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Fall 2012 Tutorial
Description: Coxeter groups arise naturally in many different areas of mathematics such as algebra, geometry and representation theory. Despite a relatively simple definition, Coxeter groups turn out to be a rich and beautiful subject which is indispensable in many branches of mathematics. For example, Weyl groups arising from Lie groups form an important class of Coxeter groups and build the heart of the classification of compact Lie groups. Coxeter groups were first introduced by Coxeter in 1934 as a generalization of reflection groups. Reflection groups are - as their name indicates - groups generated by reflections across hyperplanes of Rn which contain the origin. Common examples include symmetry groups of regular polygons or more generally those of regular polytopes. Finite Coxeter groups correspond precisely to finite reflection groups, and it is natural to investigate these groups first. One of our goals will be a classification of finite reflection groups which will lead us to the study of their combinatorial as well as geometric properties. Along the way we will introduce the notions of root systems and Weyl groups. We will go on to investigate not necessarily finite Coxeter groups. Hereby topics may include the Bruhat ordering and Kazhdan-Lusztig polynomials.
Texts: A main reference for this course is the textbook ``Reflection Groups and Coxeter Groups" by James E. Humphreys. ``Combinatorics of Coxeter Groups" by Andres Bjorner and Francesco Brenti might be also a nice source. A more advanced text on this topic is ``The Geometry and Topology of Coxeter Groups" by Micheal W.Davis, and another standard reference is ``Groupes et Algèbres de Lie, Chapitres 4,5,6" by N. Bourbaki.
Prerequisites: Some knowledge of linear algebra, e.g. the notion of a vector space and an inner product. In addition, some prior exposure to basic group theory might be helpful for this tutorial, but is not mandatory.
Contact:: Jessica Fintzen (fintzen@math.harvard.edu)
Spring 2013 Tutorials
Markov Chains
Description: How many times does one need to shuffle a deck of card to destroy all organization? How does this relate to the Representation Theory of the Symmetric group and to biologist's efforts to estimate the evolutionary distance between two organisms? What can a random walk on a Cayley graph tell you about the properties of a finitely generated group and how are random walks related to electrical networks? How to generate a random object of a kind without having to find all objects of this kind? Which methods can be employed to study the Ising model from statistical mechanics? What is the math behind the algorithmic music composition, the Google Internet Search engine, asset pricing models, and information processing?If you find at least one of these questions interesting, you are welcome to attend the Spring 2012 tutorial on Markov Chains where we will discuss them and much more interesting mathematics.
In 1906 Andrey Markov published a twenty two page paper in the Proceedings of the Kazan University Mathematical Society. In that paper with a long and unremarkable title, he first introduced the stochastic processes that later came to be known as Markov chains, and thus not only had a tremendous influence on the development of mathematics, but discovered a model that might well be the most ``real world" useful mathematical concept after that of a derivative. A Markov chain is a mathematical system that undergoes random transitions from one state to another between a certain number of possible states. Its future behavior depends only on the current state and not on the sequence of the preceding events, thus it gives us the simplest model of the random evolution of a certain system. Yet as simple as they are, Markov chains are profoundly interesting objects from both purely mathematical and applied perspectives. This tutorial will be designed to touch upon both perspectives and will be tailored to the interests of the participants. We will talk about the general theory of Markov Chains, mixing and the cutoff phenomenon and try to discuss as many interesting examples and applications as possible.
Text: Markov Chains and Mixing Times by David Levin, Yuval Peres and Elizabeth Wilmer. Available online. There will also be other references.
Prerequisites: Mathematics 121 is necessary. Mathematics 122 is desirable but not required. Previous exposure to Probability would be helpful but is not necessary. Contact: Konstantin Matveev (kmatveev@math.harvard.edu)
Morse Theory
Description: The essence of Morse theory is simple: study the topology of a manifold in terms of the critical locus of a function on it. This simple idea has been an extremely powerful and recurring theme in geometry, with applications including the Bott periodicity theorem, the topological Poincar'e conjecture in dimensions greater than or equal to 5, and the construction of invariants of smooth manifolds.In this tutorial, we will begin by studying classical Morse theory, describing how to use a Morse function to decompose a manifold into simpler pieces. We will also discuss the notion of homology of a topological space, and see how to recover the homology of a manifold using Morse theory. There will be explicit examples and applications to bring the theory to life. In the second part of the course, we will focus on one of the deeper applications of Morse theory, such as those mentioned in the previous paragraph.
Prerequisites: Topology at the level of Math 131. I will discuss the basics of manifolds in class.
Contact: Alex Perry (aperry@math.harvard.edu)