## Mathematisches Kolloquium

### Mathematisches Kolloquium 2017

• Dr. Christos Pelekis (The Czech Academy of Sciences, Institute of Computer Science)
"A generalised isodiametric problem"
Abstract: According to Bieberbach's inequality the area of a planar set whose diameter is 2 cannot be larger than \pi. What can we say about the maximum area of a planar set A, having the property that among any three points in A at least two are at distance less than or equal to 2? In the first part of the talk I will discuss the, devious, motivation behind the formulation of this question and I will sketch proofs of certain results and bounds on the maximum area of A. In the second part I will describe how the question gives rise to a "geometric analogue" of Turán's graph theorem and I will present some recent results on the corresponding extremal problem.
08.11.2017, 15:00 Uhr, HS 228 (Ulmenstr. 69, Haus 3)
• Dr. Thomas Kalinowski (The University of Newcastle)
"Extended formulations for convex hulls of graphs of bilinear functions "
Abstract: Many methods in global optimization require the approximation of convex and concave envelopes of functions. For bilinear functions, a classic approach is the McCormick relaxation: introduce additional variables representing products of pairs of original variables and write down linear constraints approximating the bilinear terms. The McCormick relaxation can be strengthened by adding more inequalities. In this direction, the Boolean Quadric Polytope (BQP) is best possible in the sense that its projection is the convex hull of the graph of the function, in other words, it provides an extended formulation for this graph. Unfortunately, in general the BQP has exponentially many facets and no complete list is known. In the talk I will discuss a method which in certain cases allows the identification of a small subset of facets of the BQP describing an extended formulation for the graph of the function.
(joint work with Natashia Boland, Akshay Gupte, Fabian Rigterink and Hamish Waterer)
19.07.2017, 15:45 Uhr, HS 228 (Ulmenstr. 69, Haus 3)
• Prof. Dr. Uwe Leck (Europa-Universität Flensburg)
"Problems and conjectures related to possible sizes of maximal antichains"
Abstract: We will address the following two problems:
1. For given n, which cardinalities are attainable by maximal antichains in the Boolean lattice B_n?
2. For given m and k, which cardinalities can the shadow of a k-uniform family of m sets in B_n have?
Some general conjectures will be stated and motivated. Strategies and partial results will be presented.
19.07.2017, 15:15 Uhr, HS 228 (Ulmenstr. 69, Haus 3)
• Prof. Dr. Jerry Griggs (University of South Carolina)
"Poset-free Families of Subsets"
Abstract: Given a finite poset P, we consider the largest size La(n,P) of a family of subsets of [n]:={1,...,n} that contains no (weak) subposet P. Early theorems of Sperner and Katona solve this problem when P is the k-element chain (path poset) P_k, where it is the sum of the middle k-1 binomial coefficients in n. Katona and his collaborators investigated La(n,P) for other posets P. It can be very challenging, even for small posets.
Based on earlier results we conjectured with Lu (2008) that pi(P), which is the limit as n goes to infinity, of La(n,P)/{n\choose{n/2}}, exists for general posets P. Moreover, it is an integer obtained in a specific way. The conjecture has been verified for various families of posets.
For most k at least 2, our work with Li verifies the conjecture for D_k, which is the k-diamond poset {A< B_1,...,B_k < C}. Yet, the case k=2 remains open, after considerable effort by researchers. We expect pi(D_2)=2, the easy lower bound. Recently, Grosz, Methuku, and Tompkins brought the upper bound down below 2.21. Tools used on these problems include probabilistic reasoning, such as bounding the average number of times a random full chain meets a P-free family F, called the Lubell function of F.
19.07.2017, 14:00 Uhr, HS 228 (Ulmenstr. 69, Haus 3)
• Prof. Dr. Alois Kneip (Universität Bonn)
"On the Optimal Reconstruction of Partially Observed Functional Data"
Abstract: We propose a new linear prediction operator that aims to recover the missing parts of a function given the observed parts. The structure of an optimal linear predictor is analyzed theoretically. Our estimation theory allows for autocorrelated functional data and considers the practically relevant situation where each function (in total n many) is observed at m discretization points. We derive uniform rates of consistency for our nonparametric estimation procedures using a double asymptotic that allows investigate all data scenarios from almost sparse to dense functional data. The finite sample properties are investigated through simulations and a real data application.
12.07.2017, 15:00 Uhr, HS 326/327 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. A. Meister
• Dr. rer. nat. habil. Michael Maiwald (BAM)
"Low field NMR spectroscopy for process control - robust automated data preparation and analysis as prerequisites"
12.07.2017, 10:15 Uhr, Raum 427 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. Klaus Neymeyr
• Dr. Romanos Malikiosis (TU Berlin)
"Formal Duality in Finite Cyclic Groups"
05.07.2017, 15:00 Uhr, HS 228 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. A. Schürmann
• apl. Prof. Heidemarie Bräsel (im Ruhestand)
"Faszination Mathematik"
Abstract: „Faszination Mathematik – Malerei und mehr“, so hieß die Ausstellung, die ich anlässlich des 50. Jahrestages der Gründung der Spezialklasse für Mathematik und Naturwissenschaften an der damaligen Technischen Hochschule Magdeburg im Herbst 2014 gestaltet habe. Die Ausstellung war so erfolgreich, dass mein Mann und ich beschlossen haben, mit ihr auf Wanderschaft zu gehen.
In meinem Vortrag möchte ich Ihnen einen Überblick über die Inhalte der Bilder und Computergrafiken, über die mathematischen Experimente und über das Zahlensammelsurium geben. Dabei spannt sich der Bogen vom Satz des Pythagoras, dem Goldenen Schnitt und den Fibonaccizahlen und Fibonaccispiralen hin zu den Fraktalen, der Geometrie der Natur.
Bei den Experimenten zu magischen und lateinischen Quadraten sollen neue Erfahrungen zur Konstruktion und Auffüllbarkeit solcher Quadrate und auch über ihren Zusammenhang gewonnen werden. Ein Modell aus 49 Würfeln, die 6 paarweise orthogonale lateinische Quadrate der Ordnung 7 enthalten, lädt zum Staunen ein. Natürlich darf auch mathematisch gepuzzelt werden.
Schließlich wird noch auf die Plakatserie des Sammelsurium eingegangen, das Kurioses, Wissenswertes und Symbolisches über die Zahlen von 0 bis 12 enthält.
Ich bin sicher, dass ich Ihnen nicht viel Neues aus der Mathematik erzählen werde, schließlich will ich vor einem Fachpublikum vortragen, aber ich erzähle es Ihnen anders: Mathematik zum Anschauen, Staunen, neugierig und aktiv werden, Begreifen und Lernen. Seien Sie neugierig!
17.05.2017, 15:00 Uhr, HS 228 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. G. Kyureghyan
• Prof. Dr. Benjamin Klopsch (Universität Düsseldorf)
"Zetafunktionen von zulässigen Darstellungen kompakter p-adischer Liegruppen"
Abstract: Die gewöhnliche’ Darstellungszetafunktion einer kompakten p-adischen Liegruppe G ist eine Dirichlet-Erzeugendenfunktion, mittels derer (endlich dimensionale) irreduzible komplexe Darstellungen von G abgezählt werden. Zunächst werde ich diese Art von Zetafunktionen motivieren und einige zugehörige Resultate skizzieren.  Allgemeiner läßt sich jeder geeigneten’ unendlich dimensionalen Darstellung von G eine Zetafunktion zuordenen; die `gewöhnliche’ Darstellungszetafunktion ist dann - bis auf Skalierung - gerade die Zetafunktion der regulären Darstellung von G. In meinem Vortrag werde ich über gemeinsame Ergebnisse mit Steffen Kionke berichten und dabei den Schwerpunkt auf Zetafunktionen induzierter Darstellungen setzen. Eine sehr einfache und schöne Quelle von expliziten Beispielen erschließt sich aus distanztransitiven Wirkungen von pro-endlichen Gruppen auf verwurzelten Bäumen, weitere dann schon kompliziertere Beispiele lassen sich mit Hilfe der Kirillovschen Bahnenmethode und Werkzeugen aus der p-adischen Integrationstheorie gewinnen.
03.05.2017, 15:00 Uhr, SR 228 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. J.-C. Schlage-Puchta
• Prof. Dr. Angelika Rohde (Albert-Ludwigs-Universität Freiburg)
Abstract: We develop honest and locally adaptive confidence bands for probability densities. They provide substantially improved confidence statements in case of inhomogeneous smoothness, and are easily implemented and visualized. The article contributes conceptual work on locally adaptive inference as a straightforward modification of the global setting imposes severe obstacles for statistical purposes. Among others, we introduce a statistical notion of local Hölder regularity and prove a correspondingly strong version of local adaptivity. We substantially relax the straightforward localization of the self-similarity condition in order not to rule out prototypical densities. The set of densities permanently excluded from the consideration is shown to be pathological in a mathematically rigorous sense. On a technical level, the crucial component for the verification of honesty is the identification of an asymptotically least favorable stationary case by means of Slepian's comparison inequality. This is a joint work with Tim Patschkowski.
21.04.2017, 15:00 Uhr, HS 125 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. A. Meister
• Prof. Dr. Kim Jong-Min (University of Minnesota-Morris)
"Mixture of D-vine copulas or modeling dependence"
Abstract : The identification of an appropriate multivariate copula for capturing the dependence structure in multivariate data is not straightforward. The reason is because standard multivariate copulas (such as the multivariate Gaussian, Student-t, and exchangeable Archimedean copulas) lack flexibility to model dependence and have other limitations, such as parameter restrictions. To overcome these problems, vine copulas have been developed and applied to many applications. In order to reveal and fully understand the complex and hidden dependence patterns in multivariate data, a mixture of D-vine copulas is proposed incorporating D-vine copulas into a finite mixture model. As a D-vine copula has multiple parameters capturing the dependence through iterative construction of pair copulas, the proposed model can facilitate a comprehensive study of complex and hidden dependence patterns in multivariate data. The proposed mixture of D-vine copulas is applied to simulated and real data to illustrate its performance and benefits.
Keywords: Dependence, Multivariate data, Pair-copula, Vines.
16.02.2017, 10:30 Uhr, HS 125 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. W.-D.Richter
• Dr. Francesco Chiacchio (Universität Neapel)
"Sharp Poincar´e inequalities"
Abstract: Let µ1(Ω) be the ﬁrst nontrivial Neumann eigenvalue for the Laplace operator in a Lipschitz, bounded domain Ω of Rn. We will present two type of lower bounds for µ1(Ω), involving diﬀerent geometrical features of Ω. In the ﬁrst estimate, obtained in [1] via “symmetrization arguments”, it appears Kn(Ω) the isoperimetric constant relative to Ω. In [2] we consider planar domains consisting of the points on one side of a ﬁxed curve γ, within a suitable distance δ from it. In this last case the estimate is given in terms of the length of γ, its curvature and δ.
References:
[1] B. Brandolini, F. Chiacchio, C. Trombetti, Optimal lower bounds for eigenvalues of linear and nonlinear Neumann problems. Proc. Roy. Soc. Edinburgh Sect. A 145 (2015), no. 1, 31-45.
[2] B. Brandolini, F. Chiacchio, E. B. Dryden, J. J. Langford, Sharp Poincar´e inequalities in a class of non-convex sets, arXiv:1608.01236v1.
25.01.2017, 17:00 Uhr, HS 125 (Ulmenstr. 69, Haus 3)
Kolloquiumsleiter: Prof. Dr. Friedemann Brock
• Prof. Dr. Sergej Bezrukov (University of Wisconsin - Superior)
"New families of edge-isoperimetric graphs"
Abstract: We present new infinite families of regular graphs whose all cartesian powers admit nested solutions in the edge-isoperimetric problem. For a given graph the problem is to specify a subgraph of a given order m that has maximum number I(m) of induced edges among all subgraphs of that order. Our results include as special cases most previously published results in this area. The graphs are specified by means of so-called delta-sequences of the length given by the number of vertices in the graph. The m-th element of the sequence d(m) is the difference I(m) - I(m-1). We also present a construction for regular graphs admitting these sequences. We show that by ordering the vertices of the n-th cartesian power of our graphs lexicographically (where n is at least 2), the subgraph induced by any initial segment of this order spans maximum number of edges.
As a byproduct, based on a special representation of graphs as a union of disjoint cliques, we introduce a new technique for extending a graph admitting nested solutions in the edge-isoperimetric problem to a larger one with that property.
12.01.2017, 15:00 Uhr