Math 497A - Honors MASS Algebra

Polynomials

Instructor: Sergei Tabachnikov, Professor of Mathematics and MASS Director

TA: Nick Early

MWRF 11:15 a.m. - 12:05 p.m.

**Description**: The course will cover various algebraic, analytic and geometrical properties of polynomials and their applications: discriminants and resultants, solving polynomial equations, estimating the number of roots, Fundamental Theorem of Algebra, interpolation, symmetric polynomials, trigonometric polynomials.

**Readings**: No textbook will be required. Possible reading: V. Prasolov, Polynomials, Springer.

Math 497B - Honors MASS Analysis

Random walk and Brownian motion

Instructor: Alexei Novikov, Associate Professor of Mathematics

TA: Brian Nowakowski

MWRF 10:10 - 11:00 a.m.

**Description**: The objective is to study properties of two basic examples in probability theory: random walk and Brownian motion.

**Readings**: Possibly – G. Lawler Random Walk and the Heat Equation, P. Doyle & J. Snell Random walks and electric networks

Math 497C - Honors MASS Geometry

An introduction to geometric topology in dynamics

Instructor: Federico Rodriguez-Hertz, Professor of Mathematics

TA: Kurt Vinhage

MWRF 1:25 - 2:15 p.m.

**Description**: The objective of the course is to build geometric tools that enables to give an accurate description of a dynamical system. We will start with circle diffeomorphisms and covering maps and their classification through topological invariants like rotation number and degree. Then we shall move to surface diffeomorphisms (particularly when the surface is the torus) and give some consequence of their homotopy type. In the mean time we shall introduce the notion of fundamental group, covering space, homotopy invariants, conjugacy invariants, suspension of a diffeomorphism, foliations, flows on surfaces.

**Readings**: The main reference we will use is the book “Introduction to the Modern Theory of Dynamical Systems” B. Hasselblatt and A. Katok, Cambridge University Press . We shall use some input from the books: A. Casson and S. Bleiler, “Automorphisms of Surfaces After Nielsen and Thurston”, Cambridge Univ. Press; A. Hatcher, “Algebraic topology” http://www.math.cornell.edu/~hatcher/#ATI; and/or “J. Milnor Topology from the “Differentiable Viewpoint”, Princeton University Press; and/or V. Guillemin and A. Pollack. “Differential Toplogy” Prentice Hall