MAC 0105, at IME/USP, the University of São Paulo
February–June 2020
Lista 1, aqui
Objetivos:
Familiarizar o estudante com a linguagem matemática e a estrutura das declarações matemáticas, bem como com alguns fatos e noções básicas sobre números, conjuntos, funções e relações.
(Familiarize the student with mathematical language and the structure of mathematical statements, as well as some basic facts and notions about numbers, sets, functions, and relations.)
Programa:
Discurso matemático: leitura e escrita matemática. Estratégias de demonstração. Princípio da indução finita. Sequências, somas, recorrências e contagem. Algoritmo de Euclides. Divisibilidade em números inteiros. Sistemas de numeração. MDC e MMC. Teorema de Bézout. Teorema fundamental da aritmética. Congruências. O anel dos inteiros módulo m. Relações de equivalência, conjunto quociente, definição de funções e operações no conjunto quociente. Ordem, fecho transitivo de relações. Conjuntos infinitos.
(Mathematical discourse: reading and writing mathematics. Proof strategies. Principle of finite induction. Sequences, sums, recurrences, and counting. Euclid's algorithm. Divisibility in the integers. Numeral systems. GCD and LCM. Bézout's theorem. Fundamental theorem of arithmetic. Congruences. The ring of integers modulo m. Equivalence relations, quotient sets, definition of functions and operations on the quotient set. Orders, transitive closure of relations. Infinite sets.)
Avaliação:
Método: aulas teóricas e de exercícios.
Critério: Listas = 50%; Prova 1 = 25%; Prova 2 = 25%.
Bibliografia:
K. Houston, How to Think Like a Mathematician: A Companion to Undergraduate Mathematics, Cambridge University Press, 2009.
M. Hutchings, Introduction to Mathematical Arguments, notes, pp. 1–27.
Michael Spivak, Calculus, 4th edition, 2008 (selected topics from the first two chapters, pages 3–35).
L. Lovász, J. Pelikán, and K. Vesztergombi, Discrete Mathematics, Brazilian Mathematical Society, 2006.
D. J. Velleman, How to Prove It: A Structured Approach, 2nd edition, Cambridge University Press, 2006.
Yoshiharu Kohayakawa, An exercise list: Exercicios.pdf
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MAC 6915, at IME/USP, the University of São Paulo
March–June 2020
Lista 1, aqui
Objectives:
We will study Fourier transforms of polytopes and Fourier–Laplace transforms of cones, and some of their applications to Minkowski's geometry of numbers, enumeration problems in geometric combinatorics (including Ehrhart polynomials and solid-angle polynomials of rational polytopes), and some more applied topics.
Evaluation:
Exercises = 80%; Exam (Prova) = 20%.
Bibliografia:
Matthias Beck and Sinai Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, 2nd edition, Springer, 2015.
Sinai Robins, Fourier Analysis on Polytopes and the Geometry of Numbers, Part I: A Friendly Introduction, American Mathematical Society, Student Mathematical Library, 2024. A preliminary version is available on arXiv: arxiv.org/abs/2104.06407
Elias Stein and Rami Shakarchi, Fourier Analysis: An Introduction, Princeton Lectures in Analysis, Vol. 1, 2003.
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MAT 0122, at IME/USP, the University of São Paulo
August–December 2019
Prova 2, em Português, aqui
( Prova 2, in English, aqui )
Objetivos:
Familiarizar o estudante com os conceitos de espaço vetorial real e transformações lineares, e com aplicações de operadores diagonalizáveis.
(Familiarize the student with the concepts of real vector spaces and linear transformations, and with applications of diagonalizable operators.)
Programa:
Espaços vetoriais: definição, subespaços, dependência linear, bases, dimensão.
Cálculo matricial, determinantes, sistemas lineares.
Transformações lineares e matrizes, núcleo, imagem, posto.
Espaços com produto interno: produto interno, norma, ortogonalidade, processo de Gram–Schmidt, complemento ortogonal, projeção, autovalores e autovetores.
Avaliação:
Método: aulas teóricas e de exercícios.
Critério: Listas = 10%; Prova 1 = 45%; Prova 2 = 45%.
Bibliografia:
Gilbert Strang, Introduction to Linear Algebra, 4th edition, 2016: math.mit.edu/~gs/linearalgebra
Howard Anton and Chris Rorres, Elementary Linear Algebra: Applications Version, 11th edition, Wiley, 2014.
M. Barone Jr., Álgebra Linear, 3rd edition, IME-USP, São Paulo, 1988.
Débora Cristiane Barbosa Kirnev and Renata Karoline Fernandes, Álgebra Linear e Vetorial, 2015.
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At IME/USP, the University of São Paulo · March–June 2018
Homework assignments:
We will use the following references, along with some additional ones as the semester progresses:
Dan Spielman's course notes: 2009 edition and 2015 edition
Norman Biggs, Algebraic Graph Theory, 2nd edition, Cambridge University Press, 1993.
Chris Godsil and Gordon Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, Springer, 2001.
Dragoš Cvetković, Peter Rowlinson, and Slobodan Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, 2010.
Matthias Beck and Sinai Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, 2nd edition, Springer, 2015.
We will choose sub-topics from these sources, with some flexibility depending on students' interests.
Grading: Homework = 30%; Midterm = 30%; Final exam = 40%.
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Graduate course at IME/USP, the University of São Paulo · March–June 2018
This graduate course is an introduction to analytic number theory. Topics include advanced linear algebra, the finite Fourier transform, Dirichlet L-functions with their functional equations and analytic continuation, Dirichlet's theorem on primes in arithmetic progressions, group characters of finite abelian groups, theta functions, and applications.
Homework assignments:
HW 1 (due March 21)
From Apostol's book (reference 1 below), please do the following exercises: Chapter 3, numbers 4a, 4b, 5a, and 5b.
Also, prove that ⌊x⌋ + ⌊x + 1/n⌋ + ⌊x + 2/n⌋ + … + ⌊x + (n−1)/n⌋ = ⌊nx⌋ for all real numbers x and all positive integers n.
Grading: Homework = 30%; Midterm = 30%; Final exam = 40%.
References:
Tom Apostol, Introduction to Analytic Number Theory, Springer, Undergraduate Texts in Mathematics, 1998.
Tom Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer, Graduate Texts in Mathematics, 2nd edition, 2012.
Matthias Beck and Sinai Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, 2nd edition, Springer, 2015.
Edmund Hlawka, Johannes Schoissengeier, and Rudolf Taschner, Geometric and Analytic Number Theory, Springer, 2012.
M. Ram Murty, Problems in Analytic Number Theory, Springer, Graduate Texts in Mathematics, 2007.
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At IME/USP, the University of São Paulo · August–December 2017
Instructor: Sinai Robins
Objectives: To study topics in number theory, with an emphasis on algorithms and their applications.
Rationale: Many algorithms developed for number theory have proved important both in mathematical research and in applications, and the subject remains an area of active research and development.
Topics include:
The gcd algorithm, modular arithmetic, fast exponentiation, and big-O notation.
Solving linear congruences and the Chinese Remainder Theorem.
Euler's phi function, primitive roots, Euler's theorem, and Fermat's little theorem.
Prime numbers, Chebyshev's theorem, pseudoprimes, and primality testing.
RSA and factoring algorithms; quadratic residues and Gauss's law of quadratic reciprocity.
A geometric introduction to continued fractions, Pell's equation, and factoring with continued fractions.
The LLL algorithm for finding a short vector in a lattice, a natural generalization of the gcd algorithm to higher dimensions.
The main reference is the book by Bressoud and Wagon below, which uses Mathematica to support the material. We will also use Sage for practical exercises throughout the course.
Bibliography:
David Bressoud and Stan Wagon, A Course in Computational Number Theory, Wiley, 2000.
Murray Bremner, Lattice Basis Reduction: An Introduction to the LLL Algorithm and Its Applications, CRC Press, 2011.
John Harris, Karen Kohl, and John Perry, Peering into Advanced Mathematics through Sage-Colored Glasses, an open-source text freely available online, based on Sage calculations.
Tom Apostol, Introduction to Analytic Number Theory, Springer, Undergraduate Texts in Mathematics, 1976.
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Computational Geometry
At IME/USP, the University of São Paulo · August–December 2016
We will draw on a range of references, mostly books along with some research papers, choosing topics mainly from the following:
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars, Computational Geometry: Algorithms and Applications, 3rd edition, Springer, 2008.
Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK, 4th edition, Springer, 2010.
Matthias Beck and Sinai Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, 2nd edition, Springer, 2015.
Satyan Devadoss and Joseph O'Rourke, Discrete and Computational Geometry, Princeton University Press, 2011.
Richard Stanley, Enumerative Combinatorics, Volume 1, 2nd edition, Cambridge University Press, 2012.
Grading: Homework = 30%; Midterm = 30%; Final exam = 40%.
For an excellent introduction to rational functions of one variable and their relationship to linear recurrence sequences, see Chapter 4 of Richard Stanley's Enumerative Combinatorics, 2nd edition.
An animation involving graphs: Möbius ladder (animated)
Homework
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Algebraic Graph Theory
At IME/USP, the University of São Paulo · August–December 2016
References:
Norman Biggs, Algebraic Graph Theory, 2nd edition, Cambridge University Press, 1993.
Chris Godsil and Gordon Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, Springer, 2001.
Dragoš Cvetković, Peter Rowlinson, and Slobodan Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, 2010.
Matthias Beck and Sinai Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, 2nd edition, Springer, 2015.
We will choose sub-topics from these sources, with some flexibility depending on students' interests.
Grading: Homework = 30%; Midterm = 30%; Final exam = 40%.
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Earlier Courses:
Brown University, Fall 2014: Harmonic Analysis on Polytopes (graduate course).
NTU, Singapore, 2014, Semester 2: Topics in Applied and Discrete Mathematics (undergraduate/graduate seminar).
NTU, Singapore, 2010–2013: "It's a Discreetly Discrete World," a large undergraduate course taught several times (347 students in 2010, 301 in 2011).
NTU, Singapore, 2012: Multivariable Calculus with Linear Algebra, for CN Yang Scholars.
NTU, Singapore, 2011: Complex Analysis (62 students).
NTU, Singapore, 2010, Semester 2: Discrete Methods (graduate course), MAS 711.
NTU, Singapore, 2009, Semester 1: Real Analysis I, MAS 311.
NTU, Singapore, 2009, Semester 2: Topics in Number Theory III: Discrete Geometry and Number Theory, MAS 742.
NTU, Singapore, 2008, Semester 1: Graph Theory, MAS 324.
NTU, Singapore, 2008, Semester 2: Discrete Mathematics with Elementary Number Theory, MAS 214 (158 students).
MSRI/BIRS, Banff, Canada, 2005: two-week summer course for 40 graduate students, co-taught with Matthias Beck.
Temple University: Number Theory (PhD level); Analytic Number Theory; Algebraic Number Theory; Number Theory (undergraduate).
Temple University: Combinatorial Geometry, based on Computing the Continuous Discretely (with Matthias Beck).
Temple University: Discrete Geometry; Combinatorial Topology and Knots.
Temple University: Cryptography and Information Theory for IST majors (undergraduate).
Temple University: Calculus I, II, III; Real Analysis; Differential Equations; For All Practical Purposes (for non-majors).
UC San Diego, 1995–1996: Calculus for the Life Sciences I, II, III.
University of Northern Colorado: Calculus I, II, III; Real Variables I, II; Linear Algebra; Abstract Algebra (master's level); Analytic Number Theory.
University of Northern Colorado: Seminar in Number Theory; Topics in Finite Mathematics; Seminar on Mathematical Discovery using Pólya's books.
University of Northern Colorado: History of Mathematics; Math for Elementary School Teachers; Math for the Liberal Arts.
Videos
Google Tech Talk, Googleplex, "Bay Area Discrete Math Day XII: Integer Linear Programming," April 2006.
Masterclasses for School Students
Books
Matthias Beck and Sinai Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, Springer, Undergraduate Texts in Mathematics, 2nd edition, 2015. Get the PDF and see reviews of the book here.
Sinai Robins, Fourier Analysis on Polytopes and the Geometry of Numbers, Part I: A Friendly Introduction, American Mathematical Society, Student Mathematical Library, 2024: bookstore.ams.org
Undergraduate thesis students, Nanyang Technological University, Singapore
Czarina Ann Marifosque, 2008
Nhat Le Quang, 2009
Stephanie Gabriela, 2009
Danh Nguyen Luu (Danny), 2011
Zhang Yichi, 2012-2013