## Topics

*Explorations in Number Theory*

Instructor: George Andrews, Evan Pugh Professor of Mathematics

TA: Lawrence Sze*Introduction to Dynamical Systems*

Instructor: Anatole Katok, Shibley Professor of Mathematics, Director of the Center for Dynamical Systems at Penn State

TA: Serge Ferleger, 1997-98 Sloan Dissertation Fellow*Linear Algebra in Geometry*

Instructor: Victor Nistor, Professor of Mathematics

TA: Alexander Mezhirov*MASS Seminar*

Instructor: Anatoli Kouchnirenko

## Course Outline

MATH 497E: EXPLORATIONS IN NUMBER THEORY (4:3:1)

TIME: MWRF 10:10 am - 11:00 am

INSTRUCTOR: GEORGE ANDREWS

TEXT: NUMBER THEORY by G.Andrews, published by Dover

This course was based on the standard syllabus for a beginning senior course in number theory approached from a combinatorial point of view.

Topics and theorems covered included: principles of mathematical induction; basis representation theorem; Fundamental Theorem of Arithmetic; Euclid's Division Lemma; Euclid's Algorithm; divisibility; linear diophantine equations; permutations/combinations; Fermat's Little Theorem; Wilson's Theorem; generating functions; congruence properties; residue systems; linear congruences; Chinese Remainder Theorem; polynomial congruences; multiplicative arithmetic functions; Mobius inversion formula; primitive roots of reduced residue systems; Prime Number Theorem; Euler's Criterion; Legendre/Jacobi Symbol; Law of Quadratic Reciprocity; distribution of quadratic/nonquadratic residues; partition theory; Ferrers diagrams; Euler's Partition theorem; infinite products and sums as generating functions for partition functions; identities between infinite series and products; Euler's Pentagonal Number Theorem; Rogers-Ramanujan Identities; and Schur's Theorem.

Students were expected to use a symbolic math package such as Maple, Mathematic or Maxima to complete some homework assignments.

MATH 497F: INTRODUCTION TO DYNAMICAL SYSTEMS (4:3:1)

TIME: MWRF 11:15 am - 12:05 pm

INSTRUCTOR: ANATOLE KATOK

TEXT: A FIRST COURSE IN DYNAMICS by B.Hasselbatt and A.Katok; Manuscript of book to be published by Cambridge University Press; certain parts of existing more advanced book Introduction to the modern theory of dynamical systems

, by A.Katok and B.Hasselblatt, published by Cambrigde University Press, were used

This course introduced and rigorously developed principal concepts and methods of the modern theory of dynamical systems and the necessary apparatus of real analysis. The topics included contraction mapping principle in metric spaces and its applications, simple behavior in one-dimensional systems including the quadratic family, differential equations as continous-time dynamical systems, detailed analysis of the quasi-periodic behavior including the uniform distribution and its applications; rotation numbers, topological conjugacy, basic paradigms and examples of complicated (chaotic) behavior including exponential growth and density of periodic orbits, topological mixing and coding.

MATH 497C: LINEAR ALGEBRA IN GEOMETRY (4:3:1)

TIME: MWRF 1:25 pm - 2:15 pm

INSTRUCTOR: VICTOR NISTOR

TEXT: LECTURES ON LINEAR ALGEBRA by I. M. Gelfand, published by Interscience

This course was divided into two equal parts. The first part covered the basics of linear algebra at an advanced undergraduate level. In the second part the class explored various applications of linear algebra. This included, among other things, projective geometry and a brief introduction to Lie groups.

MATH 497D: MASS SEMINAR (2:2:0)

TIME: T 10:10 am - 12:05 pm

INSTRUCTOR: ANATOLI KOUCHNIRENKO