Department of Mathematics and Statistics, Binghamton University

The Combinatorics Seminar

2020-01-29T14:03:07-04:00

The Combinatorics Seminar

SPRING 2018

Organizers: Laura Anderson, Michael Dobbins, and Thomas Zaslavsky.

Tuesday, January 16
No meeting planned at present.
Tuesday, January 23
Speaker: Steven Simon (Bard) Cancelled
Title: Hyperplane Equipartitions Plus Constraints
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, January 30
No seminar today.
Tuesday, February 6
Speaker: Michael Dobbins (Binghamton)
Title: Shellability is NP-Complete
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, February 13
Speaker: Ting Su (Binghamton)
Title: A Classification of Stringent Hyperfields
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, February 20
Speaker: Florian Frick (Cornell)
Title: Intersections of Finite Sets: Geometry and Topology
Time: 1:15 - 2:15
Room: WH-100E
Thursday, February 22 (in the Geometry/Topology Seminar; note special day and times)
Speaker: Olakunle Abawonse (Binghamton)
Title: Topology of the Grünbaum–Hadwiger–Ramos Hyperplane Mass Partition Problem
Time: 2:50 - 3:50
Title: Hyperplane Mass Partitions Via Relative Equivariant Obstruction Theory
Time: 4:15 - 515
Room: WH-100E
Tuesday, February 27
Speaker: Benjamin Blum-Smith (NYU)
Title: When Do Integer Permutation Invariants Form a Free Module Over the Symmetric Polynomials? An Application of Combinatorics to Invariant Theory
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, March 13 (joint with the Algebra Seminar)
Speaker: Victor Reiner (Minnesota)
Title: Finite General Linear Groups and Symmetric Groups
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, March 20
Speaker: Thomas Zaslavsky (Binghamton)
Title: Circle Problems in Signed Graphs
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, March 27 (joint with the Algebra Seminar)
Speaker: Farbod Shokrieh (Cornell)
Title: Effective Divisor Classes on Graphs
Time: 1:15 - 2:15
Room: WH-100E
Monday, April 2
Speaker: Stefan van Zwam (Louisiana State)
Title: A Stroll through Partial Fields
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, April 10
Speaker: Jacob Matherne (U. Mass. Amherst)
Title: Singular Hodge Theory of Matroids
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, April 17
Speaker: Richard Behr
Title: Edge Coloring and Special Edges of Signed Graphs
Time: 12:00 - 1:00 and 1:15 - 2:15
Room: OR-100D and WH-100E (respectively)
Tuesday, April 24 (joint with the Geometry/Topology Seminar)
Speaker: Robert Connelly (Cornell)
Title: Tensegrities: Geometric Structures Suspended in Midair
Time: 1:15 - 2:15
Room: WH-100E
Tuesday, May 1 (joint with the Geometry/Topology Seminar)
Speaker: Boris Bukh (Carnegie Mellon)
Title: Topological Version of Pach's Overlap Theorem
Time: 1:15 - 2:15
Room: WH-100E

Consider the collection of all the simplices spanned by some n-point set in R^d. There are several results showing that simplices defined in this way must overlap very much. In this talk I focus on the generalization of these results to 'curvy' simplices.

Specifically, Pach showed that every d+1 sets of points, Q₁, …, Q_d+1, in R^d contain linearly-sized subsets P_i in Q_i such that all the transversal simplices that they span intersect. In joint work with Alfredo Hubard, we show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size C(log n)^1/(d-1). We show that this is tight in dimension 2, for all surfaces other than S². Surprisingly, the optimal bound for S² is (log n)^1/2. This improves upon results of Barany, Meshulam, Nevo, Tancer.
Tuesday, May 8
Speaker: Jim Lawrence (George Mason)
Title: Interval Posets, Parity Representations, Binary Partitions, and Antiprisms
Time: 3:00 - 4:00 (Note special time.)
Room: WH-100E

Given a poset (a partially ordered set), one obtains another poset by considering the collection of intervals of the first, partially ordered by inclusion. (There are various possibilities, depending, for instance, upon whether one considers the empty set as being an “interval.”) This construction has found use in the study of convex polytopes and other places. I describe a new method of representation of posets by utilizing certain geometric complexes in R^d having vertices in Z^d. The striking feature of this method of representation is that taking the interval poset corresponds to dilation by a factor of 2 of the geometric complex. I explore connections with the integer partitions of powers of 2 into powers of 2.

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The Combinatorics Seminar

2020-01-29T14:03:07-04:00

The Combinatorics Seminar

SPRING 2004

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Organizers: Laura Anderson and Thomas Zaslavsky.

Thursday, January 29 (joint with the Geometry and Topology Seminar)

Speaker: Thomas Zaslavsky
Title: Homotopy in Biased Graphs: Combinatorics vs. Topology
Time: 2:50 - 3:50
Room: LN-2205

Thursday, February 5 Colloquium

Speaker: Bruce Sagan (Michigan State)
Title: Graph Coloring and Symmetric Functions
Room: LN-2205

Friday, February 6 (Note special day.)

Speaker: Bruce Sagan (Michigan State)
Title: Topological Properties of Activity Orders for Matroid Bases
Time: 3:30 - 4:30
Room: LN-2205

Monday, February 23

Speaker: Greg Blekherman (Univ. of Michigan)
Title: There are Significantly More Nonnegative Polynomials than Sums of Squares
Time: 3:30 - 4:30
Room: LN-2205

Friday, March 5

Speaker: Caroline Klivans (Cornell)
Title: The Bergman Complex of a Matroid and Phylogenetic Trees
Time: 3:30 - 4:30
Room: LN-2201

Monday, March 15

Speaker: Kristin Camenga (Cornell)
Title: Characterizing Inscribable Polyhedra
Time: 3:30 - 4:30
Room: LN-2201

Friday, March 26

Speaker: Sandra Kingan (Penn State Harrisburg)
Title: Excluded Minor Results in Matroids
Time: 3:30 - 4:30
Room: LN-2205

Friday, April 16

Speaker: Robert Connelly (Cornell)
Title: Comments on Generalized Heron Polynomials and Robbins' Conjectures
Time: 1:10 - 2:10
Room: LN-2205 (Note the room!)

Tuesday, April 20

Speaker: Steven J. Tedford (Franklin & Marshall)
Title: Greedoids and their Basis Graphs
Time: 1:30 - 2:30
Room: LN-2206

Friday, July 23 (Special Departmental Seminar)

Speakers: Chris Fearnley (B.A. in Mathematics from Binghamton, 1989) and Jeannie Moberley
Title: Supercircles: Expanding Buckminster Fuller's Foldable Circle Models
Time: 2:00-3:00 p.m.
Room: LN-2205

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The Combinatorics Seminar

2020-01-29T14:03:07-04:00

The Combinatorics Seminar

SPRING and SUMMER 2005

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Organizers: Laura Anderson and Thomas Zaslavsky.

Scheduled Talks in the Spring Semester

Thursday, February 3 (Colloquium)

Speaker: Mark Watkins (Syracuse)
Title: A Characterization of Infinite Planar Imprimitive Graphs
Time: 4:30 - 5:30
Room: LN-2205

Thursday, February 10

Speaker: Thomas Zaslavsky (Binghamton)
Title: Quasipolynomials and their Convolution
Time: 1:15 - 2:15
Room: LN-2201

Thursday, February 17

Speaker: Rigoberto Florez (Binghamton)
Title: Harmonic Conjugation in Harmonic Matroids
Time: 1:15 - 2:15
Room: LN-2201

Thursday, February 24

Speaker: David Forge (Orsay and Binghamton)
Title: Orlik-Solomon Algebras and Other Algebras of Hyperplane Arrangements
Time: 1:15 - 2:15
Room: LN-2201

Thursday, March 3

Speaker: Thomas Zaslavsky (Binghamton)
Title: Signed and Biased Graphs: A Survey
Time: 1:15 - 2:15
Room: LN-2205

Thursday, March 10 (joint with the Algebra Seminar)

Speaker: Franco Saliola (Cornell)
Title: Face Algebras of Hyperplane Arrangements, Lattice Cohomology, and Quivers
Time: 1:15 - 2:15
Room: LN-2205

Thursday, March 17 (joint with the Algebra Seminar)

Speaker: Lauren Rose (Bard College)
Title: Algebraic Combinatorics of Piecewise Polynomials
Time: 1:15 - 2:15
Room: LN-2205

Thursday, March 31

Speaker: Cristina Ruiz (Binghamton)
Title: A Stratification of the Middle-level MacPhersonian
Time: 1:15 - 2:15
Room: LN-2205

Thursday, April 7 (Colloquium)

CANCELLED
Speaker: Vladislav Goldberg (New Jersey Institute of Technology)
Title: Solution of Blaschke's Problem on the Linearization of Planar Webs
Time: 4:30 - 5:30
Room: LN-2205

Thursday, April 14 (Colloquium)

Speaker: Jack Graver (Syracuse)
Title: When Does a Curve Bound a Distorted Disk?
Time: 4:30 - 5:30
Room: LN-2205

Friday, April 22

Speaker: Bruce Sagan (Michigan State)
Title: GCD Determinants
Time: 2:20 - 3:20
Room: LN-2205

Friday, April 22 (Colloquium)

Speaker: Bruce Sagan (Michigan State)
Title: Congruences for Combinatorial Sequences
Time: 4:40 - 5:40
Room: LN-2205

Thursday, May 5 (Colloquium)

Speaker: Robert Connelly (Cornell)
Title: The Kneser-Poulsen Conjecture in the Plane
Time: 4:30 - 5:30
Room: LN-2205

Tuesday, May 10

Speaker: Alice Dean (Skidmore)
Title: Characterizations of Unit-Bar Visibility Graphs
Time: 2:00 - 3:00 (note unusual time)
Room: LN-1402 (note unusual room)

Scheduled Talks in the Summer

Wednesday, June 8
Speaker: Greg Kuperberg (University of California at Davis and Cornell University)
Title: Numerical Cubature from Geometry and Coding Theory
Time: 11:00 - 12:00
Room: LN-2205
Monday, June 27
Speaker: N.M. Singhi (Tata Institute)
Title: Studying t-Designs and Other Families of Subsets of a Finite Set Algebraically
Time: 11:00 - 12:00 (Note unusual time.)
Room: LN-2205
Thursday, July 14
Speaker: Rigoberto Flórez
Title: Four Studies in the Geometry of Biased Graphs
Time: 1:00 - 2:15 and 3:00 - 3:30
Room: LN-2205

This talk is the public part of Mr. Flórez's thesis defense. The committee is Thomas Zaslavsky (chair), Laura Anderson, Marcin Mazur, and Gary Gordon (Lafayette College). All persons are welcome to attend the talk.
Friday, July 15
Speaker: Gary Gordon (Lafayette College)
Title: Symmetries of the Icosahedral Matroid
Time: 11:00 - 12:00
Room: LN-2205

Departmental home page.

Spring 2018

2019-02-18T08:49:57-04:00

Spring 2018

January 16 (algebra crosspost - meets in WH-100E at 2:50)
Speaker: Jonas Deré (KU Leuven Kulak)
Title: Which manifolds admit expanding maps

Abstract: In 1981, M. Gromov completed the proof that every manifold admitting an expanding map is, up to finite cover, homeomorphic to a nilmanifold. Since then it was an open question to give an algebraic characterization of the nilmanifolds admitting an expanding map. During my talk, I will start by introducing the basic notions of expanding maps and nilmanifolds. Then I explain how the existence of such an expanding map only depends on the covering Lie group and on the existence of certain gradings on the corresponding Lie algebra. One of the applications is the construction of a nilmanifold admitting an Anosov diffeomorphism but no expanding map, which is the first example of this type.

February 1
Speaker: Jonathan Williams (Binghamton University)
Title: Sewing a homotopy into pieces

Abstract: In this talk, I will try to explain the title. There will be 4-manifolds, many pictures, and very little background needed.

February 8
Speaker: Russell Ricks (Binghamton University)
Title: A Rank Rigidity Result for Certain Nonpositively Curved Spaces via Spherical Geometry

Abstract: To understand the geometry of nonpositively curved (NPC) spaces, it is natural to classify the various types of spaces that can occur. The Rank Rigidity Theorem for compact NPC manifolds separates the class of compact NPC manifolds into three very distinct types, and proves that nothing else can exist.

A version of Rank Rigidity has been conjectured for more general NPC spaces (CAT(0) spaces). In this talk, we discuss some progress toward this general conjecture, by reducing the problem to looking at patterns on spheres. In particular, we prove the conjecture for certain NPC spaces with one-dimensional boundary. Unlike previous results in this area, there are no additional constraints on the CAT(0) space (such as a manifold or polyhedral structure).

February 15
Speaker: Carlos Vega (Binghamton University)
Title: Null Distance on a Spacetime

Abstract: In contrast to the Riemannian setting, a Lorentzian manifold (M,g) is not known to possess any naturally induced distance function. I will first try to explain why that is, starting with some of the basics of spacetime (Lorentzian) geometry. We will then discuss a `null distance function' introduced in joint work with Christina Sormani, some of its properties, examples, and some open questions.

February 22 (special two-part talk: see next entry)
Speaker: Olakunle Abawonse (Binghamton University)
Title: Topology of the Grunbaum-Hadwiger-Ramos Hyperplane Mass Partition Problem

Abstract: In this talk, we will discuss a problem raised by Ramos that asks for the smallest dimension $d=\Delta(j,k)$ such that for any $j$ measures in $\mathbb{R}^d$, there are $k$ affine hyperplanes that simultaneously cut each measure into $2k$ equal parts. We will give a general configuration space/test map scheme for this problem and show how the theory of relative equivariant obstruction theory applies to this problem.

This is part of a candidacy talk, with committee Laura Anderson (chair), Ross Geoghegan and Michael Dobbins. It is open to all.

February 22 (Special time: 4:15pm)
Speaker: Olakunle Abawonse (Binghamton University)
Title: Hyperplane Mass Partitions Via Relative Equivariant Obstruction Theory

Abstract: We will give solutions to some of the few cases in which the minimum value $d=\Delta(j,k)$ is known. We will achieve this by showing the non-existence of a certain $G$-equivariant map. By the theory of relative equivariant obstruction theory, this problem reduces to evaluating some obstruction classes.

This is part of a candidacy talk, with committee Laura Anderson (chair), Ross Geoghegan and Michael Dobbins. It is open to all.

March 1
Speaker: Jun Li (University of Michigan)
Title: The symplectomorphism groups of rational surfaces

Abstract: This talk is on the topology of $Symp(M,\omega)$, where $Symp(M,\omega)$ is the symplectomorphism group of a symplectic rational surface $(M,\omega)$. We will illustrate our approach with the 5 point blowup of the projective plane. For an arbitrary symplectic form on this rational surface, we are able to determine the symplectic mapping class group (SMC) and describe the answer in terms of the Dynkin diagram of Lagrangian sphere classes. In particular, when deforming the symplectic form, the SMC of a rational surface behaves in the way of forgetting strand map of braid groups. We are also able to compute the fundamental group of $Symp(M, \omega)$ for an open region of the symplectic cone. This is a joint work with Tian-Jun Li and Weiwei Wu.

March 8
Speaker: Lisa Piccirillo (UT-Austin)
Title: Stein Knot Traces

Abstract: Four-manifolds which admit a Stein structure have many nice properties, for example the Stein structure gives bounds on the genus function of the manifold and Stein cobordisms induce nontrivial maps on the Heegaard Floer homology of the boundary. However, handed an arbitrary four-manifold it can be difficult to determine whether it admits a Stein structure. A question in the field asked whether it is ever straightforward to detect Stein structures on particularly simple manifolds; more technically it asked whether the four manifold $X_n(K)$ obtained by attaching an $n$-framed 2-handle to $B^4$ along $K$ is Stein if and only if $n<\overline{tb}(K)$. We answer this in the negative, and in fact show that a Stein $X_n(K)$ can have $n$ arbitrarily much larger than $\overline{tb}(K)$. This talk will focus on the constructive part of our proof, a technique due largely to Osoinach for building knots $K$ and $K’$ with $X_n(K)$ diffeomorphic to $X_n(K’)$. This is joint work with Tom Mark and Faramarz Vafaee.

March 15 (Special time 2:30 pm)
Speaker: Gili Golan (Vanderbilt University)
Title: Invariable generation of Thompson groups

Abstract: A subset $S$ of a group $G$ invariably generates $G$ if for every choice of $g(s)\in G$, $s\in S$ the set $\{s^{g(s)}:s\in S\}$ is a generating set of $G$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably generates $G$. In this talk, we study invariable generation of Thompson groups. We show that Thompson group $F$ is invariable generated by a finite set, whereas Thompson groups $T$ and $V$ are not invariable generated. This is joint work with Tsachik Gelander and Kate Juschenko.

March 22
Speaker: Yair Hartman (Northwestern University)
Title: Stationary C*-Dynamical Systems

Abstract: We introduce the notion of stationary actions in the context of C*-algebras, and prove a new characterization of C*-simplicity in terms of unique stationarity. This ergodic-theoretic characterization provides an intrinsic understanding for the relation between C*-simplicity and the unique trace property, and provides a framework in which C*-simplicity and random walks interact. Joint work with Mehrdad Kalantar.

March 29
Speaker: Yuhang Liu (Penn)
Title: Closed 6-manifolds with Positive Sectional Curvature and Non-Abelian symmetry

Abstract: Understanding the structure of Riemannian manifolds with strictly positive sectional curvature remains a fundamental problem in Riemannian geometry. In this talk, I will briefly go over the history of the classification of positively curved manifolds in low dimensions under certain symmetry assumptions on the isometry group. Then I will focus on dim 6 and discuss positively curved 6-manifolds whose isometry groups are non-Abelian Lie groups. Examples of such manifolds will be given together with the isometric group actions, and if time permits I will present some results I got in this direction. This is ongoing work on my thesis problem.

April 12 2018 Hilton Memorial lecture
Speaker: Vaughan Jones (Vanderbilt)
Title: Local scale transformations in one dimension

Special time and place: 3pm–4pm in AA G023

Abstract: In two dimensional conformal field theory, local scaling symmetry means invariance of some kind under conformal transformations. The quantum theory splits into two one dimensional theories called the “chiral halves”. Conformal invariance then gives a projective representation of the the diffeomorphism group (of the line or the circle) on each of the chiral halves. In an attempt to approximate this local scaling invariance we have considered the Thompson groups F and T as approximations to the diffeomorphism groups. Though this does not work perfectly, it has yielded a kind of “topsy turvy” version of chiral CFT including an interesting family of unitary representations of F and T whose coefficients give, among other things, a way to construct all knots and links from elements of F and T, analogous to the standard construction from the braid groups.

April 12
Speaker: Vaughan Jones (Vanderbilt)
Title: The Wysiwyg representations of the Thompson groups

This is a topology seminar talk in WH-100E at a special time, 1:15pm. See the entry immediately below for the Hilton lecture, immediately following this seminar.

Abstract: I will describe a general construction of actions of Thompson’s groups F, T and V and focus on a special kind - unitary representations on a (necessarily infinite dimensional) Hilbert space, coming from very simple combinatorial data. They can be approached via their matrix coefficients which are literally visible. There are many open questions but at least for one family of combinatorial data we can decide equivalence and irreducibility.

April 19
Speaker: Akram Alishahi (Columbia)
Title: Khovanov homology and unknotting number

Abstract: Khovanov homology is a combinatorially-defined knot invariant which refines the Jones polynomial. In this talk we will recall the definition of Khovanov homology and one of its refined versions called Bar-Natan homology, and we will show that the order of h-torsion classes in Bar-Natan homology gives a lower bound for unknotting number.

Special date and time: April 24, 1:15pm in WH-100E (joint with combinatorics)
Speaker: Robert Connelly (Cornell)
Title: Tensegrities: Geometric Structures Suspended in Midair

Abstract: Suppose you have a finite collection of points in Euclidean space or the plane. Some pairs are connected by inextendible cables, others by incompressible struts, and some by fixed length bars. The artist Kenneth Snelson constructed several large structures, made of cables and bars, that hold their shape under tension, where the struts appear to be suspended in midair. Buckminster Fuller, the architect and inventor, called them “tensegrities” because of their “tensional integrity”. But why do they hold their shape? There is a very simple principle using quadratic energy functions that provides the key to their stability. I will show a catalog of highly symmetric tensegrities, created with the help of a little bit of representation theory, as well as tangible models, where you can feel their rigidity first-hand.

Special date and time: May 1, 1:15pm in WH-100E (joint with combinatorics)
Speaker: Boris Bukh (Carnegie Mellon)
Title: Topological Version of Pach's Overlap Theorem

Abstract: Consider the collection of all the simplices spanned by some n-point set in $\mathbb{R}^d$. There are several results showing that simplices defined in this way must overlap very much. In this talk I focus on the generalization of these results to 'curvy' simplices.

Specifically, Pach showed that every $d+1$ sets of points $Q_1, \ldots, Q_{d+1}$ in $\mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. In joint work with Alfredo Hubard, we show, by means of an example, that a topological extension of Pach's theorem does not hold with subsets of size $C(\log n)^{1/(d-1)}$. We show that this is tight in dimension 2, for all surfaces other than $S^2$. Surprisingly, the optimal bound for $S^2$ is $(\log n)^{1/2}$. This improves upon results of Bárány, Meshulam, Nevo, and Tancer.

May 3
Speaker: Jamie Conway (UC Berkeley)
Title: Classifying Contact Structures on Hyperbolic 3-Manifolds

Abstract: Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but notably not on any hyperbolic manifolds. In this talk, I will discuss a new classification result on an infinite family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot. This is joint work with Hyunki Min.

Robert Connelly (Cornell)

2020-01-29T14:03:07-04:00

Robert Connelly (Cornell)

Comments on Generalized Heron Polynomials and Robbins' Conjectures

Abstract for the Combinatorics Seminar 2004 March 29

In High School we learn Heron's formula for the area of a triangle in terms of the lengths of its three sides. Less well known is a very similar formula, due to Brahmagupta, for the area of a cyclic quadrilateral in terms of the lengths of its four sides. (A polygon is cyclic if its vertices lie on a circle.) In both cases the square of 4 times the area is a polynomial of the square in the lengths of its edges. David Robbins showed that for any cyclic polygon with n edges the square of 4 times its area satisfies a polynomial whose coefficients are themselves polynomials in the edge lengths, and he calculated this polynomial for n = 5 and n = 6. He conjectured the the degree of this polynomial for all n, and recently Igor Pak and Maksym Fedorchuk have shown that this conjecture of Robbins is true. We have no comments about that proof. But Robbins also conjectured that his polynomial is monic, and that is what will be shown, along with comments about a proof of this and related results in a paper by Varfolomeev.

Robert Connelly (Cornell)

2020-01-29T14:03:07-04:00

Robert Connelly (Cornell)

Tensegrities: Geometric Structures Suspended in Midair

Abstract for the Combinatorics and Geometry/Topology Seminars 2018 April 24

Suppose you have a finite collection of points in Euclidean space or the plane. Some pairs are connected by inextendible cables, others by incompressible struts, and some by fixed length bars. The artist Kenneth Snelson constructed several large structures, made of cables and bars, that hold their shape under tension, where the struts appear to be suspended in midair. Buckminster Fuller, the architect and inventor, called them “tensegrities” because of their “tensional integrity”. But why do they hold their shape? There is a very simple principle using quadratic energy functions that provides the key to their stability. I will show a catalog of highly symmetric tensegrities, created with the help of a little bit of representation theory, as well as tangible models, where you can feel their rigidity first-hand.

Robert Connelly (Cornell)

2020-01-29T14:03:07-04:00

Robert Connelly (Cornell)

The Kneser-Poulsen Conjecture in the Plane

Abstract for the Colloquium 2005 May 5

If a finite set of disks in the plane is rearranged so that the distance between each pair of centers does not decrease, then the area of the union does not decrease, and the area of the intersection does not increase. This very basic geometric property of the Euclidean plane was conjectured by Kneser and Poulsen in the 1950's and described in Chapter 3 of Klee and Wagon's book on unsolved problems in plane geometry and number theory. The proof with Károly Bezdek not only provided a solution to the problem in the plane, but also introduced at least three new techniques for this and related Kneser-Poulsen type problems. There are several related problems and extensions to higher dimensions that are still open, including the original Kneser-Poulsen problem for dimensions greater than 2.