FB 6 Mathematik/Informatik

Institut für Mathematik

# Hauptinhalt

## Topinformationen

### Oberseminar Stochastik

#### Weitere Angaben

Ort: 69/E15: Di. 12:00 - 14:00 (12x), 69/118: Freitag. 03.11., Freitag. 24.11. 12:00 - 14:00, 32/131: Donnerstag. 01.02. 14:00 - 16:00
Zeiten: Di. 12:00 - 14:00 (wöchentlich), Ort: 69/E15, Termine am Freitag. 03.11., Freitag. 24.11. 12:00 - 14:00, Donnerstag. 01.02. 14:00 - 16:00, Ort: 69/118, 32/131
Erster Termin: Di , 24.10.2017 12:00 - 14:00, Ort: 69/E15
Veranstaltungsart: Oberseminar (Offizielle Lehrveranstaltungen)

#### Studienbereiche

• Mathematik > Arbeitsgemeinschaften und Kolloquien
• Mathematik > Master Mathematik
• Mathematics/Computer Science

### Short CV

• 1990: Master degree (Dipl.-Ing.), Vienna University of Technology
• 1993: Doctor of Technical Sciences, Vienna University of Technology
• 1993 - 1999: Vertragsassistent, Vienna University of Technology
• 1996: Actuary, full member of the Aktuarvereinigung Österreich
• 1999 - 2001: Assistant Professor, Vienna University of Technology
• 2001: Habilitation, Vienna University of Technology
• 2001 - 2009: Associate Professor, Vienna University of Technology
• 2001 - 2003: Visiting Scholar, University of Freiburg
• 2003 - 2007: Guest lecturer for Actuarial Mathematics at Universität Salzburg
• since 2009: Professor of Mathematics, University of Osnabrück
• since 2011: Director of the Institute for Mathematics
• since 2013: Spokesman DFG-Graduiertenkolleg "Combinatorial Structures in Geometry"
• DAV-Correspondent at the University of Osnabrück

### Research Interests

I am interested in problems on the borderline between stochastic and geometry:
stochastic geometry, geometric probabilites, and various connections to probability theory and convex geometry.

Extreme points and convex hull of random samples: Given a random sample consisting of n iid. realizations of a random variable in. Rd. Extreme points of this sample are those points which are on the boundary of the convex hull of the sample. What is the distribution of the number of extreme points of this sample (which is in general something between d+1 and n)?

Poisson-Voronoi tesselations: Given a Poisson point process, associate to each point the corresponding Voronoi cell (that part of Rd closer to this point than to any other point of the point process). What is the probability of large Voronoi cells (large 'gaps' in the point process)? What is the shape of large Voronoi cells?

Characterization of valuations: Valuations are additive functionals on certain classes of sets. In particular, they fulfill the inclusion-exclusion property and thus all measures are valuations. Can we characterize valuations having natural invariance properties?

### Projects

• 2013 - 2018: DFG Graduiertenkolleg “Combinatorial Structures in Geometry”
• 2008 - 2010: FWF Project “Combinatorial Properties of Random Mosaics”
• 2007 - 2008: Scientist in charge of the Austrian node of "Phenomena in High Dimensions"
• 2005 - 2007: FWF Project "Valuations on convex bodies" (jointly with Monika Ludwig)

• 2001 - 2003: FWF Project “The geometric structure of random polytopes”

### Recent Talks

• Random Polytopes: Limit Theory. Kolloquium on Random Convex Hulls, UCL London, August 2016
• Random Points on a Halfsphere. AMS sectional meeting, Univ. Athens, USA, März 2016
• Voronoi trifft Poisson: Zufällige Approximation von Mengen, Weihnachtskolloquium, Universität Bochum, Dezember 2013
• Large deviation inequality for the Gilbert Graph, Oberwolfach, Februar 2013

### Publications

Stars of Empty Simplices. Submitted (Gem. mit D. Temesvari)

Crossing Numbers and Stress of Random Graphs. Submitted (Gem. mit M. Chimani, H. Döring)

Facets of high-dimensional Gaussian polytopes. Submitted (Gem. mit K. Böröczky, G. Lugosi)

Gigantic random simplicial complexes. Submitted (Gem. mit J. Grygierek, M. Juhnke-Kubitzke, T. Römer, O. Röndigs)

Production Shock Distribution and Risk Aversion in Multi-Agent-Tournaments. Submitted (Gem. mit B. Hartig)

On the cells in a stationary Poisson hyperplane mosaic. Adv. Geom., to appear (Gem. mit R. Schneider)

Cells with many facets in a Poisson hyperplane tessellation. Adv. Math. 324, 203-240 (2018) (Gem. mit G. Bonnet, P. Calka)

Random Approximation of Convex Bodies: Monotonicity of the Volumes of Random Tetrahedra.Discrete Comput. Geom. 59, 165-174 (2018) (Gem. mit S. Kunis und B. Reichenwallner)

Monotonicity of functionals of random polytopes. (Gem. mit M. Beermann) In: Ambrus, G., Böröczky, K., and Füredi, Z. (eds.): Discrete Geometry and Convexity. p. 23-28 (2017) A. Renyi Institute of Mathematics, Hungarian Academy of Sciences, Hungary

Expected Sizes of Poisson-Delaunay Mosaics and Their Discrete Morse Functions. Adv. Appl. Probab. 49, 745-767 (2017) (Gem. mit H. Edelsbrunner, A. Nikitenko)

Affine invariant valuations on polytopes. Discrete Comput. Geom. 57, 571-581 (2017) (Gem. mit M. Ludwig)

Random points in halfspheres. Random Structures & Algor. 50, 3-22 (2017),  (Gem. mit I. B\'ar\'any, D. Hug, und R. Schneider)

Limit theory for the Gilbert graph. Adv. Appl. Math. 88, 26-61 (2017) (Gem. mit M. Schulte und C. Thäle)

Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry. Editors: Peccati, G., Reitzner, M., Bocconi & Springer Series 7; Bocconi University Press, Springer, Milan (2016)

Introduction to Stochastic Geometry. (Gem. mit D. Hug) In: Peccati, G. and Reitzner, M. (eds.): Stochastic Analysis for Poisson Point Processes. p. 145--184 (2016) Bocconi & Springer Series 7; Bocconi University Press, Springer, Milan

U-statistics in stochastic geometry. (Gem. mit R. Lachieze-Rey) In: Peccati, G. and Reitzner, M. (eds.): Stochastic Analysis for Poisson Point Processes. p. 229--253 (2016) Bocconi & Springer Series 7; Bocconi University Press, Springer, Milan

On the monotonicity of the moments of volumes of random simplices. Mathematika 62, 949–958 (2016) (Gem. mit B. Reichenwallner)

Beyond the Efron-Buchta identities: distributional results for Poisson polytopes. Discrete Comput. Geom.  53, 226-244 (2015),  (Gem. mit M. Beermann)

Poisson polyhedra in high dimensions. Adv. Math. 281, 1-39 (2015) (Gem. mit J. Hörrmann, D. Hug und C. Thäle)

Central limit theorems for U-statistics of Poisson point processes. Ann. Prob. 41, No. 6, 3879-3909 (2013), (Gem. mit M. Schulte).

The monotonicity of f-vectors of random polytopes. Electron. Commun. Probab. 18, No. 23, 1-8 (2013) (Gem. mit O. Devillers, M. Glisse , X. Goaoc, G. Moroz)

Many empty triangles have a common edge. Discrete Comp. Geom. 50, 244-252 (2013), (Gem. mit I. Barany, J.-F. Marckert).

Set Reconstruction by Voronoi cells. Adv. Appl. Probab. 44, 938-953 (2012) (Gem. mit E. Spodarev, D. Zaporozhets).

On the variance of random polytopes. Adv. Math. 225, 1986-2001 (2010) (Gem. mit I. Barany).

A classification of SL(n) invariant valuations. Ann. of Math. (2) 172, 1219-1267 (2010) (Gem. mit M. Ludwig).

Poisson Polytopes. Ann. Prob. 38, 1507-1531 (2010) (Gem. mit I. Barany).

Random polytopes (survey). In: Molchanov, I., and Kendall, W. (eds.): New perspectives in stochastic geometry, p. 45--76 (2010),Oxford University Press, Oxford.

Mean width of inscribed random polytopes in a reasonable smooth convex body. J. Multivariate Anal. 100, 2287--2295 (2009), (Gem. mit K. Böröczky jr., F. Fodor, V. Vigh).

Poisson-Voronoi Approximation. Ann. Appl. Probab. 19, 719-736 (2009), (Gem. mit M. Heveling).

Elementary moves on triangulations. Discrete Comput. Geom. 35, 527-536 (2006) (Gem. mit M.Ludwig).

Central limit theorems for random polytopes. Probab. Theory Relat. Fields 133, 483-507 (2005). (Thanks to J. Pardon who pointed out an error in my proof, see Central limit theorems for uniform model random polygons, p.6).

The combinatorial structure of random polytopes. Adv. Math. 191, 178-208 (2005).

Gaussian polytopes: variances and limit theorems. Adv. Appl. Probab. 37, 297-320 (2005) (Gem. mit D.Hug).

The limit shape of the zero cell in a stationary Poisson hyperplane tesselation. Ann. Prob. 32, 1140-1167 (2004), (Gem. mit D. Hug und R. Schneider).

Asymptotic mean values of Gaussian polytopes. Beitr. Algebra Geom. 45, 531-548 (2004), (Gem. mit D.Hug und G.O. Munsonius).

Electromagnetic wave propagation and inequalities for moments of chord lengths. Adv. Appl. Probab. 36, 987-995 (2004), (Gem. mit J. Hansen).

Large Poisson-Voronoi cells and Crofton cells. Adv. Appl. Probab. 36, 667-690 (2004), (Gem. mit D. Hug und R. Schneider).

Efficient indoor radio channel modeling based on Integral Geometry. IEEE Transactions on Antennas and Propagation 52, 2456-2463 (2004), (Gem. mit J. Hansen).

Approximation of smooth convex bodies by random circumscribed polytopes. Ann. Appl. Probab. 14, 239-273 (2004) (Gem. mit K. Böröczky)

Stochastical approximation of smooth convex bodies. Mathematika 51, 11-29 (2004).

Random polytopes and the Efron-Stein jackknife inequality. Ann. Prob. 31, 2136-2166 (2003).

Random polytopes are nearly best-approximating. Rend. Circ. Mat. Palermo Suppl. 70, 263-278 (2002).

Random points on the boundary of smooth convex bodies. Trans. Amer. Math. Soc. 354, 2243-2278 (2002).

The floating body and the equiaffine inner parallel curves of a convex body. Geom. Dedicata 84, 151-167 (2001).

The convex hull of random points in a tetrahedron: Solution of Blaschke's problem and more general results. J. Reine Angew. Math. 536, 1-29 (2001) (Gem. mit C. Buchta).

Inequalities for convex hulls of random points. Monatsh. Math. 131, 71-78 (2000).

On Gram-Charlier expansion in risk theory. Working Paper of the Lab. of Act. Math., Univ. Copenhagen 161, 1-30 (1999) (Gem. mit C. Buchta).

A characterization of affine surface area. Adv. Math. 147, 138-172 (1999) (Gem. mit M. Ludwig).

On a theorem of G. Herglotz about random polygons. Rend. Circ. Mat. Palermo Suppl. 50, 89-102 (1997) (Gem. mit C. Buchta).

Equiaffine inner parallel curves of a plane convex body and the convex hulls of randomly chosen points. Probab. Theory Relat. Fields 108, 385-415 (1997) (Gem. mit C. Buchta).

Optimaler Selbstbehalt in der Kraftfahrzeug-Haftpflichtversicherung. Nicht zur Veröffentlichung bestimmter Forschungsbericht, Wien, 1995, 59 Seiten (Gem. mit C. Buchta).

Zufällige Polytope im Tetraeder. Dissertation, TU-Wien (1993).