School of Mathematics/Computer Science

Institute for Mathematics


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WS 2018/19

24.10.2018 um 17:15 Uhr in 69/125:

Dr. Matthias Wendt  (Universität Osnabrück)

Oriented Schubert Calculus: Using Quadratic Forms to Count Subspaces

Classical Schubert calculus is a set of tools for computing degrees of intersections of Schubert varieties in Grassmannians. It can be used to answer enumerative geometry questions concerning intersections of subspaces of a vector space, at least over algebraically closed fields; the most classical such question is "How many lines in projective 3-space meet four given lines in general position?" Over general fields, like the real numbers, the answer is more complicated because some of the solution vector spaces may not be defined over the given field. To solve this issue, real enumerative geometry counts solutions with signs depending on additional orientations of the vector spaces to obtain refined information, such as lower bounds for the numbers of solutions. In the talk I will explain a refinement of the classical Schubert calculus which allows to compute the Chow-Witt rings of Grassmannians. In this refinement, the "number" of solution subspaces is a quadratic form which encodes both the classical answer (the number of solutions over the algebraic closure) as well as the signed counts from real enumerative geometry.

07.11.2018 um 17:15 Uhr in 69/125:

Prof. Dr. Esther Brunner (Pädagogische Hochschule Thurgau)

"Wie es mir gefällt oder wie es für die Klasse passt?“ Beweistypen in der Sekundarstufe I

Beweisen und argumentieren gewinnt im Zusammenhang mit Bildungsstandards (Leiss & Blum, 2006; Wittmann & Müller, 1988) eine neue Bedeutung, nachdem insbesondere die Kritik am formalen Beweis und seiner Strenge für den Unterricht in der Volksschule entschärft und durch weitere Konzepte wie beispielsweise präformales (oder operatives) Beweisen ergänzt werden konnte. Dadurch stehen den Lehrpersonen unterschiedliche Möglichkeiten zur Verfügung, mit ihren Schülerinnen und Schülern zu beweisen bzw. mathematisch zu argumentieren. Dass dies gelingen kann, zeigen verschiedene aktuelle Studien für unterschiedliche Schulstufen auf. Im Vortrag werden zunächst kurze Einblicke in verschiedene aktuelle Projekte der Referentin gegeben, die sich mit der Erforschung von mathematischem Argumentieren und Beweisen für unterschiedliche Alters- und Schulstufen (Vorschule, Primarschule, Sekundarstufe I) befassen und sowohl die Perspektive der Schülerinnen und Schüler als auch diejenige der Lehrpersonen in den Blick nehmen.

Vertiefend wird im zweiten Teil anhand einer Analyse der Daten aus der Pythagoras-Studie (Klieme, Pauli, & Reusser, 2009) der Frage nachgegangen, inwiefern bestimmte Personenmerkmale der Lehrpersonen wie Alter, Berufserfahrung, Überzeugungen usw. eine Rolle spielen für die Unterrichtsgestaltung beim Beweisen und ob die durchgeführte Beweisstrategie mit den mittleren Voraussetzungen der Klasse zusammenhängen. Analysiert wurde in 32 Klassen des 8./9. Schuljahrs, welche Beweistypen bei der Bearbeitung derselben innermathematischen Aufgabenstellung aus der elementaren Zahlentheorie realisiert werden. Es interessierte ferner, inwiefern sich Zusammenhänge zwischen dem in den Klassen bearbeiteten Beweistyp einerseits und Merkmalen der Lehrpersonen sowie Lernvoraussetzungen der Klasse andererseits beschreiben lassen. Dazu wurde die videografierte Bearbeitung der Beweisaufgabe mit Personenvariablen der Lehrpersonen und mit den Leistungsdaten der Schülerinnen und Schüler in Beziehung gebracht. Die Ergebnisse zeigen, dass der beobachtete Beweistyp zum Teil mit den Eingangsvoraussetzungen der Klassen zusammenhängt, dass sich aber die Lehrpersonen, die formal-deduktiv arbeiten, auch bezüglich einzelner Personenmerkmale unterscheiden und eine besondere Gruppe darstellen. Die Ergebnisse aus den Daten der Sekundarstufe I werden in der Diskussion genutzt, um weiterführende offene Fragen – auch für andere Schulstufen – zu formulieren.

14.11.2018 um 17:15 Uhr in 69/125:

Prof. Dr. Franz-Viktor Kuhlmann (Universität Szczecin, Polen)

Resolution of Singularities and the Defect

In 1964 Hironaka proved resolution of singularities for algebraic varieties of arbitrary dimension over fields of characteristic 0. For this result, which has applications in many areas of pure and applied mathematics, he received the Fields Medal. In positive characteristic, resolution has only been proven for dimensions up to 3 by Abhyankar and recently by Cossart and Piltant. The general case has remained open although several working groups of algebraic geometers have attacked it. Following ideas of Zariski, one can also consider a local form of resolution, called local uniformization. Already in 1940 he proved it to hold for algebraic varieties of all dimensions in characteristic 0. But again, the case of positive characteristic has remained open. By its definition, local uniformization is a problem of valuation theoretical nature. In my talk, I will give a quick introduction to valuations and sketch the main idea of local uniformization. In positive characteristic, finite extensions of valued fields can show a nasty phenomenon, the defect. It has been identified as one of the main obstacles to local uniformization. I will present examples of defects and some results, as well as some main open problems. 

21.11.2018 um 17:15 Uhr in 69/125:

Prof. Dr. Rob Eggermont (University of Technology Eindhoven)

Finitely Generated Spaces in High-Dimensional Settings with Symmetry

In a high-dimensional space with the Zariski topology, it is generally difficult to find equations generating a
given subspace. If the space has additional symmetries, the problem becomes easier, and it is sometimes
possible to use equations describing a smaller space in a lower-dimensional setting. As an example, to describe
matrices of rank at most 1 it suffices to know that any 2 by 2 determinant vanishes. This does not depend on
the size of the matrix. In this talk, we give some examples of high-dimensional settings with symmetry, and
describe more formally what we mean by finitely generated spaces in these settings. We will also talk about
some recent results.

28.11.2018 um 17:15 Uhr in 69/125:

Prof. Dr. Frank den Hollander (Universität Leiden)

Exploration on Dynamic Networks

Search algorithms on networks are important tools for the organisationof large data sets. A key example is Google PageRank, which assigns a numerical weight to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of measuring its relative importance within the set. The weighting is achieved by exploration. 
The mixing time of a random walk on a random graph is the time it needs to approach its stationary distribution (also called equilibrium distribution). The characterisation of the mixing time has been the subject of intensive study. Many real-world networks are dynamic in nature. It is therefore natural to study random walks on dynamic random graphs. 
In this talk we focus on a random graph with prescribed degrees. We 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: fast, moderate, slow dynamics. These regimes exhibit surprising behaviour.
The talk is aimed at a general mathematics audience. No prior knowledge of probability theory or graph theory is required.

05.12.2018 um 17:15 Uhr in 69/125:

Prof. Dr. Nina Gantert (Techn. Universität München)

Decisiveness Versus Flexibility: Who Wins?

We discuss this question on the example of biased random walks in random environments. In particular, we consider biased random walk on percolation clusters or among random conductances. They serve as a model for transport in a inhomogeneous medium. We investigate the speed of the walk as a function of the bias, and discuss its monotonicity. We state the Einstein relation which is proved in the random conductance case. As time permits, we also mention recent results about Mott walks. No prior knowledge is assumed. The results we present come from joint work(s) with Noam Berger, Alessandra Faggionato, Xiaoqin Guo, Matthias Meiners, Sebastian Müller, Jan Nagel and Michele Salvi

12.12.2018 

interne Weihnachtsfeier Fachbereich Mathematik/Informatik 

19.12.2018 um 17:15 Uhr in 69/125:

Dr. Martin Strobel (National University of Singapore)

Exploring Voting Paradoxes via Ehrhart Theory and Computer Simulations

09.01.2019 um 17:15 Uhr in 69/125:

Dr. Felix Voigtländer (Katholische Universität Eichstätt-Ingolstadt)

Approximation Theoretic Properties of Deep ReLU Neural Networks

Studying the approximation theoretic properties of neural networks with smooth activation function is a classical topic.The networks that are used in practice, however, most often use the non-smooth ReLU activation function. Despite the recent incredible performance of such networks in many classification tasks, a solid theoretical explanation of this success story is still missing.

In this talk, we will present recent results concerning the approximation theoretic properties of deep ReLU neural networks which help to explain some of the characteristics of such networks; in particular we will see that deeper networks can approximate certain classification functions much more efficiently than shallow networks, which is not the case for most smooth activation functions.
We emphasize though that these approximation theoretic properties do not explain why simple algorithms like stochastic gradient descent work so well in practice, or why deep neural networks tend to generalize so well; we purely focus on the expressive power of such networks.

As a model class for classifier functions we consider the class of (possibly discontinuous) piecewise smooth functions for which the different "smooth regions" are separated by smooth hypersurfaces.
Given such a function, and a desired approximation accuracy, we construct a neural network which achieves the desired approximation accuracy, where the error is measured in L2. We give precise bounds on the required size (in terms of the number of weights) and depth of the network, depending on the approximation accuracy, on the smoothness parameters of the given function, and on the dimension of its domain of definition. Finally, we show that this size of the networks is optimal, and that networks of smaller depth would need significantly more weights than the deep networks that we construct, in order to achieve the desired approximation accuracy.

23.01.2019 um 17:15 Uhr in 69/125:

Prof. Dr. Ulrich Bauer (Technische Universität München)

The Morse Theory of Čech and Delaunay Complexes

Given a finite set of points in ℝⁿ and a radius parameter, we consider the Čech, Delaunay–Čech, Delaunay
(alpha shape), and wrap complexes in the light of generalized discrete Morse theory. We prove that the four
complexes are simple-homotopy equivalent by a sequence of simplicial collapses, and the same is true
for their weighted versions. Our results have applications in topological data analysis and in the reconstruction
of shapes from sampled data.

30.01.2019 um 17:15 Uhr in 69/125:

Dr. Nick Vannieuwenhoven (Katholieke Universiteit Leuven)

Tensor Decompositions and their Sensitivity

The tensor rank decomposition or CPD expresses a tensor as a minimum-length linear combination of elementary rank-1 tensors. It has found application in fields as diverse as psychometrics, chemometrics, signal processing and machine learning, mainly for data analysis purposes. In these applications, the theoretical model is oftentimes a low-rank CPD and the elementary rank-1 tensors are usually the quantity of interest. However, in practice, this mathematical model is always corrupted by measurement errors. In this talk, we will investigate the numerical sensitivity of the CPD using techniques from algebraic and differential geometry.

06.02.2019 um 16:15 Uhr in 69/125:

Prof. Dr. Hedwig Gasteiger und Prof. Dr. Martina Juhnke-Kubitzke

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