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

Ivan Izmestiev*

Louis Theran*

 Ph.D. students

 Karim Adiprasito

 Bernd Gonska

 Carsten Schultz

Student assistants

 Merle Breitkreuz*

 Gregor Lagodzinski*

 Miriam Schlöter*

Ph.D. students

David Chubelaschwili*

Bruno Benedetti*

Benjamin Matschke

Cesar Ceballos

Bernd Schulze

Emerson Leon*

Student assistants

Martin Weidner*

Postdocs

Dirk Frettlöh*

Current highlight: 

• Work by Adiprasito and Ziegler on projective uniqueness of polytopes

• Work by Jochemko and Sanyal on marked order polytopes, http://arxiv.org/abs/1206.4066

Current highlight: 

• Work by Gonska and Ziegler on f-vectors of Delaunay polytopes,   http://arxiv.org/abs/1111.5322, to appear in Adv. Gemetry

Current highlights: 

• Work by Blagojevic and Ziegler on the Nandakumar & Ramana Rao partition problem, http://arxiv.org/abs/1202.5504

• Work by Blagojevic, Lück, and Ziegler on the equivariant topology of configuration spaces, http://arxiv.org/abs/1207.2852 

• "Many non-equivalent realizations of the associahedron"
    Cesar Ceballos, Francisco Santos, Günter M. Ziegler
    Preprint September 2011, 28 pages, http://arxiv.org/abs/1109.5544  

•  "Face numbers of centrally symmetric polytopes from split graphs"
    Ragnar Freij, Matthias Henze, Moritz W. Schmitt, Günter M. Ziegler
    Preprint, January 2012, 10 pages, http://arxiv.org/abs/1201.5790

• "Arithmetic of marked poset polytopes, monotone triangle reciprocity,
    and partial colorings"
    Katharina Jochemko, Raman Sanyal
    Preprint, June 2012
    http://arxiv.org/abs/1206.4066

• "Smooth Hyperbolicity Cones are Spectrahedral Shadows"
    Tim Netzer, Raman Sanyal
    Preprint, August 2012, 8 pages, http://arxiv.org/abs/1208.0441
  

•  "Foldable Triangulations of Lattice Polygons"
    Michael Joswig, Günter M. Ziegler
    Preprint, July 2012, 4 pages, http://arxiv.org/abs/1207.6865

• "Parallelogram tilings, worms and finite orientations"
    Dirk Frettlöh, Edmund Harriss, submitted
    http://arxiv.org/pdf/1202.4686v1

• "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

• "Generic rigidity of reflection frameworks"
    Justin Malestein, Louis Theran
    Preprint 3.2012 
    http://arxiv.org/abs/1203.0772
   

• "Generic combinatorial rigidity of periodic frameworks"
    Justin Malestein, Louis Theran
    Preprint 8.2010, Revised 9.2011, to appear in Advances in Math.
    http://arxiv.org/abs/1008.3710
   

• "Topological Designs"
    Justin Malestein, Igor Rivin, Louis Theran
    Preprint 8.2010, Revised and submitted  5.2012 
    http://arxiv.org/abs/1008.1837
   

• "Generic rigidity of frameworks with orientation-preserving crystallographic symmetry"
    Justin Malestein, Louis theran
    Preprint 8.2011, Revised 2.2012 
    http://arxiv.org/abs/1108.2518
   

• "Henneberg constructions and covers of cone-Laman graphs"
    Louis Theran
    Submitted preprint 4.2012 
    http://arxiv.org/abs/1204.0503
   

• "Generic rigidity with forced symmetry and sparse colored graphs"
    Justin Malestein, Louis Theran
    Submitted preprint 3.2012
    http://arxiv.org/abs/1203.2276

• "The Algebraic Combinatorial Approach for Low-Rank Matrix Completion"
  Franz Kiraly, Louis Theran, Ryota Tomioka, Takeaki Uno
  http://arxiv.org/abs/1211.4116
  

•  "Inscribable stacked polytopes"
    Bernd Gonska, Günter M. Ziegler
    Preprint, November 2012, 15 pages, http://arxiv.org/abs/1111.5322

•  "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://opus4.kobv.de/opus4-matheon/files/808/ecke.pdf
   

•  "Discrete Morse Theory is at least as perfect as Morse Theory"
    Bruno Benedetti
    Preprint, Oct. 2010, revised Jan. 2012, submitted
    http://arxiv.org/pdf/1010.0548

•  "Discrete Morse Theory  for manifolds with boundary"
    Bruno Benedetti
    Transactions of the AMS 364 (2012), 6631-6670
    http://arxiv.org/pdf/1007.3175  

•  "Metric geometry and collapsibility"
    Karim Adiprasito, Bruno Benedetti
    Preprint, July 2011, revised Jan. 2012, submitted
    http://arxiv.org/abs/1107.5789
  

•  "Convex Equipartitions via Equivariant Obstruction Theory"
    Pavle V. M. Blagojević, Günter M. Ziegler
    Preprint, January 2012; revised February 2012, 17 pages, 
    http://arxiv.org/abs/1207.5504
  

• "Equivariant Topology of Configuration Spaces"
  Pavle V. M. Blagojević, Wolfgang Lück, Günter M. Ziegler
  Preprint, July 2012, 40 pages,  http://arxiv.org/abs/1207.2852
  

• "Extensions of theorems of Rattray and Makeev"
   Pavle V. M. Blagojevic, Roman Karasev
   http://arxiv.org/pdf/1011.0869
    Topological Methods in Nonlinear Analysis
   http://www.tmna.ncu.pl/, to appear

• "The Schwarz genus of the Stiefel manifold and counting geometric configurations"
  Pavle Blagojević, Roman Karasev
  Preprint, November 2012, 7 pages
  http://arxiv.org/pdf/1211.5003v1.pdf
  

• "Tight complexes in 3-space admit perfect discrete Morse functions"
  Karim Adiprasito, Bruno Benedetti
  http://arxiv.org/abs/1202.3390
  

   

   

  
  

•  "Realizing the associahedron: Mysteries and questions"
     Cesar Ceballos, Günter M. Ziegler
     in: “Associahedra, Tamari Lattices and Related Structures”,
     Tamari Memorial Festschrift
     (F. Müller-Hoissen, J. Pallo, J. Stasheff, eds.),
     Progress in Mathematics, Vol. 299, Springer Basel 2012, 119-127;
     http://arxiv.org/abs/1110.4059
  

• "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
   

• "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
   

• "Optimal bounds for a colorful Tverberg-Vrecica type problem"
  Pavle Blagojevic Benjamin Matschke, Günter M. Ziegler
  Advances in Math. 226 (2011), 5198-5215
  http://arxiv.org/abs/0911.2692

• "Turán numbers for K(s,t)-free graphs: topological obstructions and algebraic
  constructions"
  Pavle Blagojević, Boris Bukh, Roman Karasev
  http://arxiv.org/abs/1108.5254
  Israel Journal of Mathematics, to appear
  http://www.ma.huji.ac.il/~ijmath/
  

• "Functions, measures, and equipartitioning convex k-fans"
  Imre Barany, Pavle Blagojevic,
  Aleksandra Dimitrijevic Blagojevic
  Discrete and Computational Geometry, to appear
  http://www.springerlink.com/content/0179-5376
  
  

• "A problem related to Barany-Grünbaum conjecture"
  Pavle Blagojevic, Aleksandra Dimitrijevic Blagojevic
  Filomat, to appear
  http://www.pmf.ni.ac.rs/pmf/publikacije/filomat/filomat_pocetna.php