Research Group Lattice Polytopes 


class date 
Contents 
Notes  last modified 
04/14 
Basics lattices Hilbert bases subdivisions abd triangulations 

04/21 
Shellings and Euler characteristic Ehrhart Theory generating functions examples integer point generating function and series Ehrhart's theorem 

04/28  no class  
05/05 
Ehrhart's theorem h^{*}polynomial degree and codegree, normalized volume Stanley's reciprocity theorem Ehrhart reciprocity 

05/12 
Ehrhart reciprocity Stanley's nonnegativity theorem BrianchonGram identity Brion's theorem signed decompositions of simplicial cones 

05/19 
signed decompositions of simplicial cones Barvinok's algorithm Geometry of Numbers Blichfeldt's theorem Minkowski's first theorem 

05/26  
06/02  
06/09  
06/16  
06/23 
Unimodular triangulations pulling subdivisions regular full triangulations exist unimodular triangulations don't Paco's Lemma 

06/30 
applications of Paco's Lemma KnudsenMumfordWaterman Theorem 

07/07  Gorenstein polytopes with RUT have unimodal h^{*} vector  
07/14 
date 
Title 
speaker  references 
04/15  polygons and the number 12  Benjamin  [PR] 
04/22  polygons and onion skins  Therese  [HS] 
05/06  lattice width  Matthias  [HZ] 
05/13  Gröbner bases, Graver bases and integer programming  Benjamin  [AWW] 
05/20  Roots of Ehrhart polynomials  Marianne  
05/27  applications of Minkowski's theorems  Moritz  [B1], Ch. VII.4 
06/03  Short vectors in lattices: LLL and PSLQ  Moritz  [B2], Ch. 12 & [PSLQ] 
06/10  Fourier series and periodic functions on Z  Felix  [BR], Ch. 7 
06/17  Dedekind sums  Felix  [BR], Ch. 8 
06/24  integer Carathéodory  Kaie  [BGHMW] & [Seb] 
07/01  unimodular triangulations of 3polytopes  Kaie  [KS] 
07/08  Algorithms to compute Hilbert bases  Lars  [H] 
07/15  What I did this summer  Benjamin 
date 
Contents 
due 
04/14  lattices and shellings  04/21 
04/21  Ehrhart polynomials  05/05 
05/05  Ehrhart polynomials II  05/12 
05/12  Reciprocity  05/19 
05/19  Minkowski  05/26 
05/26  Lattices  06/02 
06/02  Width  06/09 
06/09  Finiteness Theorem  06/16 
06/16  Gorenstein and Reflexive Polytopes  06/23 
06/23  Unimodular Triangulations, Pulling  
06/30  Unimodular Triangulations, Facet Width One  
07/07  
07/14 
Text Books  
[AZ]  Aigner, Martin; Ziegler, Günter: Proofs from THE BOOK. SpringerVerlag, Berlin. viii+199. (1998). [3540636986] 
[B1]  Barvinok, Alexander: A course in convexity. Graduate Studies in Mathematics. 54. Providence, RI: American Mathematical Society (AMS). x, 366 p. (2002). [ISBN 0821829688/hbk; ISSN 10657339] 
[B2]  Barvinok, Alexander: Integer Points in Polyhedra. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), 2008. 
[BG]  Bruns, Winfried; Gubeladze Joseph: Polytopes, Rings, and KTheory. Springer to appear 2009. 
[BR]  Beck, Matthias; Robins, Sinai: Computing the continuous discretely. Integerpoint enumeration in polyhedra. Springer Undergraduate Texts in Mathematics, to appear. 
[H]  Hemmecke, Raymond: Representations of lattice point sets: Theory, Algorithms, Applications. Habilitation thesis, University Magdeburg, 2006. 
[Sch1]  Schrijver, Alexander: Theory of linear and integer programming. Repr. Chichester: Wiley. xi, 471 p. (1998). [ISBN 0471982326/pbk] 
[Sch2]  Schrijver, Alexander: Combinatorial Optimization. Polyhedra and Efficiency. Repr. Springer 
[Stu]  Sturmfels, Bernd: Gröbner bases and convex polytopes. University Lecture Series. 8. Providece, RI: American Mathematical Society (AMS). xi, 162 p. (1996). [ISBN 0821804871] 
[Zie]  Ziegler, Günter, Lectures on Polytopes, Springer 
Journal Articles  
[AWW]  Aardal, Karen; Weismantel, Robert; Wolsey, Laurence: Nonstandard approaches to integer programming. Discrete Applied Mathematics 123 (2002) 5 74 [PDF] 
[BGHMW]  Bruns, Winfried; Gubeladze, Joseph; Henk, Martin; Martin, Alexander; Weismantel, Robert: A Counterexample to an Integer Analogue of Carathéodory's Theorem J. Reine Angew. Math. 510 (1999), 179185 [PDF] 
[PSLQ]  Ferguson, Helaman; Bailey, David; Arno, Stephen: Analysis of PSLQ, An Integer Relation Finding Algorithm. Math. Comput. 68, 351369, 1999. 
[HM]  Haase, Christian; Ilarion Melnikov: The reflexive dimension of a lattice polytope. Annals of Combinatorics, 10:211217, 2006. [PDF] 
[HS]  Haase, Christian; Schicho, Josef: Lattice Polygons and the number 2i+7. Am. Math. Mon. February 2009. math.CO/0406224 
[HZ]  Haase, Christian; Ziegler, Günter: On the maximal width of empty lattice simplices. European J. Combinatorics , 21(1):111119, 2000. [PDF] 
[KS]  Kantor, JeanMichel; Sarkaria, Karanbir: On primitive subdivisions of an elementary tetrahedron Pacific J. Math. 211 (2003), 123155 PJM 
[PR]  Poonen, Bjorn; RodriguezVillegas, Fernando: Lattice polygons and the number 12. Am. Math. Mon. 107, No.3, 238250 (2000) [PS] 
[Sca]  Scarf, Herbert: Integral polyhedra in three space. Math. Oper. Res. 10 (3) 403438 (1985). [ISSN 0364765X] 
[Seb]  András Sebö: Hilbert bases, Carathéodory's Theorem and Combinatorial Optimization in Ravindran Kannan and William R. Pulleyblank (eds.) Integer Programming and Combinatorial Optimization Math. Prog. Soc., Univ. Waterloo Press 1990, 431456 [PS] 
[Ver]  R. Vershynin Integer Cells in Convex Sets. math.FA/0403278 