# Mathematisches Kolloquium

geplante Vorträge für das SS 2017: 27.4.2017 / 18.5.2017 / 8.6.2017 / 29.6.2017

Die Vorträge der vergangenen Semester finden Sie in den Archiven unter Veranstaltungen-Mathe.

Donnerstag, 27. April 2017, 16:30 Uhr, Carl-Zeiß-Straße 3, SR 308 - FÄLLT wegen Krankheit AUS!!

### Thema: An abstract convexity approach to set relations and set optimization

Abstract: Many order relations are defined or can be expressed via families of scalar functions, for example stochastic dominance orders, non-total preferences in economics and vector orders generated by closed convex cones by means of the bipolar theorem. It is shown that such an abstract convexity approach to order relations

(a) covers a vast variety of known and produces new examples,

(b) very naturally leads to new set relations as well as to

(c) intrinsic hull operators and complete lattices of sets,

(d) can be used to transform a (vector-type) minimization problem with respect to a non-lattice preorder into a set optimization problem with a complete lattice as image space.

Concepts for exact and approximate solutions of the resulting set optimization problems are introduced, and Weierstrass-type theorems are given. Several examples from economics, finance, statistics and multi-criteria decision making are discussed. The approach leads to new answers of questions which are subject of ongoing discussions in the respective communities such as the following. What is the (lower or upper) quantile of a multivariate random variable? How should utility maximization for incomplete preferences be done? What is the Legendre-Fenchel conjugate of a vector- or even set-valued convex function?

The talk starts with a brief introduction of set relations and set optimization with many examples. The main part is devoted to the abstract convexity approach to set optimization and includes the main results using tools from lattice theory and nonlinear functional analysis. The third part deals with a few examples in more detail: duality for vector/set optimization, multivariate statistics and risk measures for markets with transaction costs.

Joint work with Giovanni Crespi, Matteo Rocca, Carola Schrage

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Donnerstag, 18. Mai 2017, 16:30 Uhr, Carl-Zeiß-Straße 3, SR 317

### Thema: Ein übersehenes Problem: Bettizahlen geschlossener Mannigfaltigkeiten

Abstract: t.b.a.

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Donnerstag, 8. Juni 2017, 16:30 Uhr, Carl-Zeiß-Straße 3, SR 308

### Thema: Was läuft im Mathematikunterricht und in der Lehrerbildung falsch? Wie könnte umgesteuert werden?

Abstrakt: Den ersten Teil des Vortrags bilden operative Beweise mit Plättchen, die vom Kindergarten bis zur Universität reichen. Diese Beweise verkörpern das, was dem heutigen Mathematikunterricht im Kern fehlt: aufbauendes, fachliches Lernen von Grund auf.

Im zweiten Teil des Vortrags werden Vorschläge zu einer Umsteuerung in den Lehrplänen und in der Lehrerbildung vorgestellt, die sich um die Wiedereinsetzung der Elementarmathematik als Basis des Unterrichts, der Mathematikdidaktik und der Mathematik ranken. Diese Vorschläge erfordern die aktive Mitarbeit der mathematischen Community, die dazu aber von ihrem hohen Ross heruntersteigen bzw. aus ihrer Klausur in den Spezialgebieten heraustreten müsste.

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Donnerstag, 29. Juni 2017, 16:30 Uhr, Carl-Zeiß-Straße 3, SR 309

### Thema: Digits and dynamics -A tour of Benford's Law

Abstract: Benford's Law (BL), a notorious gem of mathematics folklore, asserts that leading digits of numerical data are usually not equidistributed, as might be expected, but rather follow one particular logarithmic distribution. Since first recorded by Newcomb in 1881, this apparently counter-intuitive phenomenon has attracted much interest from scientists and mathematicians. This talk will introduce and discuss some of the intriguing aspects of BL, and relate them to problems in probability and number theory and, above all, dynamics.
In view of their pivotal role as models of many real-world processes, it is natural to ask whether dynamical systems can actually comply with BL in some sense or other and, if so, whether in turn something about dynamics
can be learned from this. The talk will answer both questions in the affirmative. Moreover, all real data sets, such as e.g. data recorded from a dynamical system, necessarily are finite, and determining exactly what (and what not) BL means for such data will emerge as a formidable challenge in itself.