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Events for October 2008 – February 2009

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Oct. 07, 2008 1pm (Math 350)	PDE/Analysis Arlan Ramsay X There are several sets of axioms for both Euclidean and hyperbolic plane geometry. If the axioms in each such set are combined, we could say that we have definitions of Euclidean or hyperbolic planes. My goal is to describe one formulation and how the uniqueness theorems can be proved in that context. In this formulation, the real numbers are taken as a starting point, rather than deriving the real numbers from other properties of planes. The two uniqueness theorems have a number of steps in common, consisting of what are called neutral theorems, i.e., ones that do not depend on any assumption about the number of lines parallel to a given line through a given point. Indeed one of the theorems is that there is always at least one such parallel line. I will only mention some of the theorems more central to the development. Axiomatic geometry, Euclidean and otherwise, part 1
Oct. 07, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X Many finite Coxeter groups occur naturally as groups of automorphisms of polytopes (higher dimensional polyhedra). Some of my recent work considers subcomplexes of these polytopes, that is, spaces obtained by deleting interiors of certain faces. If these subcomplexes are chosen carefully, their homology groups (don't worry, I'll explain this) afford representations of the Coxeter group. I will discuss how one might find the characters of such representations, concentrating mostly on the case of type D. This will be a good excuse to discuss the representation theory of finite Coxeter groups of classical type (A, B, D) over the complex numbers. Homology representations of Coxeter groups I
Oct. 14, 2008 1pm (Math 350)	PDE/Analysis Arlan Ramsay X There are several sets of axioms for both Euclidean and hyperbolic plane geometry. If the axioms in each such set are combined, we could say that we have definitions of Euclidean or hyperbolic planes. My goal is to describe one formulation and how the uniqueness theorems can be proved in that context. In this formulation, the real numbers are taken as a starting point, rather than deriving the real numbers from other properties of planes. The two uniqueness theorems have a number of steps in common, consisting of what are called neutral theorems, i.e., ones that do not depend on any assumption about the number of lines parallel to a given line through a given point. Indeed one of the theorems is that there is always at least one such parallel line. I will only mention some of the theorems more central to the development. Axiomatic geometry, Euclidean and otherwise, part 2
Oct. 14, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X Many finite Coxeter groups occur naturally as groups of automorphisms of polytopes (higher dimensional polyhedra). Some of my recent work considers subcomplexes of these polytopes, that is, spaces obtained by deleting interiors of certain faces. If these subcomplexes are chosen carefully, their homology groups (don't worry, I'll explain this) afford representations of the Coxeter group. I will discuss how one might find the characters of such representations, concentrating mostly on the case of type D. This will be a good excuse to discuss the representation theory of finite Coxeter groups of classical type (A, B, D) over the complex numbers. Homology representations of Coxeter groups II
Oct. 28, 2008 2pm (MATH 350)	Lie Theory Hugh Denoncourt (CU) X Stanley's formula for the number of reduced expressions of a permutation regarded as a Coxeter group element raises the question of how to enumerate the reduced expressions of an arbitrary Coxeter group element. We provide a framework for answering this question by constructing combinatorial objects that represent the inversion set and the reduced expressions for an arbitrary Coxeter group element. It will be shown that this construction generalizes the balanced labellings of Fomin et al. The framework also provides a formula for the length of an element formed by deleting a generator from a Coxeter group element. Some combinatorial models for reduced expressions
Oct. 29, 2008 1pm (Math 350)	PDE/Analysis Lawrence Baggett X Let L be a linear differential operator, and let f be a smooth function with compact support K. Is there a solution to the equation Lu = f? The answer is "no" in general, but the answer to a related question is often "yes." The related question is, does there exist a function g, for which g restricted to K equals f, and for which there is a solution to the equation Lu = g? In this talk I will give some beginning hints about how noncommutative harmonic analysis (representation theory) can be used to show that the answer to the second question is "yes" for certain left-invariant differential operators on Lie groups. Local solvability of certain differential and difference operators
Nov. 04, 2008 2pm (MATH 350)	Lie Theory Matt Ondrus (Weber State) X The module theory of complex finite-dimensional semisimple Lie algebras, such as sl_2, includes several interesting families such as highest weight modules and Whittaker modules. Other algebras, both Lie and associative, have families of modules that are similar to those of sl_2. In this talk, we briefly discuss Whittaker modules for sl_2, and we describe some similar modules for several algebras having a triangluar decomposition. Representations of algebras with a triangular decomposition
Nov. 11, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X For reasons that are not completely clear, the combinatorics of the action of the Coxeter group of type $D6$ on the cosets of certain of its subgroups has applications to hypergeometric series. I will describe some of the combinatorial features that are most useful in applications. The aforementioned applications are collaborative work with M. Formichella, I. Mishev and E. Stade. Some combinatorial properties of the Coxeter groups of type D I
Nov. 18, 2008 2pm (MATH 350)	Lie Theory Richard Green (CU) X For reasons that are not completely clear, the combinatorics of the action of the Coxeter group of type $D6$ on the cosets of certain of its subgroups has applications to hypergeometric series. I will describe some of the combinatorial features that are most useful in applications. The aforementioned applications are collaborative work with M. Formichella, I. Mishev and E. Stade. Some combinatorial properties of the Coxeter groups of type D II
Dec. 01, 2008 2pm (MATH 350)	Lie Theory Dan Daly (DU) X This talk will be an introduction to permutation patterns. A permutation p in Sn is said to contain a pattern permutation q in Sm if there exists a subsequence of p that is order-isomorphic to q. If p does not contain q, then p avoids q. One of the central questions in the study of patterns is: given a pattern p in Sm, how many permutations in Sn avoid p? This question has a beautiful answer for the case m=3 and we will prove the result in this case. When m=4, the problem gets extraordinarily more complicated. We will survey what is known for m=4 and discuss what is still left to be solved. Permutation Patterns
Dec. 02, 2008 2pm (MATH 350)	Lie Theory Rachel Krieger (CU) X Understanding molecular structure is a keystone concept in chemistry. Knowledge of structure precludes understanding molecular interactions, and therefore chemical reactions. Once the composition of a molecule is determined (i.e. how many and what type of atoms comprise a molecule), there are many possible configurations of the atoms. However, only one configuration provides an accurate model of the molecule. Different molecular structures have different groups of symmetries, and certain representations of this group correspond to physical, measurable phenomena. This talk will describe what kinds of representations are physically identifiable, and how such observations lead to an understanding of the symmetry structure of a molecule. Divination of molecular structure through an application of character theory
Dec. 09, 2008 1pm (Math 350)	PDE/Analysis James D. Meiss X Just as one can generate an exact, symplectic map by an implicit generating function, exact volume-preserving maps can be obtained by implicit generating forms. Hector Lomeli and I have shown recently how to generalize this idea, originally due to Carroll, to obtain generating forms of different "types" analogous to the four standard canonical generators. A first step in a theory of transport is computation of the flux though lobes of turnstiles. For the area preserving case, these fluxes are given by differences between the actions of the heteroclinic orbits that bound the lobes. We show how to generalize this to the exact volume-preserving case. In this case the generating forms give rise to an "action" given by an integral along heteroclinic curves. Generating Forms and Flux for Volume Preserving Maps
Dec. 09, 2008 2pm (MATH 350)	Lie Theory Josh Wiscons (CU) X The class of groups of finite Morley rank consists of groups that can be equipped with a (model theoretic) notion of dimension, called Morley rank, that applies to the definable subsets of the group in much the same way that the notion of dimension of an algebraic group applies to its constructible sets. In fact, much effort has been put into the Cherlin-Zil'ber conjecture which states that a group of finite Morley rank which is simple is isomorphic (as an abstract group) to an algebraic group over an algebraically closed field. We will discuss how BN-pairs, which have been very helpful in understanding simple algebraic groups, may be helpful in approaching this conjecture. BN-pairs and groups of finite Morley rank
Feb. 03, 2009 2pm (MATH 350)	Lie Theory Richard Green (CU) X Regular polytopes are the n-dimensional analogs of the Platonic solids. In this talk, I will sketch how the theory of Coxeter groups can be used to classify the regular polytopes. The classification includes higher dimensional analogs of the tetrahedron, the cube and the octahedron, and several exceptional cases. Coxeter groups and regular polytopes I
Feb. 10, 2009 2pm (MATH 350)	Lie Theory Richard Green (CU) X Regular polytopes are the n-dimensional analogs of the Platonic solids. In this talk, I will sketch how the theory of Coxeter groups can be used to classify the regular polytopes. The classification includes higher dimensional analogs of the tetrahedron, the cube and the octahedron, and several exceptional cases. Coxeter groups and regular polytopes II
Feb. 17, 2009 2pm (MATH 350)	Lie Theory Siye Wu (CU) X In this talk, we review Coxeter and dual Coxeter numbers and related concepts. We also establish a relations between these numbers of a simple Lie algebra and its Langlands dual. Coxeter and dual Coxeter numbers of simple Lie algebras
Feb. 24, 2009 2pm (MATH 350)	Lie Theory Nat Thiem (CU) X Random walks on groups have come up in a surprising number of models. If they happen to also "respect" the group's representation theory, these walks become algebraically understandable. This talk explores some of the ideas of what makes an algebraically nice walk, and how one can exploit this knowledge. Nice random walks on groups

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