History of the Seminar

Abstract: This talk is aimed at graduate students. I'll introduce property T (due to D.Kazhdan) and discuss some basic properties. Many examples will be provided.

Abstract: In this talk I focus on a particular type of discrete dynamical system defined over an underdirected base graph. In some cases, a morphism of this graph can lead to a natural reduction of the dynamical system defined over it. I will present necessary and sufficient conditions for this to happen, give some examples, and will discuss some interesting open questions.

Abstract: The first explicit family of expanders have been found by G.Margulis, using property T. Since then several more examples have been discovered. I'll discuss some old and recent examples. Hopefully, I'll provide some pictures too. Only some basic notions of group theory and graph theory is required.

Abstract: The theory of Simplicial Non Positive Curvature (SNPC) was developed by Tadeusz Januszkiewicz and Jacek Swiatkowski. SNPC is a combinatorial condition on simplicial complexes that is similar to the metric non positive curvature condition. SNPC and traditional non positive curvature have many similar properties, but are distinct. Neither condition implies the other in general.

In this talk I will give an introduction to the theory of SNPC including examples of SNPC spaces and some of the properties associated with them.

Abstract: What makes 2 and 3 different from all other primes? Why is it that they are the only primes that divide the order of EVERY simple non-abelian finite group? At least part of the answer lies in the structure of projective geometries and the torision elements inside GL₂(Z). This talk will be elementary and accessible to graduate students.

Abstract: Many real-world markets are subject to network constraints, and can naturally be modeled over a graph, where the goods are to be delivered as a concurrent feasible network flow. Simple examples of this are electricity, oil, and water. In this talk, I will give a purely mathematical definition of market power in this setting, show how it arises, and how it behaves under collusion. I will discuss the market power spectrum, which describes the role that the topological constraints play in the market. Finally, I will present some experimental results using the Portland electrical grid and show how they match the theory, some interesting trends that arise, and what this all implies. This is an ongoing research project that began at Los Alamos National Laboratory, and now continues at the Virginia Bioinformatics Institute.

Abstract: Every finite simple group is either cyclic, alternating, of Lie type or sporadic. The first three are collections of infinite families, and the sporadic ones are the 26 exceptions that don't fit elsewhere. The first sporadic groups to be discovered were the 5 Mathieu groups. In this talk, I'll introduce the largest of the Mathieu groups, M₂₄, (and its 244 million elements) via some very pretty finite projective geometry. We will also answer the question how does one find a collection of 8 element subsets of {1,2,...,24} so that every 5 element subset is contained in exactly one of the 8 element subsets. Collections of this type are called Steiner systems, and this one is an S(5,8,24). This particular collection is crucial to the construction of the Mathieu group M₂₄

Abstract: Simple groups have been (almost) classified in the theory of finite groups, and in the theory of Lie Groups and Algebraic Groups. Moreover, simple groups form building blocks of the structure theory of groups mentioned above. In the context of finitely generated groups, simple groups seem less important, nevertheless, in recent years (decades) some very interesting examples of such groups have been found. I will describe some examples of finitely generated(presented) infinite simple and almost simple groups.

Abstract: A group is called just-infinite if it is infinite but all proper homomorphic images are finite. Arithmetic lattices in simple Lie groups are one of the major sources of such groups. Another source is the class of so-called branch groups. I will discuss a structure theorem due to Wilson and Grigorchuk. The talk is aimed at graduate students.

Abstract: This talk will be a gentle introduction to several specific highly symmetric graphs (such as Heawood's graph and the Tutte-Coxeter graph) as well as to the general notion of a diagram (or Buerkenhout) geometry. Potential applications include spherical buildings for finite groups of Lie type, geometries for the sporadic groups, and a one-relator Coxeter presentation for the Monster finite simple group.

Abstract: (Continuation of the previous talk) This talk will be a gentle introduction to several specific highly symmetric graphs (such as Heawood's graph and the Tutte-Coxeter graph) as well as to the general notion of a diagram (or Buerkenhout) geometry. Potential applications include spherical buildings for finite groups of Lie type, geometries for the sporadic groups, and a one-relator Coxeter presentation for the Monster finite simple group.

Abstract: The girth of graphs is an old notion. Finite graphs with large girth and small chromatic number are interesting to mathmeaticians and computer scientists. The girth of groups is a new notion. I will discuss some results. The talk is aimed at graduate students.

Abstract: Let G be a finite group, and d=d(G) be the minimal number of generators. For any natural number k>= d(G) one can define a Schreier graph S(k;G). The graph S(d+1;G) is particularly interesting. It has been conjectured by R.Graham that for finite simple groups S(3;G) is always connected (it is known that every finite simple group H is generated by 2 elements, so d(H)+1=3). This conjecture of R.Graham is still widely open. It is proved only in the case of SL_2 over a finite field. Interestingly, the same question remains open also for arbitrary finite groups. I will discuss some recent results.

Abstract: Let G be a finite group, and d=d(G) be the minimal number of generators. For any natural number k>= d(G) one can define a Schreier graph S(k;G). The graph S(d+1;G) is particularly interesting. It has been conjectured by R.Graham that for finite simple groups S(3;G) is always connected (it is known that every finite simple group H is generated by 2 elements, so d(H)+1=3). This conjecture of R.Graham is still widely open. It is proved only in the case of SL_2 over a finite field. Interestingly, the same question remains open also for arbitrary finite groups. I will discuss some recent results.

Abstract: Quantum calculus is ordinary calculus without taking limits. The talk will be a brief overview of this perspective will an emphasis on the various combinatorial objects that arise naturally along the way.

Abstract: Classical computers perform logical operations on strings of bits; quantum computers act by unitary operators and projections on vectors in a Hilbert space. Quantum logic, roughly speaking, is one attempt to create something analogous in form and function to propositional boolean logic, but for quantum computation. "Propositions" are replaced with "closed subspaces", "and" is replaced with "intersection", "or" is replaced with "(closure of) span of union", and "not" is replaced with "orthogonal complement". The resulting logic behaves in some ways like a classical logic but in other ways not. For example, the distributive laws do not hold.

In this talk I'll give a little bit of motivational background, define quantum logic on a finite dimensional vector space, say a wee bit about decidability, discuss some notions that can be expressed, and show that the logic can in a specific sense detect the dimension of the underlying vector space. I won't assume any background in logic or physics; the talk should be accessible to anyone who has had a course in linear algebra.

The talk is based on joint work with Michael Dunn, Larry Moss, and Zhenghan Wang.

Abstract: Suppose one has a sequence of homomorphisms from a fg group G into the group of isometries of a Gromov negatively curved space. If there is an element of g whose translation length goes to infinity then a subsequence converges to an action of G on an R-tree T. If the translation length of every element for the action on T is always an integer then T is simplicial. We will explain an application to representions of G into SL(2,C) involving valuations, Newton polygons and tropical geometry if time permits.

Abstract: Two n-node binary trees are related by a sequence of local restructuring called rotations. The set of n-node binary trees is a metric space with the distance between two trees is the minimum number of rotations needed to transform one into the other. In this talk I will discuss a paper of Sleator, Tarjan and Thurston giving a bound on the maximum rotation distance between two n-node trees of 2n - 6 for n > 10. The proof relies on relating rotation distance to the volume of a polyhedron in hyperbolic 3-space.

Abstract: The quasi-isomteric rigidity of lattices in semisimple Lie groups has been proved in the late 90s thanks to the efforts of many mathematicians. The same question for lattices in solvable Lie groups, however, remains open. In the solvable case, the strong rigidity (or Mostow rigidity) and quasi-isometric rigidity seem much less related compared with the semisimple case. I'll mention and discuss some very recent results.

Abstract: This will be a quick introduction to the notion of polytope duality. If you think that finding the dual of a polytope is as easy as picking a vertex from each facet and taking a convex hull, then this talk is for you. At the end of the talk, the groundwork will be laid for a discussion of non-rational polytopes later in the quarter through the introduction of the formalism of oriented matroids.

Abstract: This talk is aimed at graduate students. I will discuss the wellknown "ping-pong argument", and using it, demonstrate the Tits Alternative for some classes of groups (complete proof for hyperbolic groups will be given). I also will mention some interesting (and elementary) open problems.

Abstract: After introducing/reviewing the elementary notions of affine and projective geometry, I will discuss their finite versions and some of applications to the classification of finite simple groups and the construction of the octonions.

Abstract: If you're a bit rusty on Hamilton's quaternions (or have never encountered them) and would like to brush up on them in preparation for Atiyah's colloquium on Tuesday, then today's seminar is for you. We will be discussing the close connection between the quaternions and the rotations of the 2-sphere.

Abstract: This talk is aimed at graduate students. I will prove the famous Erdos-Szekeres Theorem and discuss some very interesting generalizations of it. Let me remind the classical Erdos-Szekeres Theorem: for any natural number n, there exisits a number g(n) such that for any natural number m not less than g(n), a finite set of m points in the plane, located in a general position, contains a subset of n points which are the vertices of a convex n-gon.

Abstract: Take a set of triangles, squares or pentagons and identify three around each vertex. This gives a construction of the Platonic solids: a regular tetrahedron, a cube and a regular dodecahedron. If you do the same with a set of regular hexagons and choose the proper side identifications, you get a hexagonal tiling of a torus. What if you try this with regular heptagons?

With proper choices, the surface you get is Klein's Quartic Curve. In this talk I will discuss a constrction of this highly symmetric object and some of it's properties, including its relation to the Fano Plane.

Abstract: In this talk we'll discuss the following topics:
1) the simple closed curves on a closed orientable surface S,
2) the curve complex of S,
3) the mapping class group of S, and
4) how the mapping class group acts on the curve complex.

Abstract: Essential simple closed curves on surfaces are important tools in the study of surfaces and their fundamental groups. The most basic homeomorphism invariant of two curves on a surface is their intersection number. This number is, in fact, an algebraic property of the fundamental group of the surface. In an arbitrary group the analogue of a simple closed curve is a splitting of the group as an amalgamated product or HNN extension. Peter Scott defined an intersection number for two splittings of a group. In this talk I will present his definitions and give lots of topological motivation. I will also state several applications (due to Scott, Swarup, and others) of the intersection number to group theory and 3-manifold topology.

Abstract: In this talk I will define intrinsic knotting and intrinsic linking, explore which graphs are knotted and linked, and discuss how the 2 notions are related to one another. I will also mention some work I did on the subject as an undergrad.

Abstract: Everyone knows that the sum of the angles in a Euclidean triangle is pi, but what can be said about the sum of the dihedral angles in an Euclidean tetrahedron? Elementary examples show that the answer is not constant. In this talk I will present a very elegant generalization of the triangle angle sum formula to arbitrary polytopes due to Peter McMullen. If there is enough time, I will also discuss the 2-dimensional and n-dimensional combinatorial Gauss-Bonnet theorem.

Abstract: This is an introductory talk. A group acts on a tree without a global fixed point iff the group is a non-trivial amalgamated free-product. We will talk about some cute examples of this.

Abstract: Recently, Professor D.Cooper gave a talk on aperiodic tilings of the Euclidean plane(Penrose tiling). I will discuss aperiodic tilings of the hyperbolic plane by sketching the paper of G.Margulis and Sh.Mozes.

Abstract: This will be a modified, but quite different version of a talk I gave a year and a half ago in the student seminar, and it should be widely accessible to all. We say that two sets are equidecomposable if one can be partitioned and rearranged via isometries into the other one. The most famous such problem is the celebrated Banach-Tarski paradox. Equally impressive is Laczkovich's recent sucess of "squaring the circle", which he proved in a 40-page paper using high-powered number theoretical techniques, and roughly 10^50 cuts. One may also add the restriction that the decomposition must be into polygons or straight-line cuts. I will discuss numerous decomposition problems, and present at least a sketch of the proofs of several of them, making use of vastly different techniques. First, I'll prove a pure mathematical decomposition problem by using nothing more than the physical properties of resistors -- Ohm's law and Kirchhoff's theorem. Next, I'll present a proof of the Bolyai-Gerwien theorm using geometric invariants, and show how Max Dehn made clever use of the axiom of choice to extend this argument and resolve Hilbert's third problem. Finally, I'll show how these techniques might be used to solve some open problems about polygon decomposition, number theory, and magic squares.

Abstract: The Catalan numbers are a sequence of natural numbers that occur in various counting problems in combinatorics. The n-th Catalan number is defined by C_n=(2n)!/(n+1)! n! (n>=0) and the first few are (1, 2, 5, 14, 42, 132,  429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, ...). They are named after the Belgian mathematician Eugène Catalan (1814--1894).

These numbers for example describe the number of ways a polygon with n+2 sides can be cut into n triangles by means of non-intersecting diagonals, or the number of ways in which parentheses can be placed in a sequence of numbers to be multiplied, two at a time, or the number of rooted, trivalent trees with n+1 nodes, as well as the number of paths of length 2n through an n-by-n grid that do not rise above the main diagonal.

Perhaps one of the most interesting occurences of these numbers is their relevance to the solution of the Ballot Problem: Suppose A and B are candidates for office and let A receive a votes, and B, b votes, with a > b. Assuming the random ordering of the votes, what is the probability that A always stays ahead of B as the votes are being counted? The solution turns out to be (a-b)/(a+b), as was first shown by M. Bertrand  and then more elegantly by D. Andre' in 1887.

In this talk we will outline several properties of the Catalan Numbers and show their relevance in a few of the above-mentioned problems as well as to the Ballot Problem.

Abstract: This group was introduced by Richard Thompson (who is not a mathematician). Like Grigorchuk's group, or SL(n,Z), it arises very naturally, nevertheless answers many difficult questions in combinatorial group theory. I will mention some open problems which might interesting to graduate students.

Abstract: The finite groups generated by reflections that act faithfully by isometries on Euclidean space have long been classified. These groups are the finite Coxeter groups and they include the symmetric groups. In the fall, Aaron Galbraith described two classes of polytopes in the seminar are (secretly) closely connected to the symmetric group: permutahedra and associahedra. In this talk I will describe how these these constructions can be generalized to arbitrary finite Coxeter groups. The generalization of the associahedra, in particular, is only about 5 years old (due to Fomin and Zelevinsky) and it arises in their invention of cluster algebras. The particular elementary construction of associahedra that I will discuss is even more recent.

Abstract: Viewing the simple n-associahedron as an object describing the Catalan-many triangulations of the (n+3)-gon, I will explain how a combination of labels and relations gives a correspondence between the simple n-associahedron and an n-dimensional cluster algebra. Fun take-home diagrams for all who attend.

Abstract: In the first talk I briefly described (with basic examples) representation theory, Lie algebras, Hopf algebras and quantum groups. This second talk will be an exposition of the path model of Littelmann, which gives a simple decomposition formula for tensor products of irreducible representations. According to Littelmann, "only some basic knowledge in weight theory and in the combinatorics of Weyl groups is required", and hopefully there will be enough time to cover these prerequisites AND the main item of interest this time. Crystal bases were the main motivation behind Littelmann's constructions, but they require even more prerequisites, and these will most likely be covered in an upcoming algebra seminar.

Abstract:

Abstract: In topology, the Seifert-Van Kampen theorem is a useful tool for expressing the fundamental group of a space in terms of fundamental groups of subspaces. The analogous operation in group theory is expressing a group as an amalgamated product or HNN-extension. After a brief review of these concepts, we will define the notion of a "graph of groups" and explain how Bass-Serre theory gives a characterization of those groups which can be decomposed as a graph of groups. Fundamental groups and covering space theory will turn out to be key to developing an intuition for the beautiful relationship between groups, graphs, and trees.

Abstract: Dynkin diagrams are easily classified finite graphs that satisfies certain properties that arose out of Lie theory, but since then have appeared in many seemingly non-related areas of mathematics. I will present them first as the solutions to a simple combinatorial problem, and also as the precise set of graphs that admit a particular positive definite quadratic form. Time permitting, I will discuss their relevance to the classification of semisimple Lie Algebras, regular polytopes, representations of finite type, and finite Weyl groups.

Abstract: Associated to any finite reflection group, there is a combinatorial object called the lattice of noncrossing partitions. In type A, these are just the classical noncrossing partitions. In this talk, I will discuss a new approach to the lattice of noncrossing partitions for simply-laced finite reflection groups, using the representation theory of quivers. This yields a new proof that the noncrossing partitions do indeed form a lattice for these groups. This approach also clarifies and makes precise connections between noncrossing partitions, Cambrian lattices, and clusters. I hope eventually to apply these results to understand the noncrossing partitions in affine types, where much less in know. This is work in progress, joint with Colin Ingalls.

Abstract: This will be an introductory talk aimed at graduate students with no prior knowledge of combinatorics or discrete geometry. The goal will be to give a quick tour of the types of topics and ideas which arise in the more combinatorially oriented aspects of geometry and the more geometrically oriented aspects of combinatorics. One particular emphasis will be to highlight the main objects of study in geometric group theory and in geometric combinatorics. I've been informed by a (reliable) friend of mine that Australians call these types of whirlwind tours, Tiki tours.

Abstract:I will discuss several definitions of hyperbolic spaces (and groups) and mention some of their basic properties. Many examples of hyperbolic groups will be given.

Abstract: (see above)

Abstract: The concept of a Finite State Automaton (FSA) is a useful tool in the study of specfic groups by providing an efficient tool for doing compuitations on the set of words on the generating sets. In this talk I will define a finite state automata and give some examples of their applications. In particular I will discuss the study of Automatic Groups.

Abstract: In this talk I will introduce the two polyhedra in the title. The first has vertices in one-to-one correspondence with the permutations of a finite set and the second has vertices in one-to-one correspondence with the ways to associate a finite list. Lots of colored pictures will be drawn.

Abstract: Classical noncrossing partitions are partitions p such that a certain planar diagram of p has non-intersecting parts. These form a lattice under refinement order. Algebraically, noncrossing partitions relate to factorizations of an n-cycle in Sn into transpositions. The construction can be generalized to an arbitrary finite Coxeter group W (i.e. a finite group generated by reflections) by considering factorizations of a Coxeter element c into reflections. One obtains a "noncrossing partition lattice" for W which is instrumental in constructing finite K(pi,1) spaces and monoid structures for the Artin group associated to W.

In this talk we study noncrossing partitions by way of the Coxeter plane, a certain plane fixed (as a set) by the action of c. We give a general recursive formula for counting maximal chains in noncrossing partition lattices. We also show how the planar diagrams for classical noncrossing partitions and their counterparts of types B and D arise from a uniform construction, rather than in an ad hoc manner and discuss this construction in the case of the exceptional groups.

The talk will begin with a brief review of noncrossing partitions and Coxeter groups.

Abstract: Given a complete graph on n-vertices (an n-clique), a 2-coloring of the graph is an assignment of either "red" or "blue" to each edge. The Ramsey number of p and q, denoted R(p,q), is the smallest positive integer n such that a two coloring of an n-clique results in either a red p-clique or a blue q-clique. I will talk about the calculation of Ramsey numbers, bounds of Ramsey numbers, and the odd fact that R(p,q) exists, but R(7,7) still is not determined.

Also, I will have baked goods.

Abstract: The purpose of this talk is to give an elementary introduction to crystal basis theory, a powerful combinatorial tool in the study of representations of quantum groups. Along the way I hope to mention (and briefly describe, with basic examples) representation theory, Lie algebras, Hopf algebras and quantum groups, crystal bases, and Littelmann paths, although this plan may turn out to be a bit too ambitious for a one-hour talk.