KOI Combinatorics Lectures
See schedule
Date:
Location:
CB 114
Event Series:
Math Conference
KOI Combinatorics Lectures
See schedule
Date:
Location:
CB 114
Event Series:
KOI Combinatorics Lectures
KOI Combinatorics Lectures
KOI Combinatorics Lectures
KOI Combinatorics Lectures
Speaker: Richard Ehrenborg, University of Kentucky
Title: Sharing pizza in n dimensions
Abstract:
We introduce and prove the n-dimensional Pizza Theorem. Let H be a real n-dimensional hyperplane arrangement. If K is a convex set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement H. We prove that if H is a Coxeter arrangement different from A1n such that the group of isometries W generated by the reflections in the hyperplanes of H contains the negative of the identity map, and if K is a translate of a convex set that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H that we call the even restricted arrangement. We get stronger results in the case of balls. We prove that the pizza quantity of a ball containing the origin vanishes for a Coxeter arrangement H with |H|-n an even positive integer.
This is joint work with Sophie Morel and Margaret Readdy.
https://www.ms.uky.edu/~readdy/KOI/
Date:
Location:
CB 114
Event Series:
KOI Combinatorics Lectures
KOI Combinatorics Lectures
Speaker: Richard Ehrenborg, University of Kentucky
Title: Sharing pizza in n dimensions
Abstract:
We introduce and prove the n-dimensional Pizza Theorem. Let H be a real n-dimensional hyperplane arrangement. If K is a convex set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement H. We prove that if H is a Coxeter arrangement different from A1n such that the group of isometries W generated by the reflections in the hyperplanes of H contains the negative of the identity map, and if K is a translate of a convex set that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H that we call the even restricted arrangement. We get stronger results in the case of balls. We prove that the pizza quantity of a ball containing the origin vanishes for a Coxeter arrangement H with |H|-n an even positive integer.
This is joint work with Sophie Morel and Margaret Readdy.
https://www.ms.uky.edu/~readdy/KOI/
Date:
Location:
CB 114
Event Series:
KOI Combinatorics Lectures
KOI Combinatorics Lectures
Speaker: Eric Katz, Ohio State University
Title: Models of Matroids
Abstract:
Matroids are known for having many equivalent cryptomorphic definitions, each offering a different perspective. A similar phenomenon holds in the algebraic geometric approach to them. We will discuss different ways of viewing matroids, motivated by group actions, intersection theory, and K-theory, each of which has been generalized from the realizable case to a purely combinatorial approach.
Date:
Location:
CB 114
Event Series:
KOI Combinatorics Lectures
KOI Combinatorics Lectures
Speaker: Eric Katz, Ohio State University
Title: Models of Matroids
Abstract:
Matroids are known for having many equivalent cryptomorphic definitions, each offering a different perspective. A similar phenomenon holds in the algebraic geometric approach to them. We will discuss different ways of viewing matroids, motivated by group actions, intersection theory, and K-theory, each of which has been generalized from the realizable case to a purely combinatorial approach.
Date:
Location:
CB 114
Event Series:
KOI Combinatorics Lectures
KOI Combinatorics Lectures
Speaker: Lei Xue, University of Michigan
Title: A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds
Abstract:
In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s ≤ 2d vertices has at least φk(d+s, d) = {d+1 choose k+1} + {d choose k+1} - {d+1-k \choose k+1} \] k-faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on d-dimensional polytopes with 2d+1 or 2d+2 vertices.
Date:
Location:
CB 114
Event Series:
KOI Combinatorics Lectures
KOI Combinatorics Lectures
Speaker: Lei Xue, University of Michigan
Title: A proof of Grünbaum's Lower Bound Conjecture for polytopes, lattices, and strongly regular pseudomanifolds
Abstract:
In 1967, Grünbaum conjectured that any d-dimensional polytope with d+s ≤ 2d vertices has at least φk(d+s, d) = {d+1 choose k+1} + {d choose k+1} - {d+1-k \choose k+1} \] k-faces. In the talk, we will prove this conjecture and discuss equality cases. We will then extend our results to lattices with diamond property (the inequality part) and to strongly regular normal pseudomanifolds (the equality part). We will also talk about recent results on d-dimensional polytopes with 2d+1 or 2d+2 vertices.
Date:
Location:
CB 114
Event Series: