these advantages. Taking a step to address this, in this talk we will analyze how the FISTA algorithm behaves with inexact discretizations. In doing so, we also prove new convergence results beyond the capabilities of the original algorithm. We will finish with numerical experiments which demonstrate the potential for improved efficiency by using adaptive finite elements in convex optimization problems.\n
Therapeutic genome editing and its need for artificial intelligence
Friday, 14.1.22, 12:00-13:00, online: Zoom
Die Weierstraßdarstellung von Minimalflächen
Monday, 17.1.22, 16:15-17:15, BBB-Raum (s. Diffgeo-Liste)
In diesem Vortrag wird die Weierstraß-Darstellung für konform parametrisierte Minimalflächen hergeleitet, in welcher eine holomorphe und eine meromorphe Funktion auftreten, die eine solche Fläche unter geringen Zusatzbedingungen beschreiben. Anfangs wird dafür an einige Begriffe der Elementaren Differentialgeometrie und der Funktionentheorie erinnert. Besonderes Augenmerk liegt im weiteren Verlauf auf der Korrespondenz zwischen der Menge der konform parametrisierten Minimalflächen und der Menge der holomorphen, isotropen Funktionen auf demselben Definitionsbereich, da diese Beziehung den Ausgangspunkt der Weierstraß'schen Konstruktion darstellt.
Optimal stochastic control of a path-dependent risk indicator for insurance companies
Tuesday, 18.1.22, 08:30-09:30, Zoom Meeting
The drawdown of a stochastic process (modelling the surplus of a company) is the absolute distance to its historical high water mark. It can therefore be interpreted as a "relative loss" and\nis a risk and performance measure widely used in financial applications: whilst large and long-\nlasting drawdowns might manifest existing financial and reputational risks, small and infrequent\ndrawdowns can be considered a sign of economic strength and stability. For this reason,\nminimising drawdowns is desirable for companies - especially in insurance, where customer trust\nis the basis for success. In this talk, we consider a stochastic control problem inspired by the\nminimisation of the drawdown size and "recovery time" for insurance companies. By exploiting\nconnections to Laplace transforms of passage times, Hamilton-Jacobi-Bellman equations and\nreflected stochastic differential equations, we find value functions and optimal strategies. We\ndiscuss our results and implications of the model in explicit examples.
On Absence of Arbitrage and Propagation of Chaos
Tuesday, 18.1.22, 10:30-11:30, Zoom Meeting
In the talk I discuss two recent research projects. The first part is related to mathematical finance. More precisely, I consider a single asset model whose (discounted) price process is assumed to be a non-negative semimartingale diffusion. The important new feature of this model is that the diffusion is not assumed to have an SDE representation, which allows possible local time effects such as sticky points. For this financial model I discuss explicit deterministic sufficient and necessary conditions for the existence and absence of arbitrage in the sense of NFLVR. The proof of the result is based on the concept of separating times, which I also shortly explain. In the second part of my talk I discuss a propagation of chaos result for a system of (weakly) interacting stochastic PDEs. More precisely, under quite mild continuity and linear growth conditions, I present a law of large numbers and the corresponding McKean-Vlasov limit. The first part of my talk is based on joint work with Mikhail Urusov (U Duisburg-Essen).
Dependence Structures in Finance: Applications in Credit Risk Modeling and Pairs Trading
Tuesday, 18.1.22, 14:00-15:00, Zoom Meeting
In the first part of the talk, we develop a generalized interacting intensity-based contagious credit risk model with hidden Markov state process. The main contribution is that the model, as well as the closed-form default distributions derived are applicable to a wide class of default intensities with various forms of dependence structures. A number of practical problems can then be solved efficiently with these explicit formulas for the distribution of default times. In the second part of the talk, we discuss optimal pairs trading strategies under symmetric and non-symmetric trading constraints. Under the assumption that the price spread of a pair of correlated securities is mean-reverting and follows a Ornstein-Uhlenbeck process, closed-form trading strategies under each of the constraints are obtained in a mean-variance framework. Numerical results indicate that our pairs trading strategies have fairly good performance.
Density of compressibility
Tuesday, 18.1.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
Compressibility is a certain isolation notion suited to NIP theories. One definition of distality of a theory (a crucial notion with useful combinatorial consequences) is that every type is compressible. I will discuss some good properties of compressibility and their consequences, which include the existence of "compressibly atomic" models over arbitrary sets in countable NIP theories, and uniform honest definitions for an NIP formula. \n\nJoint work with Itay Kaplan and Pierre Simon.
Zeichne eine Skizze. (K)eine Hilfe bei der Bearbeitung von mathematischen Modellierungsaufgaben im Bereich der Geometrie?
Tuesday, 18.1.22, 19:30-20:30, Zoom-Link: https://uni-freiburg.zoom.us/j/3852066250 Kennwort NZUWh12NY
Selbst erstellte Skizzen haben das Potential, Schüler:innen beim mathematischen Modellieren zu unterstützen. Jedoch zeichnen Schüler:innen selten spontan eine Skizze. Die Aufforderung zum Zeichnen einer Skizze ist daher ein vielversprechendes Instrument, die Modellierungsleistung der Schüler:innen zu verbessern. Im DFG-Projekt ViMo wird der Frage nachgegangen, unter welchen Bedingungen Zeichenaufforderungen die Modellierungsleistungen im Bereich der Geometrie verbessern. Im Vortrag werden zentrale Ergebnisse aus drei ViMo-Studien präsentiert: einer Laborstudie zur Nutzung von Skizzen, einer experimentellen Studie zu Effekten unterschiedlicher Zeichenaufforderungen und einer Unterrichtsstudie zu Effekten der Vermittlung von Strategiewissen. Zusammengenommen zeigt sich, dass für die Wirksamkeit von Zeichenaufforderungen kognitive Voraussetzungen (u.a. Strategiewissen) sowie vermittelnde Variablen (u.a. Skizzenart und Skizzenqualität) eine Rolle spielen. Schlussfolgerungen für die weitere Forschung und die Unterrichtspraxis werden diskutiert.
Zeichne eine Skizze. (K)eine Hilfe bei der Bearbeitung von mathematischen Modellierungsaufgaben im Bereich der Geometrie?
Tuesday, 18.1.22, 19:30-20:30, Hörsaal II, Albertstr. 23b
Selbst erstellte Skizzen haben das Potential, Schüler:innen beim mathematischen Modellieren zu unterstützen. Jedoch zeichnen Schüler:innen selten spontan eine Skizze. Die Aufforderung zum Zeichnen einer Skizze ist daher ein vielversprechendes Instrument, die Modellierungsleistung der Schüler:innen zu verbessern. Im DFG-Projekt ViMo wird der Frage nachgegangen, unter welchen Bedingungen Zeichenaufforderungen die Modellierungsleistungen im Bereich der Geometrie verbessern. Im Vortrag werden zentrale Ergebnisse aus drei ViMo-Studien präsentiert: einer Laborstudie zur Nutzung von Skizzen, einer experimentellen Studie zu Effekten unterschiedlicher Zeichenaufforderungen und einer Unterrichtsstudie zu Effekten der Vermittlung von Strategiewissen. Zusammengenommen zeigt sich, dass für die Wirksamkeit von Zeichenaufforderungen kognitive Voraussetzungen (u.a. Strategiewissen) sowie vermittelnde Variablen (u.a. Skizzenart und Skizzenqualität) eine Rolle spielen. Schlussfolgerungen für die weitere Forschung und die Unterrichtspraxis werden diskutiert.
Does hyperbolic 3-geometry provide an infinite family of fields with class number one?
Friday, 21.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The class number one problem dating back to Gauss' work on quadratic\nforms asks whether infinitely many number fields of ideal class number\none exist. In an exciting 2017 paper, Ulf Rehmann and Ernest Vinberg\nhave described a candidate family of fields defined in hyperbolic\n3-geometry which, in all computed examples, turned out to have class\nnumber one. In the talk, we will introduce this family of fields and\nexamine its class number phenomenon from different perspectives: by an\nanalogy to a known class number formula in hyperbolic 3-geometry, by\nempirical computations and by estimates with known class number\nstatistics.
Deep generative approaches for omics data: interpretability and sample-size constraints
Friday, 21.1.22, 12:00-13:00, online: Zoom
Deep generative models (DGMs) are promising tools, e.g., for learning latent structure in high dimensional omics data such as single-cell RNA-Seq data as well as for generating synthetic observations, e.g. for securely sharing single nucleotide polymorphism (SNP) data. Here I address the interpretability of DGMs, specifically by showing how to\nlink latent space information with observed variables (e.g. expression levels of genes). In addition I address the performance of DGMs under sample size constraints which are frequently observable when working with omics data in the biomedical context.
Yamabe Flow on Singular Spaces
Monday, 24.1.22, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
I will talk about the Yamabe flow on compact spaces with conical singularities (and more generally: smoothly stratified spaces with iterated cone-edge metrics). I will present the classical Yamabe problem, and talk about why the Yamabe flow exists for all time in our setting. I will end by discussing convergence (and failure thereof). \n\nThis is joint work with Gilles Carron and Boris Vertman, arXiv:2106.01799 .\n\n\n
HOMOGENIZATION OF DISCRETE THIN STRUCTURES
Tuesday, 25.1.22, 14:15-15:15, https://uni-freiburg.zoom.us/j/62669086921?pwd=bEdlZDNQU2plREZ3aEJ6RFpTOWNuQT09
We investigate discrete thin objects which are described by a subset \(X\) of \(\bmathbb{Z}^d\btimes \b{0,\bdots, T-1 \b}^k\), for some \(T\bin\bmathbb{N}\) and \(d,k\bgeq 1\). We only require that \(X\) is a connected graph and periodic in the first \(d\)-directions.\nWe consider quadratic energies on \(X\) and we perform a discrete-to-continuum and dimension-reduction process for such energies.\nWe show that, upon scaling of the domain and of the energies by a small parameter \(\bvarepsilon\), the scaled energies \(\bGamma\)-converges to a \(d\)-dimensional functional. The main technical points are a dimension-lowering coarse-graining process and a discrete version of the p-connectedness approach by Zhikov.\nThis is a joint work with A. Braides.
Distality-Rank
Tuesday, 25.1.22, 14:30-15:30, Raum 404, Ernst-Zermelo-Str. 1
For 1≤k<ω we introduce k-distality and strong k-distality as properties of first-order theories, which both coincide with distality for k=1. With these properties we define the (strong) distality rank of a theory, and we give examples of theories with (strong) distality rank m for all ordinal numbers m between 1 and ω. We prove that the two ranks coincide for strongly minimal theories by providing a characterization in terms of the algebraic closure.\n
The standard conjecture of Hodge type for abelian fourfolds
Friday, 28.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
The standard conjecture of Hodge type for abelian fourfolds
Friday, 28.1.22, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
Let S be a surface, V be the Q-vector space of divisors on S modulo numerical equivalence and d be the dimension of V. The intersection product defines a non degenerate quadratic form on V. The Hodge index theorem says that it is of signature (1,d-1).\nIn the Sixties Grothendieck conjectured a generalization of this statement to cycles of any codimension on a variety of any dimension. In characteristic zero this conjecture is a consequence of Hodge theory but in positive characteristic almost nothing is known. Instead of studying these quadratic forms at the archimedean place we will study them at p-adic places. It turns out that this question is more tractable, thanks to p-adic Hodge theory. Moreover, using classical product formulas on quadratic forms, the p-adic result will give non-trivial information on the archimedean place. For instance, we will prove the original conjecture for abelian fourfolds.
Friday, 28.1.22, 12:00-13:00, online: Zoom
Giant Gravitons in twisted holography
Monday, 31.1.22, 16:15-17:15, BBB (link will be distributed via the Diffgeo mailing list)
I will talk about a correspondence between solutions of certain matrix equations and holomorphic curves in SL(2,C). The correspondence is motivated by twisted holography, which is a physical duality between a chiral algebra and topological B-model on SL(2,C). Determinant operators in the chiral algebra are dual to the Giant Graviton branes in the B-model. For each saddle of the correlation functions of determinants, we will define a spectral curve in SL(2,C), which we will identify with the worldsheet of the dual Giant Graviton brane.