## Topics

*Arithmetic and Geometry of the Unimodular Group*

Instructor: Svetlana Katok, Professor of Mathematics

TA: Alexander Mezhirov*Real Analysis*

Instructor: Nigel Higson, Professor of Mathematics

TA: Dorin Dumitrascu*Explorations in Geometry*

Instructor: Dmitri Burago, Assistant Professor of Mathematics, 1997 Sloan Fellow

TA: Eric Johnson, a winner of the 1996-97 Penn State Teaching Assistant award).*MASS Seminar*

Instructor: Anatoli Kouchnirenko

## Course Outline

MATH 497A: ARITHMETIC AND GEOMETRY OF THE UNIMODULAR GROUP (4:3:1)

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

INSTRUCTOR: SVETLANA KATOK

TEXT: Instructor's Notes Arithmetic and Geometry of the Unimodular Group

by S. Katok, 60 pp.

This course was devoted to the study of the ** unimodular group of matricies** (SL(2,R)) with applications to number theory and geometry.

Particular topics included were: Algebraic Preliminaires: groups and fields, vector spaces, group representations; Linear and Affine Transformations the Plane: linear algebra in two dimensions, classification of matrices in SL(2,R), subgroups of GL(2,R) from a geometric point of view, affine transformations of the plane; Isometries of the Euclidean Plane: the Euclidean plane, classification of Euclidean isometries, geodesics in the Euclidean plane, finite groups of isometries, discrete groups of isometries, fundamental regions and quotients, geodesic lines on the torus; Hyperbolic geometry: hyperbolic metric, geodesics, classification of isometries, connections between Euclidean and hyperbolic geometries, isometric circles and inversions, hyperbolic area and Gauss-Bonnet formula; Fuchsian groups: the group PSL(2,R) revisited, fundamental regions for Fuchsian groups; the Modular group: its fundamental region, quotient space, and generators.

The course entailed a large number of homework problems with emphasis on proofs, including a lot of challenging problems, computer-related projects, and individual presentations by the students. The final presentations included the following more advanced topics: Wallpaper groups; Triangle groups in hyperbolic geometry; Geodesic flow; Coding of Closed geodesics on the modular surface; Continued fractions.

MATH 497B: EXPLORATION IN GEOMETRY (4:3:1)

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

INSTRUCTOR: DMITRI BURAGO

TEXT: CONVEX BODIES: THE BRUNN-MINKOWSKI THEORY, Chapter 1, by Rolf Schnieder, published by Cambridge University Press;

ELEMENTARY DIFFERENTIAL GEOMETRY, 2nd Edition, by Barrett O'Neill, published by Academic Press;

A COMPREHENSIVE INTRO TO DIFFERENTIAL GEOMETRY, 2nd Edition, by Michael Spivak

**No purchase was required. It was important to note that these books contained more material than was covered in the course. Weekly handouts contaied all material that was covered. These handouts included text from these books listed plus additional resources.

The course started with an elementary theory of convex bodies. Our exposition included such beautiful results as theorems of Minkowski, Krein-Milman and Helly, and a little bit of separation theory, duality and integral geometry. The second part of the course was devoted to differential geometry of curves and surfaces in Euclidean space. Some important notions related to local and global geometry of curves and surfaces were, including different kinds of curvature. Several theorems describing local and global properties of curvature were proved, including calculation of integral curvature of a closed curve. Description of boundary of a convex set in a plane was the point where both parts of the course met together.

MATH 497C: REAL ANALYSIS (4:3:1)

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

INSTRUCTOR: NIGEL HIGSON

TEXT: PRINCIPLES OF REAL ANALYSIS, 3rd or most recent edition, by Walter Rudin, published by McGraw-Hill

This course covered the fundamentals of real analysis in one variable as set out in the textbook. Among others, the following topics were included: Axioms for a completed field; Countability of Q, the Cantor's theorem and uncountability of R; Axioms for a metric space, open and closed sets, coninuity, connectedness, compactness, the Heine-Borel Theorem; Convergence of infinite series, absolute convergence, convergence tests; Mean Value Theorem, L'Hopitale's Rule; Uniform continuity, definition and properties of the Riemann integral, the fundamental theorem of calculus; Uniform convergence, different theorems involving uniform convergence (interchanging limits, derivatives and integrals); Power series: radius of convergence, differentiation and integration of power series term by term; Definitions and properties of several elementary functions; Fourier series.

MATH 497D: MASS SEMINAR (3:3:0)

TIME: T 9:30 am - 11:15 am

INSTRUCTOR: ANATOLI KOUCHNIRENKO

TEXT: Instructor's Handouts

The seminar focused on selected interdisciplinary topics in real analysis, linear algebra, multivariable calculus and geometry. These areas were related to the three basic MASS courses. Among others, the following topics were discussed during the semester: examples of basic algebraic structures; main theorem of algebra; resultant and discriminant; examples of metric spaces; compactness as a tool in n-dimensional geometry; Newton's method; homotopy method of numerical solutions of algebraic equation (including experiments with Mathlab scripts).