Math 497A - Honors MASS Algebra

Number Theory in the Spirit of Ramanujan

Instructor:  George Andrews, Evan Pugh Professor of Mathematics
TA:  Ayla Gafni

113 McAllister Building,  MWF 10:10 - 11:00 a.m.,  T 11:15 a.m. - 12:05 p.m.

Description:  The primary object of the course will be to understand those portions of elementary number theory that are closely related to the work of the Indian genius, Ramanujan. The honors objective will be to look at this mathematics from the broader perspective of the mathematical and societal influences surrounding Ramanujan's short, meteoric career. Perhaps this is explained best in the following description of Berndt's text prepared by the publisher:
    "Ramanujan is recognized as one of the great number theorists of the twentieth century. Here now is the first book to provide an introduction to his work in number theory. Most of Ramanujan's work in number theory arose out of q-series and theta functions. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the Ramanujan tau function. The majority of the results discussed here are originally due to Ramanujan or were rediscovered by him. Ramanujan did not leave us proofs of the thousands of theorems he recorded in his notebooks, and so it cannot be claimed that many of the proofs given in this book are those found by Ramanujan. However, they are all in the spirit of his mathematics. The subjects examined in this book have a rich history dating back to Euler and Jacobi, and they continue to be focal points of contemporary mathematical research. Therefore, at the end of each of the seven chapters, Berndt discusses the results established in the chapter and places them in both historical and contemporary contexts."

Math 497B - Honors MASS Analysis

Elements of Functional Analysis

Instructor:  Boris Kalinin, Associate Professor of Mathematics
TA:  Shilpak Banerjee

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

Description:  The course will introduce students to various ideas and techniques of functional analysis. The topics will include spaces of functions, Banach and Hilbert spaces, functionals and operators, and applications.

Math 497C - Honors MASS Geometry

Winding number in topology and geometry (and the rest of mathematics)

Instructor:  John Roe, Professor of Mathematics
TA:  Dong Chen

113 McAllister Building,  MWF 11:15 a.m. - 12:05 p.m.,  T 10:10 - 11:00 a.m.

Description:  The winding number is one of the most basic invariants in topology: it is an integer that measures the number of times a moving point P goes around a fixed point Q, provided that P travels on a path that never goes through Q and that the final position of P is the same as its starting position. For example, the winding number of the tip of the minute hand of a clock (P), about the center of the clock (Q), between 11 a.m. and 4 p.m. the same day, is -5. This simple idea has far-reaching applications in almost every area of mathematics. For instance, in the course we’ll learn about how the winding number (and its generalizations)
    Helps us show that every polynomial equation has a root (the fundamental theorem of algebra)
    Guarantees a fair division of three objects in space by a single cut (the ham sandwich theorem)
    Shows why every simple closed curve has an inside and an outside (the Jordan curve theorem)
    Allows you to “renormalize” the difference of two infinities and get a finite answer (Toeplitz index theory)
    Help explain why electrons fill successive “shells” around atomic nuclei (thereby giving rise to chemistry)