An asterisk by a 800-level course number indicates it is not open to mathematics graduate students.
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Announcements for Fall 2018 courses
- Math 911
The goal of this course will be an introduction to the theory of groups, with an emphasis on infinite groups.
Groups are a useful mathematical tool which originated in the study of symmetry. For example, reflections, rotations, and translations of the plane R^n that preserve infinite tilings form examples of infinite groups.
Several methods for studying groups, including topological, geometric, and combinatorial techniques, will be discussed in this course. Many of these methods have applications in other areas of algebra and mathematics in general, as well. There will be two main topics for the course:
1) Geometric and combinatorial group theory: Geometric and combinatorial group theory encompasses a wide variety of geometric, combinatorial, topological, and algorithmic methods in the study of finitely generated (and finitely presented) groups. In GGT a distance function (metric) is placed on the group, and information about the group (eg subgroup structure, complexity of algorithms for solving problems, etc.) are obtained from this function.
2) Homology of groups: Homology for groups can be defined either using algebra or using topology. In addition to being a widely applied invariant for distinguishing groups, homology gives a useful way to measure how close an infinite group is to being finite.
In all of the topics covered, I'll discuss applications to finding computational algorithms for answering questions about the groups, and to finding bounds on how efficient those algorithms can be. Prerequisites: Math 872 and Math 817, or permission of instructor. Text: There will be no formal text for the course, but much of the material we will cover in the first topic of the course can be found in the blog at berstein.wordpress.com/, and material for the second part of the course can be found in "Cohomology of Groups" by K.S. Brown (Springer, 1982).
- Math 918
- Topics in algebra: finite dimensional algebras
We plan to talk about finite dimensional algebras (both commutative and non-commutative), their modules, homomorphisms, exact sequences, projective resolutions, derived categories, and tilting theory. After touching general theories, we would consider objects like (commutative) Artinian Gorenstein rings, the rings of regular functions vanishing on fat points, (non-commutative) path algebras, and preprojective algebras. We will try our best to make everything as explicit as possible. Each student is expected to give at least two short presentations.
Prerequisites: Math 817/818, or permission of instructor. Text: There will be no formal text for the course, but part of the material will be based on the following notes.
- Math 958
Topics in discrete math: Probabilistic methods in combinatorics. This course is an invitation to explore the exciting world of probabilistic combinatorics and random discrete structures.
Motivation: The probabilistic method is a very powerful tool that can be used to obtain nonconstructive proofs of the existence of a prescribed kind of mathematical object. It works by building an appropriate probability space and showing that a random outcome in that space has the desired properties with non-zero probability. The method was pioneered by Paul Erdős and has been widely used in combinatorics and many other areas of mathematics, such as number theory, linear algebra and real analysis, as well as in computer science and information theory.
While the original method uses probabilistic ideas to prove deterministic results, it can also be applied to analyze random discrete structures, which model a large variety of objects in physics, biology, or computer science (e.g., social networks). As a result, probabilistic combinatorics has become one of the most exciting and fruitful areas of research in recent years, and the field continues to grow, in an interplay with other disciplines.
Potential topics: First and second moment methods, Lovasz local lemma, correlation inequalities, martingales, concentration of measure, pseudo-randomness and branching processes.
Text: There will be no formal text for the course, but much of the material will be based on the excellent book "The probabilistic method" by N. Alon and J. Spencer (Wiley 4th edition 2016).
Prerequisites: * Math 850/852 is recommended but not required (an exposure to discrete math and proof techniques at a graduate level will be helpful). * A basic background in probability theory (measure theory not required). * A reasonable level of mathematical maturity and enthusiasm.
If you'd like more information, please feel free to email the instructor or stop by.