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Real geometry (Spring 2018)

Real geometry, from algebraic to subanalytic and beyond.
MATH 649K, Spring 2018

We will study real algebraic sets (with emphasis on geometry over algebra), semialgebraic sets (sets
defined by polynomial equalities and inequalities) , and subanalytic sets. We will consider the basic
properties of these sets, study of appropriate classes of functions on them, topology, applications.
We may get into further classes of “tame geometry” such as o-minimal geometries.
There are no particular prerequisites, just a certain level of mathematical maturity; undergraduate
mathematical analysis and topology should be enough.

Instructor: Les Wilson

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Representation theory (Spring 2018)

Math 649: Representation theory.

This course is intended to become a regular offering in the ‘algebra and number theory’ collection of courses and so is not really a 649; rather it is the official ‘algebra and number theory’ offering this Spring. Here is a list of topics that are planned for this course.

Linear representations of finite groups: basic theory, group algebras, Schur’s lemma, Maschke’s theorem, the regular representation, character theory, induced representations, Frobenius reciprocity, examples, applications.

Representation theory of semisimple Lie algebras: the basics of Lie algebras, representations of $\mathrm{sl}_2(\mathbb C)$, representations of higher rank semisimple Lie algebras, highest weight theory, roots, relation to the representation theory of semisimple Lie groups.

Further topics may include (some but not all of): in-depth examples such as the symmetric group or finite linear groups, Burnside’s theorem, applications to combinatorics, harmonic analysis on finite groups, modular representation theory, representation theory of compact groups, the classification of complex semisimple Lie algebras and Dynkin diagrams.

Textbook: Fulton and Harris’ “Representation theory, a first course”, supplemented with other texts as necessary.
Instructor: Robert Harron

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Placement Exam & Math Boot Camp

Click here for Math Boot Camp info

The Math Department requires an exam for placement in certain courses. It is called the placement exam but also sometimes referred to as the assessment exam. Based on your math placement exam score, the Mathematics Department will enter a BMAT score into your Banner student record. Your BMAT score determines the classes in which you are allowed to register, and it stays valid for one year.
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