Archiv des Frankfurter Seminars

Veranstaltungen im WiSe 2019/ 2020

 

 

 

 

 

 

 

30. Oktober 2019

Sam Payne (UT Austin)

Tropical curves, graph homology, and top weight cohomology of M_g

I will discuss the topology of a space of stable tropical curves of genus g with volume 1. The reduced rational homology of this space is canonically identified with the top weight cohomology of M_g and also with the homology of Kontsevich's graph complex. As one application, we show that H^{4g-6}(M_g) is nonzero for infinitely many g. This disproves a recent conjecture of Church, Farb, and Putman as well as an older, more general conjecture of Kontsevich. We also give an independent proof of a recent theorem of Willwacher, that homology of the graph complex vanishes in negative degrees, using the identifications above and known vanishing results for M_g. And we prove a formula for the S_n-equivariant Euler characteristic of M_{g,n}, which was conjectured by Zagier.

 

  
                           

 

 

 

 

 

 

 

27. November 2019

Mohab Safey El Din (Sorbonne Université)

On solving polynomial systems over the reals and applications in robotics

Polynomial systems arise in many areas of engineering and computer sciences such as signal theory, cryptography, biology and robotics. In this talk, we will focus on the analysis of kinematic singularities in robotics, which is a fundamental problem in mechanism design. This analysis boils down to a core algorithmic issue in effective real algebraic geometry: given a polynomial system with real coefficients, how to count the number of connected components of their real solution set (or answer connectivity queries on this set)? We will review recent (mathematical and algorithmic) results which yield practically efficient algorithms to solve these problems using computer algebra methods and report on their implementations which can already solve concrete problems in robotics.

  
                           

 

 

 

 

 

 

 

15. Januar 2020

Hansjörg Geiges (Universität zu Köln)

The topology of global surfaces of section

Global surfaces of section were introduced by Poincaré in the context of celestial mechanics, allowing him to reduce the search for periodic orbits in the 3-body problem to finding periodic points of the return map on the surface of section. They continue to play an important role in modern symplectic dynamics. In this talk I shall concentrate on some simple topological aspects of global surfaces of section. In particular, I plan to discuss the topology of surfaces of section for the Hopf flow on the 3-sphere. This leads to an intriguing proof of the degree-genus formula for complex projective curves, using ideas motivated by symplectic dynamics.

This talk is based on joint work with Peter Albers and Kai Zehmisch.

    
                           

 

 

 

 

 

 

 

29. Januar 2020

Felix Otto (MPI Leipzig)

Effective behavior of random media

In engineering applications, heterogeneous media are often described in statistical terms. This partial knowledge is sufficient to determine the effective, i.e. large-scale behavior. This effective behavior may be inferred from the Representative Volume Element (RVE) method. I report on last years' progress on the quantitative understanding of what is called stochastic homogenization of elliptic partial differential equations: optimal error estimates of the RVE method and the homogenization error, and the leading-order characterization of fluctuations. Methods connect to elliptic regularity theory and to concentration of measure arguments.

 

  
                           

Veranstaltungen im SoSe 2019

 

 

 

 

 

 

 

24. April 2019

Karl-Theodor Sturm (Universität Bonn)

Optimal transport, heat flow, and Ricci curvature on metric measure spaces

We present a brief survey on the theory of metric measure spaces with synthetic lower Ricci bounds, initiated by the author and by Lott/Villani, and developed further by Ambrosio/Gigli/Savare and by many others. Particular emphasis will be given to recent breakthroughs concerning the local structure of RCD-spaces by Mondino/Naber and by Brue/Semola and to rigidity results. For instance, given an arbitrary RCD(N-1,N)-space (X,d,m), then

∫  ∫ cos d(x,y) dm(x) dm(y) ≤  0
 
if and only if N is an integer and (X,d,m) is isomorphic to the N-dimensional round sphere. Moreover, we study the heat equation on time-dependent metric measure spaces and its dual as gradient flows for the energy and for the Boltzmann entropy, resp. Monotonicity estimates for transportation distances and for squared gradients will be shown to be equivalent to the so-called dynamical convexity of the Boltzmann entropy on the Wasserstein space which is the defining property of super-Ricci flows. Moreover, we show the equivalence with the monotone coupling property for pairs of backward Brownian motions as well as with log Sobolev, local Poincare and dimension free Harnack inequalities.

  
                           

 

 

 

 

 

 

 

15. Mai 2019

Andrea Colesanti (Universität Florenz)

Valuations on spaces of functions

A valuation is, roughly speaking, a real-valued mapping defined on a family of sets, which is finitely additive. A class of sets for which valuations have been intensively studied is that of convex bodies (compact convex subsets of the n-dimensional Euclidean space). The beauty of the theory of valuations on convex bodies resides, in my opinion, in the elegance and the profundity of some of the results that have been proved within this area. For instance, Hadwiger’s theorem states that intrinsic volumes form a basis of rigid motion and continuous valuations; McMullen homogeneous decomposition theorem and Alesker’s density theorem are of similar fundamental importance.

Recently, the notion of valuation found a natural adaptation in the context of spaces of functions: one just needs to replace, in the finite additivity condition, union and intersection by pointwise maximum and minimum. How much of the theory of valuations on convex bodies can be extended to the functional setting? And how? These questions have not been answered yet, and are the main motivation for my research in this area.

  
                           

 

 

 

 

 

 

 

05. Juni 2019

Michel Brion (Universität Grenoble)

Automorphism groups of compact complex manifolds

The automorphism group of a compact complex manifold is known to be a complex Lie group, and hence an extension of a discrete group (the group of components) by a complex connected Lie group. The talk will discuss the group of components, which is quite mysterious. It is easy to see that this group is generally infinite. Recently, Lesieutre, Dinh and Oguiso constructed examples for which the group of components is not finitely generated, thereby answering a long-standing question by the negative. On the other hand, this group turns out to be finitely presented for some interesting classes of manifolds, starting with complex tori.

    
                           

 

 

 

 

 

 

 

10. Juli 2019

Mihyun Kang (TU Graz)

Topological aspects of random graphs

In this talk we shall discuss various topological aspects of random graphs. How quickly does the genus of a random graph grow as the number of edges increases?  How do topological constraints (such as imposing an upper bound on the genus) influence the structure of a random graph?

  
                           

Veranstaltungen im WiSe 2018/ 2019

 

 

 

 

 

 

 

24. Oktober 2018

Valentin Blomer (Universität Göttingen)

Eigenfunctions on arithmetic manifolds

It is a classical problem in analysis to study Laplace eigenfunctions on Riemannian manifolds. Number theory enters the scene if the manifold has additional arithmetic structure, such as a family of Hecke operators. In this talk, I will present methods from number theory, Lie groups and automorphic forms to obtain information on the mass distribution of joint eigenfunctions on various arithmetic spaces, and discuss some applications.		  
                           

 

 

 

 

 

 

 

12. Dezember 2018

Sandra Di Rocco (KTH Stockholm)

Discriminants: an interplay between algebra, combinatorics and geometry

The term “discriminant” is well known in relation with low degree univariate equations or ordinary differential equations and it has a well defined geometrical meaning naturally connected to the way our vision grasps geometrical shapes. The discriminant of polynomials with specified shape (when it exists) is an (irreducible) polynomial, whose variables are in coefficients of the assigned shape, vanishing when the corresponding polynomial has multiple roots. Finding the discriminant or at least estimating its degree is a classical, well explored problem in mathematics which still faces many challenges.

Besides its importance in computational mathematics, the discriminant locus carries deep and interesting geometrical and combinatorial insights and is one example of fruitful interplay between classical projective algebraic geometry and convex real geometry.

This interplay will be the main theme of the talk. Besides the classical theory of discriminants, natural generalisations associated to systems of polynomial equations will be presented.
  
                           

 

 

 

 

 

 

 

19. Dezember 2018

Jon Keating (Universität Bristol)

Characteristic Polynomials of Random Unitary Matrices, Partition Sums, and Painlevé V

The moments of characteristic polynomials play a central role in Random Matrix Theory. They appear in many applications, ranging from quantum mechanics to number theory. The mixed moments of the characteristic polynomials of random unitary matrices, i.e. the joint moments of the polynomials and their derivatives, can be expressed recursively in terms of combinatorial sums involving partitions. However, these combinatorial sums are not easy to compute, and so this does not give an effective method for calculating the mixed moments in general. I shall describe an alternative evaluation of the mixed moments, in terms of solutions of the Painlevé V differential equation, that facilitates their computation and asymptotic analysis.

    
                           

 

 

 

 

 

 

 

30. Januar 2019

Silvia Sabatini (Universität Köln)

12, 24 and beyond: a bridge from reflexive polytopes to symplectic geometry

Mathematics finds itself divided and subdivided into hyper-specialized areas of study, each of them with its own internal beauty.  However, what I find most fascinating is when one can build a bridge between two of these seemingly isolated theories. For instance, (symplectic) geometry and combinatorics have a very strong connection, due to the existence of some special manifolds admitting a torus symmetry. The latter is encoded in a map, called moment map, which "transforms" the manifold into a convex polytope. Hence many combinatorial properties of (some special types of) polytopes can be studied using symplectic techniques.
In this talk I will focus on the class of reflexive polytopes, which was introduced by Batyrev in the context of mirror symmetry, and explain how the "12" and "24" phenomenon for reflexive polytopes of dimensions 2 and 3 can be generalized to higher dimensions using symplectic geometry.

  
                           

Veranstaltungen im SoSe 2018

 

 

 

 

 

 

 

11. April 2018

Bernd Sturmfels (MPI-MIS Leipzig)

Learning Algebraic Varieties from Samples

This lecture discusses the role of algebraic geometry in data science. We report on recent work with Paul Breiding, Sara Kalisnik and Madeline Weinstein. We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods arestudied and new methods are developed. Our focus lies on topological and algebraic features, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.

  
                           

 

 

 

 

 

 

 

02. Mai 2018

Frank den Hollander (Universität Leiden)

Exploration on dynamic networks

Search algorithms on networks are important tools for the organisation of large data sets. A key example is Google PageRank. Real-world networks are modelled as random graphs and search algorithms as random walks. The mixing time of the algorithm is the time it takes the random walk to approach its equilibrium distribution.

Many real-world networks are dynamic in nature. In this talk we focus on a random graph with prescribed degrees and investigate what happens to the mixing time of the random walk when at each unit of time a certain fraction of the edges is randomly rewired. We identify three regimes in the limit as the graph becomes large, and show that these exhibit surprising behaviour.

  
                           

 

 

 

 

 

 

 

09. Mai 2018

Pierre-Emmanuel Caprace (UC Louvrain)

A taste of simple groups

The goal of this talk is to have a tour in the fascinating and eclectic universe of simple groups. A special emphasis will be put on simple locally compact groups, and their use in the study of discrete groups.

    
                           

 

 

 

 

 

 

 

04. Juli 2018

Nigel Hitchin (University of Oxford)

The geometry of surfaces — a new look

The talk will discuss compact surfaces of negative curvature from the point of view of the space of geodesics on the universal covering. The motivation comes from an attempt to describe an infinite dimensional version of Teichmüller space.

 
                           

Veranstaltungen WiSe 2017/ 2018

 

 

 

 

 

 

 

15. November 2017

Hendrik Lenstra (Universität Leiden)

Solving equations in orders

An order is a commutative ring of which the additive group is, for some non-negative integer n, the group of vectors with n integral coordinates. The lecture is devoted to the algorithmic problem of solving polynomial equations in one variable in orders.

  
                           

 

 

 

 

 

 

 

13. Dezember 2017

Camillo de Lellis (Universität Zürich)

The Onsager's Theorem and beyond

In 1949 the famous physicist Lars Onsager made the striking statement that there are continuous solutions of the incompressible Euler equations that do not preserve the kinetic energy. Such statement has been rigorously proved only very recently and in this lecture we will explore some of the related ideas.		  
                           

 

 

 

 

 

 

 

24. Januar 2018

Barbara Wohlmuth (Technische Universität München)

A priori and a posteriori modifications in finite element flow simulations

We discuss two types of local and computationally inexpensive modifications resulting in finite element approximations of higher accuracy. In case one, a local recovery of the discrete flux is performed such that mass conservation on a dual mesh can be guaranteed. The corrected flux enters then into the advective part of the energy equation. In case two, the flowability in the discrete scheme is locally enhanced such that energy can be preserved. A variational crime analysis then guarantees improved convergence rates.All theoretical results are illustrated by a series of simulations including large scale runs where a all-in-once multigrid solver is used for the flow part.

    
                           

 

 

 

 

 

 

 

31. Januar 2018

Günter Ziegler (Freie Universität Berlin)

Semi-algebraic sets of integer points

We look at sets of integer points in the plane, and discuss possible definitions of when such a set is “complicated” — this might be the case if it is not the set of integer solutions to some system of polynomial equations and inequalities. Let ’s together work out lots of examples, and on the way let’s try to develop criteria and proof techniques …
The examples that motivated our study come from polytope theory: Many question of the type “What is the possible pairs of (number of vertices, number of facets) for 4-dimensional polytopes? ” have been asked, many of them with simple and complete answers, but in other cases the answer looks complicated. Our main result says: In some cases it IS complicated! (Joint work with Hannah Sjöberg.)