COURSE DESCRIPTIONS

MMA101 Linear Algebra
Yasuo Morita,Tohoku University, Japan
(Email: morita@math.tohoku.ac.jp)

Students learn Group, rings and fields; Matrices and vectors, Determinants, Linear spaces and linear maps, Systems of linear equations, Eigenvalues and eigenvectors, Bilinear forms and metrics on linear spaces , Dual spaces and tensor spaces.

MMA102 Probability
Sonia Fourati, INSA, university of Rouen, France
(Email: sonia.fourati@upmc.fr)

Elements of Combinatorial Analysis, Finite probability spaces, Conditioning and Independence: Conditioning by an event, Denumerability and summability, Generalization to denumerable spaces of finite probability spaces and conditioning and independence, Random variables having a density, Covariance, Line of Regression, The Gaussian Case, Distribution function and characteristic function of a random variable, Different techniques of simulation.

Text books:
William Feller: Introduction to Probability Theory (Volume I and II)

MMA103 Geometry (Differential Calculus)
Kiminao Ishitoya, Aichi University of Education, Japan
(Email: kishito@auecc.aichi-edu.ac.jp)

The course includes Curves, Surfaces, Conformal Mappings, Geodesics, The Gauss-Bonnet Theorem, Introduction to Algebraic Topology

MMA 104 Introduction to real analysis
Eduardo Cattani, University of Massachusset, Amherst, USA

Basic Notion: (Properties of the real numbers. Countable and uncountable set), Sequences and Series: Basic definitions, Cauchy sequences, Bolzano-Weierstrass Theorem. Continuity: Definition, Continuous functions on closed intervals, Intermediate value Theorem, Uniform continuity.. Riemann Integral: Upper and lower sums, Existence of integral for continuity functions. Differentiation: Definition of derivative, Fundamental Theorem of calculus, Taylor’s Theorem. Sequences of Functions: Pointwise and uniform convergence, matric spaces and contraction mapping theorem. Series of functions: Power series-radius of convergence, behavior of series under differentiation and integration, Taylor series and the remainder theorem.

Text books:
   - Bernard Gelbaum and John Olmested, Counterexamples in analysis. Dover, 2003
   - Michael Reed, Fundamental Ideas of Analysis, Wiley, 1998
   - Walter Rudin, Principles of mathematical Analysis. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., 1976

MMA105 Discrete Mathematics
Michel Waldschmidt,University of Paris 6,Frace
(Email: miw@math.jussiru.fr)

   1. Counting: Integers,sets,subsets.
   2. Combinatorial probability
   3. Arithmetics: divisibility,primality,factorization
   4. Graphs, trees
   5. Cryptography

Text books:
   - László Lovász and Kati Vesztergombi, Discrete Mathematics,1999(psfile,2,4Mo). Further references:
   - MITOPENCOURSEWARE , Massachusetts Institute of Technology, Undergraduate Seminar in Discrete Mathematics, 18-304Spring-2006.zip (ZIP - 1.70 MB)
   - William Chen- Discrete Mathematics, 201 pp. (web edition, 2008).
   - Victor Shoup - A Computational Introduction to Number Theory and Algebra, Cambridge 2005. Second print editon, Fall 2008 (pdf file 3,5 Mo).

MMA 106 Introduction to statistics
Jan Hannig, University of North Carolina, US
(Email: hannig@email.unc.edu)

This is a course in statistics. The covered topic will include basics of mathematical statistics such as distribution theory, asymptotic theory, point estimate, interval estimate, hypothesis testing. We also cover some basics of linear regression.

Text Books:
   - Casella and Berger: Statistical Inference, Duxbury 2/e (Chapter 5-10) ISBN: 0 534 24312 6
   - Bickel and Doksum: Mathematical Statistics, Vol 1, 2/e, Prentice Hall, ISBN: 0-138-50363-X
   - Mood, Graybill, Boas, Introduction to the theory of Statistics
   - Ross, A First Course in Probability
   - Grimmet and Stirzaker, Probability and Random Processes

MMA107 Complex Analysis
Yan Pautrat, University of Paris Sud Orsay, France
(Email: yan.pautrat.pro@gmail.com)

This is a course on functions of complex variables and their properties. We will review :
   - The properties and geometry of the set of complex numbers
   - The notion of complex differentiability and holomorphic functions
   - Power series and real or complex analytic functions
   - Integration in the complex plane
   - Cauchy's formula and its consequences
   - Laurent series and meromorphic functions
   - The computation of integrals by the method of residues.

Textbook: Rudin, real and complex analysis and the online course by W.P. Novinger at http://www.math.uiuc.edu/~r-ash/CV.html

MMA 108 Topology
Michel Jambu, University of Nice, France
(E-mail: jambu@unice.fr)

This course is an introduction to topology. Before defining topologies, we recall the topologies defined by metrics. Then we focus on the main properties of Hausdorff spaces, connectedness and compactness. Finally, we introduce the fundamental groups and we give some examples of applications.

Major Topics by Chapter in text:
   1. Introduction and mathematical preliminaries
   2. Metric spaces
   3. Topological spaces
   4. Convergence and Hausdorff spaces
   5. Connectedness
   6. Compactness
   7. Fundamental groups; Applications

Textbooks: M A Armstrong. Basic Topology, Springer, 1983.

MMA 109 Complement 1 for teachers (Projective Geometry)
Michel Jambu, University of Nice Sophia Antipolis, France
(E-mail: jambu@unice.fr)

This course is an introduction to projective geometry. The students are supposed to master affine geometry. This is a classical course on projective geometry with a last chapter on some applications to algebraic curve where elliptic curve will be introduced.

Major Topic by Chapter in text:
   1. Projective spaces
   2. Projective frames
   3. Projective transformations, homography
   4. Affine vs projective
   5. Affine coordinates vs projective coordinates
   6. Topology of projective spaces (real and complex)
   7. The complex projective line, the cross-ratio, homography, the circular group
   8. Complexifications
   9. Cyclic points
   10. Projective duality
   11. Application to algebraic curves

Textbook: Geometry, by Michèle Audin, Universitext, Springer-Verlag 2003

MMMA 110 Combinatorial Geometry
Paul Vaderlind Rikard Bogvad, Stockholm University, Sweden
(Email: paul@math.su.se)

Part 1:
   1. Convexity in R^n. Caratheodory and Radons Theorems.
   2. Helly type theorems for finite family of countable families and for overcountable families.
   3. Generalizations of Helly’s Theorem. Basic applications to geometry.
   4. Sets of constant width and Universal covers, the Joung’s Theorem
   5. Illumination Problem and the Hadwiger Conjecture
   6. Hausdorff metric for compact sets, Blashke Selection Theorem, The existence of Maximal sets,the Area and Volume
Part 2:
   1. V and H-description of polytopes
   2. Faces and combinatorial type of polytopes
   3. Higher-dimensional polytopes (cyclic and simplicial polytopes)
   4. Number of faces: Eulers theorem and Dehn-Somerville equations, Upper bound theorem
   5. Diameter of a polytope and Hirsch conjecture
   6. Hausdorff metric for compact sets, Blashke Selection Theorem, The existence of Maximal sets, the Area and Volume.

MMA 111 PDE (Partial Differential Equation)
Will Murray, California State University, Long Beach, USA
(Email: wmurray@csulb.edu)

Prerequisites: Students should have strong calculus skills and at least basic linear algebra. Especially important topics include integration, Taylor Series, partial differentiation, matrix manipulation, and eigenvalue theory.
We will study the theory and applications of ordinary and partial differential equations. Topics include first-order equations, second and higher order equations, series solutions of second order linear equations, systems of first order linear equations, numerical methods, partial differential equations, and Fourier Series.

Text Books: - Boyce, Elementary Differential Equation and Boundary Value Problems, Ninth Edition. Selected topics from chapters 1-5, 7, 8, and 10.

MMA 112 Algebraic Equations of Higher Degree and the Galois Theory
Shun-ichi Kimura, hiroshima University, Japan
(Email: kimura@math.sci.hirosima-u.ac.jp)

In this course, we learn the development of the algebraic equations, from historical view point. We treat the formula for cubic and quartic equations, and the relation between algebraic equations and geometric constructions. By proving that geometric construction corresponds to solving a series of quadratic equations, we can formulate the impossibility of trisection an angle and doubling the cube. Using the noting of the field extension, we will rigorously prove the impossibility. At the same time, it implies that we can construct regular 17gons. We also discuss geometric construction using paper folding, mark ruler, or other interesting tools. They are equivalent to using the formula for cubic and quartic equations. Finally, by introducing the Galois group, we can give more subtle research of field extension. We mainly concentrate the subfield of the rational function field over C, to prove the impossibility of the formula for quintic equations, only using radicals and addition, subtraction, multiplication and division.

Major Topic by Chapter in Text : (subject to change)
   1. Cardano formula, Ferrai’s formula
   2.Geometric construction and quadratic equation
   3. Euclidean Algorithm and Rationalization of denominator
   4. Notion of fields
   5. Three problems from Ancient Greece
   6. Formula for extension degree
   7. Symmetric functions and invariant subfield
   8. Galois group and the Fundamental Theorem
   9.Construction of regular 17gons
   10. Radical extension and normal subgroups
   11. Impossibility for quintic equations

Textbooks: (1) Emit Artin: Galois Theory (Dover) Concise but well written textbook:Shortest path to Galois theory.
(2) Postnikov: Foundations of Galois Theory (Dover) Richer account for the background material.
(3) Jean-pierre Tignol:Galois’Theory of Algebraic Equations (World Scientific Pub). Very rich historical background for the development of Galois theory.

MMA 113 Complement 2 for teachers
Pierre Arnoux, University of Marseille, France

To be developed

MMA 114 Numerical Analysis
Angel Pineda, California State University, Fullerton, USA.
(Email: apineda@fullerton.edu)
Homepage: http://math.fullerton.edu/apineda/

This course is an introduction to numerical analysis using MALAB as the scientific computing platform. The objective of this course is to understand how to use computers to solve mathematical problems and to implement the algorithms. The course will cover both theory and applications. The following topics will be covered: solutions of nonlinear equations in one variable, solutions to linear systems of equations in several variables, polynomial interpolation, approximate solutions to problems using linear least squares and numerical differentiation and integration.

Major Topics by Chapter in Text:
   Chapter 0: Finite Precision Arithmetic and Computer Implementation of Algorithms
   Chapter 1: Solving Linear and Non-linear Equations with one Unknown
   Chapter 2: Solving Linear and Non-linear Systems of Equations
   Chapter 3: Interpolation
   Chapter 4: Least squares
   Chapter 5: Numerical Differentiation and Integration

Textbook: Timothy Sauer, Numerical Analysis, Pearson, 2006. Software: MALAB, The Math Works, Inc.

MMA 115 Compact Riemann Surfaces
Takayasu Kuwata, Tokyo Denki University, Japan
(E-mail: kuwata@im.dendai.ac.jp)

The theory of compact Riemann surfaces is attractive, because we need knowledge in various fields like algebra, geometry, complex analysis etc. After the review of manifolds and complex analysis, I want to treat an introduction to compact manifolds, especially, compact Riemann surfaces, as applications.

Major Topics by Chapter in Text:
   1. Algebraic curves in the projective plane
   2. Review of complex analysis
   3. Riemann surfaces (one dimension complex manifolds)
   4. Holomorphic functions and meromorphic functions
   5. Differential forms
   6. Divisors on compact Riemann surfaces
   7. Riemann-Roch’s Theorem of compact Riemann surfaces
   8. Some applications

Textbooks:
- Phillip A. Griffiths, Introduction to Algebraic Curves, American Mathematical Society
- M. Namba, Geometry of Projective Curves, Marcel Dekker, 1984