SUMS 2015: Math and Symmetry

Faculty Talks

Regular Polyhedra

Jeffrey Adams, University of Maryland

One of the famous results from ancient Greek mathematics is that there are exactly 5 regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron and icosahedron. These object have high degrees of symmetry, and their classification can be viewed as a result in group theory. In four dimensions there are also 5 regular polyhedra, and in dimensions 6 and above there are only 3. These are examples of Coxeter groups, a ubiquitous family of symmetry groups with many beautiful properties.

Continuous Symmetry, Shrinky Dinks, and the Poincaré Conjecture

Jeffrey Brock, Brown University

When we think of symmetry, we often think of objects with finite symmetry, such as polygons or snowflakes. Continuous symmetry can be even more striking, and even useful: a wheel is "rotationally" symmetric, and if you've tried to drive on a flat tire you quickly realize the importance of that symmetry!

Continuous symmetry plays a vital role in topology as well: each Riemann surface has a most symmetric geometry, and recently Grisha Perelman proved that William Thurston's conjecture that three dimensional spaces (manifolds) can be cut into pieces each of which has a symmetric kind of structure. This symmetric geometry is found by putting the manifold "in the oven" of Ricci flow - think of heating saran wrap with a hairdryer until it smoothes out. Like shrinky dinks, three manifolds find their optimal shape, and in the case of a manifold where every loop is contractible, this optimal shape is the round 3-sphere - this proves the famous Poincaré Conjecture. In this talk, I'll discuss the idea of continuous symmetry and its role in topology.

The Role of Symmetry in Sphere Packing

Henry Cohn, Microsoft/MIT

In this talk, we'll examine the sphere packing problem: how can we arrange congruent balls to cover as large a fraction of space as possible if they aren't allowed to overlap? In particular, we'll focus on the role of symmetry. In high dimensions, should we expect the best sphere packings to be ordered or disordered? This problem is relevant to mathematics, physics, and computer science, but the answer remains a mystery.

Symmetry: Where does it come from? Where does it go?

Marjorie Senechal

"The mathematical study of symmetry," says MathWorld, "is systematized and formalized in the extremely powerful and beautiful area of mathematics called group theory." Or is it? From flowers to crystals to bipeds like ourselves, symmetry's origins are mysterious and its behaviour dynamic, posing mathematical questions beyond group theory.

Student Talks

Coloring Techniques for Pattern Avoidance over an Infinite Sequence

Robert Laudone, Boston College

We investigate Grytzcuk’s conjecture and extend the entropy compression method to prove that long-square-free sequences (length greater than or equal to 6) can be chosen from lists of size 3. Within this proof we utilize and independently discover a bijection between plane trees and difference sequences (sequences of integers generated when running an algorithm). We also investigate Omega Sets, utilizing new strategies to surpass the results of entropy compression. We then expand known work on shuffle squares to apply to shuffle long-squares and provide a general formula for further expansion.

Counting Faces of Colorful Associahedra

Kyle Meyer, Northeastern University

The classical associahedra can be formulated in terms of flipping the diagonals of triangulations of convex polygons. Similarly the colorful associahedra, an abstract polytope, introduced by Araujo-Pardo, Hubard, Oliveros, and Schulte, is formulated in terms of flipping diagonals of triangulations whose diagonals are colored (colored triangulations). In this talk we will give a modified formulation of the colorful associahedra in terms of partial colored triangulations, and using this formulation we will count the number of faces of the colorful associahedra by dimension.

Cluster Algebra Symmetries in Scattering Amplitudes

Daniel Parker, Brown University

Cluster Algebras are a brand new field of mathematics, first introduced in 2002. Since then, cluster algebras have turned up across dozens of areas across mathematics, leading to many exciting connections. This talk will introduce cluster algebras with a visual approach and then describe one such connection: how cluster algebras can be used to describe the symmetries of scattering amplitudes in a particular Quantum Field Theory.

The K-Knuth Equivalence Relation

David Schwein, Brown University

Pairing-based cryptography, which takes advantage of the algebraic properties of elliptic curves, lies at the foundation of modern cryptosystems. In this talk, I will give an overview of the basic concepts and algorithms used in elliptic curve cryptography and I will discuss a new cryptosystem made for use in elections.

The 1-Dimensional Diagonal and the Mirror Pentagram Map

Zijian Yao, Brown University

The pentagram map sends a polygon in the projective plane to another polygon in the projective plane, and has been studied in a series of papers by Schwartz and others. Schwartz showed that an axis-aligned polygon collapses to a point under certain iterations of the pentagram map. Since then, there has been generalizations of the pentagram map in higher and lower dimensions. However the lower dimensional generalization was obtained by algebraic relations other than than geometric ones. In this talk, we will define a correct one-dimensional generalization of the pentagram map using very simple methods of symmetry. In particular, we construct an analogous notion of diagonals in dimension 1. We also discuss certain related problems of the point of collapse.

Student Poster Presentations

Characterization of Unconfinable Hexagonal Cells

Katherine Pilewski, Boston College

We address an open question posed by Alfeld, Piper, and Schumaker in 1987 regarding the characterization of unconfinable cells. For cells with 6 interior edges, we obtain a geometric characterization of confinability in terms of cross-ratios. This characterization allows us to show that a hexagonal cell in which the diagonals intersect at the interior vertex is unconfinable if and only if the lines containing opposite edges and the diagonal through the remaining points are either parallel or are concurrent.

Incorporation of PPI Information into a Statistical Association Study for Exome Sequencing Data

Dongni Zhang, Worcester Polytechnic Institute

Statistical association studies have contributed significantly in the detection of novel genetic factors associated with complex diseases. However, there are various challenges in solving the missing heritability issue solely depending on statistical evidence of the association between genotype and phenotype, especially for sequencing data. Incorporation of biological information that reflects the complex mechanism of disease development is likely to increase the power of association studies for detecting novel disease genes. In this study, we develop a statistical framework for association studies that integrates the information of the functional effect of SNPs to the disease related protein- protein interactions. The method is applied to GAW19 exome sequencing data of uncorrelated individuals for detecting novel genes associated to hypotension. Based on both the real and simulated phenotypes of hypertension, the method is compared with multiple well-known association tests for sequencing data.