Geometry and Topology Commons^™

Complex Multiplication Symmetry Of Black Hole Attractors, Monika Lynker, Vipul Periwal, Rolf Schimmrigk

Faculty Articles

We show how Moore’s observation, in the context of toroidal compactifications in type IIB string theory, concerning the complex multiplication structure of black hole attractor varieties, can be generalized to Calabi-Yau compactifications with finite fundamental groups. This generalization leads to an alternative general framework in terms of motives associated to a Calabi-Yau variety in which it is possible to address the arithmetic nature of the attractor varieties in a universal way via Deligne’s period conjecture.

Open Problems From Cccg 2002, Erik D. Demaine, Joseph O'Rourke

Computer Science: Faculty Publications

No abstract provided.

The Cohomology Of The Steendrod Algebra And Representations Of The General Linear Groups, Nguyen H. V. Hu'ng

Mathematics Research Reports

Let Tr_k be the algebraic transfer that maps from the coinvariants of certain GL_k-representation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer tr_k : pi_*^S((B[doublestrike V]_k)_+) --> pi_*^S(S^0). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Tr_k is an isomorphism for k = 1, 2, 3 and that T_r = ⊕_k(Tr_k) is a homomorphism of algebras.

In this paper, we first recognize the phenomenon that if we start from any degree d, and apply Sq^0 repeatedly at most (k- 2) …

Consensus-Halving Via Theorems Of Borsuk-Ulam And Tucker, Forrest W. Simmons, Francis E. Su

All HMC Faculty Publications and Research

In this paper we show how theorems of Borsuk-Ulam and Tucker can be used to construct a consensus-halving: a division of an object into two portions so that each of n people believes the portions are equal. Moreover, the division takes at most n cuts, which is best possible. This extends prior work using methods from combinatorial topology to solve fair division problems. Several applications of consensus-halving are discussed.

The Connective K-Theory Of Finite Groups, Robert R. Bruner, John Greenlees

Mathematics Faculty Research Publications

This paper is devoted to the connective K homology and cohomology of finite groups G. We attempt to give a systematic account from several points of view. In Chapter 1, following Quillen [50, 51], we use the methods of algebraic geometry to study the ring ku^*(BG) where ku denotes connective complex K-theory. We describe the variety in terms of the category of abelian p-subgroups of G for primes p dividing the group order. As may be expected, the variety is obtained by splicing that of periodic complex K-theory and that of integral ordinary homology, however the way these parts fit …

Partitioning Regular Polygons Into Circular Pieces I: Convex Partitions, Mirela Damian, Joseph O'Rourke

Computer Science: Faculty Publications

We explore an instance of the question of partitioning a polygon into pieces, each of which is as “circular” as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a …

Computational Geometry Column 44, Joseph O'Rourke

Computer Science: Faculty Publications

The open problem of whether or not every pair of equal-area polygons has a hinged dissection is discussed.

On The Development Of The Intersection Of A Plane With A Polytope, Joseph O'Rourke

Computer Science: Faculty Publications

Define a “slice” curve as the intersection of a plane with the surface of a polytope, i.e., a convex polyhedron in three dimensions. We prove that a slice curve develops on a plane without self-intersection. The key tool used is a generalization of Cauchy's arm lemma to permit nonconvex “openings” of a planar convex chain.

A Few Weight Systems Arising From Intersection Graphs, Blake Mellor

Mathematics, Statistics and Data Science Faculty Works

No abstract provided.

A Geometric Interpretation Of Milnor's Triple Invariants, Blake Mellor, Paul Melvin

Mathematics, Statistics and Data Science Faculty Works

Milnor's triple linking numbers of a link in the 3-sphere are interpreted geometrically in terms of the pattern of intersections of the Seifert surfaces of the components of the link. This generalizes the well known formula as an algebraic count of triple points when the pairwise linking numbers vanish.

Boundary Volume And Length Spectra Of Riemannian Manifolds: What The Middle Degree Hodge Spectrum Doesn't Reveal, Carolyn S. Gordon, Juan P. Rossetti

Dartmouth Scholarship

No abstract provided.