S-unit equation and Chabauty
Freitag, 8.1.21, 10:30-11:30, SR 404
Monopoles and Landau-Ginzburg Models
Montag, 11.1.21, 16:00-17:00, vSR318 (Kasparov)
Approximation of Integral Fractional Laplacian and Fractional PDEs via sinc-Basis
Dienstag, 12.1.21, 14:00-15:00, Hörsaal II (virtuell:Lasker)
Fueled by many applications in random processes, imaging science, geophysics, etc., fractional Laplacians have recently received significant attention. The key driving force behind the success of this operator is its ability to capture non-local effects while enforcing less smoothness on functions. In this paper, we introduce a spectral method to approximate this operator employing a sinc basis. Using our scheme, the evaluation of the operator and its application onto a vector has complexity of O(Nlog(N)) where N is the number of unknowns.\nThus, using iterative methods such as CG, we provide an efficient strategy to solve fractional partial differential equations with exterior Dirichlet conditions on arbitrary Lipschitz domains. Our implementation works in both 2d and 3d. We also recover the FEM rates of convergence on benchmark problems. We further illustrate the efficiency of our approach by applying it to fractional Allen-Cahn and\nimage denoising problems.\n
Algebraic cycles and refined unramified cohomology
Freitag, 15.1.21, 10:30-11:30, SR 404
We introduce refined unramified cohomology groups. This notion allows us to give in arbitrary degree a cohomological interpretation of the failure of integral Hodge- or Tate-type conjectures, of l-adic Griffiths groups, and of the subgroup of the Griffiths group that consists of torsion classes with trivial transcendental Abel--Jacobi invariant. Our approach simplifies and generalizes to cycles of arbitrary codimension previous results of Bloch--Ogus, Colliot-Thélène--Voisin, Voisin, and Ma that concerned cycles of codimension two or three. As an application, we give for any i>2 the first example of a uniruled smooth complex projective variety for which the integral Hodge conjecture fails for codimension i-cycles in a way that cannot be explained by the failure on any lower-dimensional variety.
Multiplication of BPS states in VOAs from string theory
Freitag, 15.1.21, 14:15-15:15, vSR217 (Steinitz)
In the first part of the talk, I will state some generalities about vertex operator algebras (VOAs). This includes a brief outline of how studying these mathematical objects is justified by their importance to conformal field theory.\n\nThe second part will contain segments of my PhD thesis project. This will make use of a generalized version of VOAs, which is needed, for instance, to formalize those field theories occurring in string theory. The project aims at a mathematically rigorous definition of an algebra structure on states of minimal energy -- so-called Bogomol'nyi-Prasad-Sommerfield (BPS) states --, which was first introduced by Harvey and Moore. While a significant amount of generalization is still work in progress or beyond the scope of the talk, I will try and demonstrate the main concept in the case of torus compactifications.
Numerical Analysis of Implicitly Constituted Incompressible Flow
Montag, 18.1.21, 08:30-09:30, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
In the classical theory of fluid mechanics a linear relationship between the shear stress\ntensor and the symmetric velocity gradient is often assumed. Even when a nonlinear\nrelationship is assumed between the shear stress and the symmetric velocity gradient,\nit is typically formulated in terms of an explicit relation. Implicit constitutive models\nprovide a theoretical framework that generalises this, allowing for an implicit\nconstitutive relation. In this talk, I will present some results dealing with the finite\nelement approximation of implicitly constituted incompressible flow, ranging from\nconvergence aspects to fast solvers. In particular, I will introduce a preconditioner\nbased on augmented Lagrangian stabilisation and a specialised multigrid algorithm\nthat exhibits robust behaviour even for models incorporating thermal effects.
The benefits of smoothness in Isogeometric Analysis
Montag, 18.1.21, 11:00-12:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
Splines are piecewise polynomial functions that are glued together with a given\nsmoothness. When using them in a numerical method, the availability of proper error\nestimates is of utmost importance. Classical error estimates for spline approximation\nare expressed in terms of\n\n(a) a certain power of the maximal grid spacing,\n\n(b) an appropriate derivative of the function to be approximated, and\n\n(c) a "constant" which is independent of the previous quantities but usually depends on the degree and smoothness of the spline.\n\nAn explicit expression of the constant in (c) is rarely available in the literature, because\nit is a minor issue in standard approximation analysis. There they are mainly\ninterested in the approximation power of spline spaces of a fixed degree. However, one\nof the most interesting features in the emerging field of Isogeometric Analysis is krefinement,\nwhich denotes degree elevation with increasing interelement smoothness.\nThe above mentioned error estimates are not sufficient to explain the benefits of\napproximation under k-refinement so long as it is not well understood how the\nconstant in (c) behaves.\nIn this talk we provide error estimates for k-refinement on arbitrary grids with an\nexplicit constant that is, in certain cases, sharp. These estimates are in fact good\nenough to cover convergence to eigenfunctions of classical differential operators. This\nforms a theoretical foundation for the outperformance of smooth spline discretizations\nof eigenvalue problems that has been numerically observed in the literature.\nSeite 2\nMoreover, we discuss how these error estimates can be used to mathematically justify the\nbenefits of spline approximation under k-refinement. Specifically, by comparing the constant\nfor spline approximation of maximal smoothness with a lower bound on the constant for\ncontinuous and discontinuous spline approximation, we show that k-refinement provides better\napproximation per degree of freedom in almost all cases of practical interest.\nThis talk is based on work performed in collaboration with Andrea Bressan and Carla Manni\nand Hendrik Speleers.
Simple Singularities and Their Symmetries
Montag, 18.1.21, 16:15-17:15, bbb Konferenzraum 1 (PW Konferenz3210)
We will study simple singularities from various points of view.\nIn the first part, I will give an introduction to the theory of unfoldings. We will see how to use unfoldings to analyse and resolve singularities. An important tool therein will be the Jacobian algebra. \nThen, we will review blowups which provide a different method to resolve singularities. Here, the type of a singularity is determined by the appearance of its exceptional divisor. \nIn both cases, the associated objects allow for actions of symmetry transformations. In the last part of the talk, we will study how to translate between the different perspectives.
Multispecies kinetic modelling and mathematical theory for physical applications
Dienstag, 19.1.21, 08:30-09:30, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
My goal is to model certain physical problems that can be described by kinetic partial\ndifferential equations. Our models allow for efficient numerical simulations. Examples\nof the physics models are inertial confinement fusion and the re-entry problem of a\nspace vehicle. Using mathematical theory we verify well posedness and essential\nphysical properties of the model. From the modeling point of view this requires\nextending existing models in the literature by extending them to gas mixtures or by\nincluding degrees of freedom in internal energy, chemical reactions and quantum\neffects. This leads to new difficulties in their theoretical study.\nIn this talk, I will present recent results on existence and large time behaviour of\nsolutions to kinetic equations for gas mixtures; entropy inequality and large-time\nbehaviour of kinetic equations with degrees of freedom in internal energy and chemical\nreactions; and results on entropy minimization problems leading to equations with\nconservation of mass, momentum and energy when we deal with velocity dependent\ncollision frequencies.
Approximation of the Willmore energy by a discrete geometry model
Dienstag, 19.1.21, 11:00-12:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
In this joint work with Heiner Olbermann (UCLouvain), we study the discrete bending\nenergy of Grinspun et al defined for triangular complexes, and show that varying over\nall complexes with the Delaunay property, the minimal bending energy converges, as\nthe size of triangles tends to zero, to a version of the Willmore energy, in the sense of\nGamma-convergence. We show also that the Delaunay property is essential to\nguarantee the lower energy bound. Our article combines results from finite difference\nmethods, discrete geometry, and geometric measure theory.
Compensated Compactness and L1-estimates
Dienstag, 19.1.21, 14:00-15:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
In the first part of the talk we will review some recent results on the Murat--Tartar\nframework of Compensated Compactness Theory, by which we mean weak (lower\nsemi-)continuity of nonlinear functionals interacting with weakly convergent\nsequences of PDE constrained vector fields. We present improvements of the original\nwork of Murat and Tartar, as well as more recent work of Fonseca--Müller. We also\npresent answers to questions of Coifman--Lions--Meyer--Semmes and De Philippis.\nThe second part of the talk will concern properties of solutions of linear systems \( L u\n= \bmu \), where \( \bmu \) is a Radon measure, a borderline case not covered by\nCalderón--Zygmund Theory. We build on the fundamental work of Bourgain--Brezis--\nMironescu and Van Schaftingen towards surprising strong interior Sobolev estimates\nfor solutions. We also discuss the start of a theory towards estimates up to the\nboundary. The final part of the talk will cover fine properties of solutions and possible\nintersections with Geometric Measure Theory.
Mathematikunterricht in einer durch Digitalisierung geprägten Welt
Dienstag, 19.1.21, 19:30-20:30, Hörsaal Rundbau, Albertstr. 21a
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.
Numerical methods for conservation laws with nonlocal and discontinuous fluxes
Freitag, 22.1.21, 08:30-09:30, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
The field of hyperbolic conservation laws is a cornerstone in PDE theory. In my talk I\nwill present recent results pertaining nonlocal conservation laws as well as\nconservation laws with discontinuous flux.\nIn the first part of the talk I will present new regularity results for a class of nonlo-cal\nconservation laws. These results motivate the design of higher-order numerical\nschemes which are asymptotically compatible with the underlying local conservation\nlaw. I will detail the construction of a second-order numerical scheme that generalizes\nthe class of second-order reconstruction-based schemes for local conservation laws. It\ncan be shown that the second-order scheme converges towards a weak solution, and—\nunder certain assumptions on the nonlocal interaction kernel—even towards the\nunique entropy solution of the nonlocal conservation law. Such a result is currently\nout of reach for local conservation laws.\nIn the second part of the talk I will focus on conservation laws with discontinuous flux\nwhich has been an active research area during the last several decades. Many selection\ncriteria to single out a unique weak solution have been proposed in this context and\nseveral numerical schemes have been designed and analyzed in the literature.\nSurprisingly, the preexisting literature on convergence rates for such schemes is\npractically nonexistent. In this talk, focusing on so-called adapted entropy solutions,\nI will present the first-ever convergence rate results for finite volume and front tracking\nmethods as well as a flux-stability result. As an application, these results can be used\nfor uncertainty quantification in two-phase reservoir simulations for reservoirs with\nvarying geological properties.
Irregular fibrations and derived categories
Freitag, 22.1.21, 10:30-11:30, SR 404
In this seminar I will show that an equivalence of derived categories of sheaves of smooth projective varieties preserves some specific classes of fibrations over varieties of maximal Albanese dimension. These types of fibrations, called chi-positive higher irrational pencils, can be thought as an extension to higher-dimension of the notion of a irrational pencil over a smooth curve of genus greater or equal to two. This is a joint work with F. Caucci and G. Pareschi.
Asymptotic Behavior of Gradient Flows and Nonlinear Spectral Theory
Freitag, 22.1.21, 11:00-12:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
In this talk I will discuss the gradient flow of absolutely p-homogeneous convex\nfunctionals on a Hilbert space and show that asymptotic profiles of the solution are\neigenfunctions of the subdifferential operator of the functional. This work applies, for\ninstance, to local and nonlocal versions of PDEs like p-Laplacian evolution equations,\nthe porous medium equation, and fast diffusion equations, herewith generalizing\nmany results from the literature to an abstract setting. Then I discuss the eigenvalue\nproblem associated to a infinity-Dirichlet energy in some more detail, show relations\nto distance functions, and speak about a discrete-to-continuum limit for this model\nusing Gamma-convergence. I conclude with perspectives and future work.
Space-time deep neural network approximations for high-dimensional PDEs
Freitag, 22.1.21, 14:00-15:00, Sonderkolloquium – Angewandte Mathematik virtueller Konferenzraum 3 (Konferenz3210)
It is one of the most challenging issues in applied mathematics to approximately solve\nhigh-dimensional partial differential equations (PDEs) and most of the numerical\napproximation methods for PDEs in the scientific literature suffer from the so-called\ncurse of dimensionality (CoD) in the sense that the number of computational\noperations employed in the corresponding approximation scheme to obtain\nan approximation precision \(\bvarepsilon >0\) grows exponentially in the PDE\ndimension and/or the reciprocal of \(\bvarepsilon\). Recently, certain deep learning\nbased approximation methods for PDEs have been proposed and various numerical\nsimulations for such methods suggest that deep neural network (DNN) approximations\nmight have the capacity to indeed overcome the CoD in the sense that the number of\nreal parameters used to describe the approximating DNNs grows at most polynomially\nin both the PDE dimension \(d \bin \bN\) and the reciprocal of the prescribed\napproximation accuracy \(\bvarepsilon >0\). There are now also a few rigorous\nmathematical results in the scientific literature which substantiate this conjecture by\nproving that DNNs overcome the CoD in approximating solutions of PDEs. Each of\nthese results establishes that DNNs overcome the CoD in approximating suitable PDE\nsolutions at a fixed time point \(T >0\) and on a compact cube \([a, b]^d\) but none of\nthese results provides an answer to the question whether the entire PDE solution on\n\([0, T] \btimes [a, b]^d\) can be approximated by DNNs without the CoD.\nIn this talk we show that for every \(a \bin \bR\), \( b \bin (a, \binfty)\) solutions\nof suitable Kolmogorov PDEs can be approximated by DNNs on the space-time region\n\([0, T] \btimes [a, b]^d\) without the CoD.
On certain lattice polarized K3 surfaces
Montag, 25.1.21, 16:15-17:15, vSR318 (Kasparov)
Let M be an even non-degenerate lattice of signature (1,t). A complex K3 surface X is M-polarized, if there exists a primitive lattice embedding of M into its Picard group Pic(X). \n\nVia such a polarization of the Picard group one is able to encode certain properties of the members of the family of M-polarized K3 surfaces. In this talk we will focus on Kummer surfaces which correspond to the product of two elliptic curves. We will discuss which kind of polarization, i.e. which lattice M, leads to those special Kummer surfaces. \n\nThe bigger picture that these polarized K3 surfaces fit into was described by Dolgachev’s influential paper „Mirror symmetry for lattice polarized K3 surfaces“. We will sketch some of Dolgachev’s insights and give an idea how they can be applied to the Kummer surfaces mentioned above.
Minimisation of the Willmore functional under isoperimetric constraint
Dienstag, 26.1.21, 10:00-11:00, virtueller Raum vWang
A Local Singularity Analysis for the Ricci flow
Dienstag, 26.1.21, 11:30-12:30, virtueller Raum vWang
In this talk, I will describe a refined local singularity analysis for the Ricci flow developed jointly with R. Buzano. The key idea is to investigate blow-up rates of the curvature tensor locally, near a singular point. Then I will show applications of this theory to Ricci flows with scalar curvature bounded up to the singular time.
Modelling mRNA counts of single-cells and small pools of cells
Dienstag, 26.1.21, 14:00-15:00, online: Zoom
Colour Image Denoising: Numerical Approximation of the Constrained Total Variation Flow
Dienstag, 26.1.21, 14:15-15:15, Hörsaal II (virtuell:Lasker)
The Rudin-Osher-Fatemi model is a well-known denoising model for greyscale images. One approach to adapt this model to colour images involves computing the total variation flow constrained to a manifold. In a recent work by Giga et al., an algorithm to approximate the constrained total variation flow is proposed and a discretisation using piecewise constant functions is presented. As piecewise constant functions are generally not suited to approximate functions of bounded variation, we present a similar algorithm using piecewise affine finite elements, in particular Crouzeix-Raviart elements. We show stability of the scheme, experimental convergence rates and some improved denoising results. This work is the result of my master thesis.
Modellbegleiter und die Definable Multiplicity Property (DMP)
Dienstag, 26.1.21, 14:30-15:30, virtueller SR 125 Anderssen
Prescribed curvature measure problem in hyperbolic space
Dienstag, 26.1.21, 15:00-16:00, virtueller Raum vWang
The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this talk, we are going to talk about our recent result about prescribed curvature measure problem in hyperbolic space.We obtained the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures (k<n) by establishing crucial C^2 regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.
Mobius Invariant Equations in Dimension Two
Dienstag, 26.1.21, 16:30-17:30, virtueller Raum vWang
Conformally invariant equations in \(n\bgeq3\) have played an important role in the study of \(\bsigma_k\)-Yamabe problem in geometric analysis. \nIn this talk, we will discuss a class of Mobius invariant equations in dimension two. We will then present related properties for such equations, including Liouville type theorems, Bocher type theorems and existence of solutions. This is a joint work with Yanyan Li and Siyuan Lu.\n
Cone structures and parabolic geometries
Freitag, 29.1.21, 10:30-11:30, SR 404
Deep Learning for Brain Signals
Freitag, 29.1.21, 15:00-16:00, online: Zoom