ERC Advanced Grant Project

The research of the SDModels project has the goal to connect traditionally quite distant fields of current mathematical research via common or structurally similar discrete (mostly: geometric) models. We try to thus make substantial contributions to mathematical research, by highlighting, developing, and exploiting theory for common structures and structural similarities that occur in problems/theories from diverse mathematical application areas. This involves, in particular, the areas of

1. Discrete Geometry
2. Discrete Differential Geometry
3. Mesh Generation/Numerics of PDEs
4. Topology, Topological Combinatorics
5. Simplicial Quantum Gravity

The work in the project is concentrated in three Focus Areas, namely
Focus Area 1: High-complexity Geometry
Focus Area 2: Delaunay Geometry: Polyhedral models with circle/sphere patterns
Focus Area 3: Topological connectivity and diameter of Discrete Structures

Project head

Günter M. Ziegler

Guest professor

Pavle Blagojevic*

Junior professor

Raman Sanyal*

Postdocs

Dirk Frettlöh*

Louis Theran*

 Ph.D. students

 Karim Adiprasito

 Bernd Gonska

 Emerson Leon*

Student assistants

 Merle Breitkreuz*

 Gregor Lagodzinski*

 Miriam Schlöter*

Ph.D. student

David Chubelaschwili*

Bruno Benedetti*

Benjamin Matschke

Martin Weidner*

Current highlight: Work on projective uniqueness of polytopes

Current highlight: Ongoing work by Gonska and Ziegler on f-vectors of Delaunay polytopes

 Current highlights:

•  Our work on the Colored Tverberg problem, as

     presented in the April 2011 Notices of the AMS

     http://www.ams.org/notices/201104/rtx110400550p.pdf

•  Work by Benedetti and Adiprasito on discrete Morse theory and collapsibility

"Metric geometry and collapsibility"

 Karim Adiprasito, Bruno Benedetti

 Preprint, July 2011

 http://arxiv.org/abs/1107.5789

 

"Discrete Morse Theory is as perfect as Morse Theory"

 Bruno Benedetti

 Transactions Amer. Math. Soc., to appear

 http://arxiv.org/pdf/1010.0548

 

"Discrete Morse Theory for manifolds with boundary"

 Bruno Benedetti

 Preprint, October 2010, submitted

 http://arxiv.org/pdf/1007.3175  

 (*ERC acknowledgement missing in preprint, will be added in publication)

 

"Extensions of theorems of Rattray and Makeev"

 Pavle V. M. Blagojevic, Roman Karasev

 http://arxiv.org/pdf/1011.0869

 

"Many non-equivalent realizations of the associahedron"

 Cesar Ceballos, Francisco Santos, Günter M. Ziegler

 Preprint August 2011, in preparation: Revision of http://arxiv.org/abs/1006.3487

 

"Optimal elliptic Sobolev regularity near three-dimensional,

 multi-material Neumann vertices"

 Robert Haller-Dintelmann, Wolfgang Höppner, Hans-Christoph Kaiser,

 Joachim Rehberg, Günter M. Ziegler.

 Matheon Preprint 808, April 2011, 27 pages

 http://vs24.kobv.de/opus4-matheon/files/808/ecke.pdf

 

"Perfect Colourings of Cyclotomic Integers"

 E. Paulo Bugarin, M. Louise A.N. de las Penas, Dirk Frettlöh

Geometriae Dedicata, to appear

http://arxiv.org/pdf/0905.4048v2

 

"Parallelogram tilings, worms and finite orientations"

Dirk Frettlöh, Edmund Harriss, submitted

http://arxiv.org/pdf/1202.4686v1

"On locally constructible spheres and balls"

 Bruno Benedetti, Günter M. Ziegler

 Acta Mathematica 206 (2011), 205-243

 http://www.springerlink.com/content/x8r6mm1155102542/

 

"Spectral sequences in combinatorial geometry: Cheeses, inscribed sets,

 and Borsuk–Ulam type theorems"

 Pavle V. M. Blagojevic, Aleksandra Dimitrijevic Blagojevic, John McCleary

 Topology and its Applications 158 (2011), 1920-1936

 http://www.sciencedirect.com/science/article/pii/S0166864111002586

 

"A tight colored Tverberg theorem for maps to manifolds (extended abstract)"

 Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler

 In Proc. FPSAC 2011 (Reykjavík, Iceland), 

 Discrete Mathematics and Theoretical Computer Science (DMTCS), Vol. AO,

 http://www.dmtcs.org, page 183–190, 2011.

 http://combinatorics.is/proceedings/dmAO0117.pdf

 

"A tight colored Tverberg theorem for maps to manifolds"

 Pavle V. M. Blagojević, Benjamin Matschke, Günter M. Ziegler

 Topology and its Applications 158 (2011), 1445–1452.

 http://www.sciencedirect.com/science/article/pii/S0166864111001635

 http://arxiv.org/abs/1107.1904

 

"Selfdual Substitutions in Dimension One"

 Valérie Berthé, Dirk Frettlöh, Victor Sirvent

 European Journal of Combinatorics 33 (2012) 981-1000

 http://arxiv.org/pdf/1108.5053v2