Math 497C - Honors MASS Geometry

Lie Groups in Two, Three and Four Dimensions

Instructor:  Nigel Higson, Evan Pugh Professor of Mathematics
Teaching Assistant:  Qijun Tan

113 McAllister Building,  MWRF 10:10 - 11:00 a.m.

Description:  Lie groups arise from the continuous symmetries that are seen in nature and mathematics. Sometimes the symmetries are easy to appreciate (such as the continuous rotational symmetries of a circle, as opposed to the discrete, 3-fold rotational symmetries of an equilateral triangle) and sometimes they are not. This course will given an introduction to the theory of Lie groups, with an emphasis on examples in low dimensions, where many of the most interesting applications to mathematics and physics are to be found. Topics will include the basic properties of matrix Lie groups, Lie algebras and the exponential map; examples from real Euclidean space, complex Hermitian space, the quaternions and the octonions; the mechanics of rotating bodies; complexification; representations; applications to spin, the eightfold way, Lorentz transformations, and other things.

Objectives:  The first objective of the course to will to teach the basic theory of Lie groups and Lie algebras. Beyond that, the course will present many examples where mathematical techniques from geometry, algebra, analysis and combinatorics interact with on another, and contribute to our understanding of the laws of nature.

Reading:  No textbook will be used. Readings will be suggested by the instructor, and lecture notes will be developed and distributed as the course progresses.

Math 497B - Honors MASS Analysis

Classical Mechanics and Calculus of Variations

Instructor:  Mark Levi, Professor of Mathematics
Teaching Assistant:  Adam Zydney

113 McAllister Building,  MWRF 11:15 a.m. - 12:05 p.m

Description:  Classical mechanics and calculus of variations lie at the foundation of the modern theory of dynamical systems. This is the field where geometry, differential equations and number theory interact with each other and with physics. I will describe examples of this interaction, along with the fundamental concepts and ideas of the subject, with numerous special examples and with many entertaining problems.

Objectives:  I would like to give a clear intuitive understanding of the subject of classical mechanics and of calculus of variations, and to show how many ideas that appear mysterious or arbitrary are actually very natural and simple, if looked at in the right way. We will also solve many intuition-enhancing problems.

Reading:  Mark Levi. Classical Mechanics with Calculus of Variations and Optimal Control: An Intuitive Introduction.  ISBN-10: 0-8218-9138-3;  ISBN-13: 978-0-8218-9138-4.

Math 497A - Honors MASS Algebra

Introduction to Applied Algebraic Geometry

Instructor:  Jason Morton, Assistant Professor of Mathematics and Statistics
Teaching Assistant:  Sara Jamshidi

113 McAllister Building,  MTWF 1:25 - 2:15 p.m.

Description:  Fundamentals of algebraic geometry. Polynomial rings, ideals, varieties. Affine and projective space, Segre varieties, secant varieties, and tensor rank. Group actions on varieties. Techniques for application including identifying algebraic varieties in nature, computational methods, and solving systems of polynomial equations. Applications to statistics, quantum information, and computational complexity.

-Learn basic concepts of algebraic geometry.
-Learn how to recognize when algebraic geometry could be used to study a problem and take the first steps in that direction.
-Acquire computational tools.
-Learn about current applications.

Reading:  Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea will be used in the first part of the course to supplement lectures so that students can acquire the needed definitions and basic techniques in algebraic geometry.