Discrete Mathematics Day At WPI

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Schedule and Abstracts:

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• 9:15-9:55 am -- Light Breakfast, Salisbury Labs Lounge, Main Floor
• 9:55-10:00 am -- Welcome and Announcements
• 10:00-11:00 am -- Edwin van Dam, Tilburg University, the Netherlands

Title: Eigenvalues and Structure in Graphs
Abstract: The eigenvalues of the adjacency matrix of a graph contain a lot --- but not always all --- information on the structure of the graph. We will review some structural properties that can be derived from the eigenvalues of a graph, and discuss when a graph is determined by its spectrum (of eigenvalues), or how different graphs with the same spectrum can be constructed.

In the second part of the talk, we will dive deeper into graphs that have a lot of combinatorial symmetry: distance-regular graphs. We will see how systems of orthogonal polynomials can help to recognize such graphs from their eigenvalues and a little extra information.

We apply this to obtain a recent characterization of the generalized odd graphs by their number of distinct eigenvalues and the length of their shortest odd cycles.

If time permits, we also show how the above led to the construction of the twisted Grassmann graphs, a family of distance-regular graphs that have the same spectrum as certain Grassmann graphs, but that are not vertex-transitive.

• 11:10-11:50 am -- Elizabeth Hartung, Syracuse University

Title: Fullerenes and Carbon Molecules
Abstract: A fullerene is a plane trivalent graph consisting of pentagonal and hexagonal faces. We will explore the connections between chemical properties of carbon molecules and the mathematical properties of fullerenes. We will then discuss the face independence number of different classes of fullerenes.

• 12:00-2:00 pm -- Lunch and Discussion Break
• 2:00-3:00 pm -- Tim Penttila, Colorado State University

Title: Incidence geometry from Klein's point of view
Abstract: Imre Lakatos, in his classic text, Proofs and Refutations, advocated teaching mathematics via a rational reconstruction of its history. This talk attempts to explain the major themes of incidence geometry from Felix Klein's point of view - as transformation geometry via groups. (Incidence geometry was born when Hilbert reorganized the foundations of geometry in 1899.)

The Jordan-Hölder theorem says that every finite group is composed of simple groups: a finite group is analogous to a molecule, and the atoms out of which that molecule is built are simple groups. The classification of finite simple groups says that every finite simple group is cyclic of prime order, an alternating group of degree at least five, a group of Lie type or one of 26 explicitly listed sporadic groups. This is the periodic table - vaguely stated, it says that most finite simple groups are of Lie type.

Groups of Lie type arise from Lie algebras, and have an associated Lie rank. That the major families of incidence geometry (projective, polar, parapolar and metasymplectic spaces and generalized polygons) arise from groups of Lie type of Lie rank at least two has been known for a long time, but this connection will be sketched in the talk.

The novelty of the talk is to use groups of Lie type of Lie rank one to produce most of the remaining themes of incidence geometry in a rational reconstruction of the subject's history a la Lakatos. For example, the study of ovals, ovoids, spreads, unitals, maximum distance separable error-correcting codes, BLT-sets, flocks all could have arisen as an outgrowth of the study of subgroups of Lie type of Lie rank one of groups of Lie type of Lie rank at least two.

This provides a unifying perspective on a diverse collection of topics, giving an aerial survey of the last fifty years of incidence geometry.

• 3:00-3:30 pm -- Coffee Break
• 3:30-4:10 pm -- Karola Mészáros, MIT

Title: Triangulations of root polytopes and subdivision algebras
Abstract: A type A_n-1 root polytope is the convex hull in Rⁿ of the origin and a subset of the points e_i - e_j, 1 ≤ i < j ≤ n . A collection of triangulations of these polytopes can be described by reduced forms of monomials in an algebra generated by n² variables x_ij, for 1 ≤ i < j ≤ n. In a closely related noncommutative algebra, the reduced forms of monomials are unique, and correspond to shellable triangulations whose simplices are indexed by noncrossing alternating trees. Using these triangulations Ehrhart polynomials are computed. Special cases of our results prove several conjectures of Kirillov regarding certain quadratic algebras.

• 4:20-5:00 pm -- Jason Williford, University of Wyoming

Title: Examples of Cometric Schemes
Abstract: An association scheme can be thought of as collection of graphs which mimic the combinatorial properties of the orbitals of a transitive group. The adjacency matrices of these graphs span a matrix algebra known as a Bose-Mesner algebra which is also closed under the Schur product. A scheme is called cometric if, after suitably reordering the idempotents, the i-th idempotent is a degree i polynomial of the first idempotent. In this talk we will describe some of the more recently discovered examples of cometric schemes, with particular emphasis on those arising from finite geometry. Some background material on the problem will be provided, as well as some open problems.

• 5:30-8:00 pm -- Reception at Higgins House
• free to all conference participants
• hot and cold hors d'œuvres
• non-alcoholic beverages
• cash bar

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