MAC 0105, at IME/USP, the University of São Paulo
Fevereiro - Junho, 2020
Lista 1, aqui
Objetivos:
Familiarizar o estudante com a linguagem matemática e a estrutura das declarações matemáticas, bem como com algumas fatos e noções básicas sobre números, conjuntos, funções e relações.
(Familiarize the student with the mathematical language and structure of mathematical statements, as well as some facts and basic 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 contando. Algoritmo de Euclides. Divisibilidade em números inteiros. Sistemas de numeração. MDC e MMC. Teorema de Bézout. Teorema fundamental da aritmética. Congruências, O anel do módulo inteiro m. Equivalência relações, conjunto de quocientes, definição de funções e operações no conjunto de quocientes. Ordem, fechamento transitivo das relações. Infinito conjuntos.
(Mathematical speech: mathematical reading and writing. Demonstration (proving) strategies. Principle of finite induction. Sequences, sums, recurrences and counting. Euclid's algorithm. Divisibility in integers. Numbering systems. MDC and MMC. Bézout's theorem. Fundamental theorem of arithmetic. Congruences. The ring of the integers modulo m. Equivalence relations, quotient set, definition of functions and operations in the quotient set. Order, transitive closure of relations. Infinite sets.)
Avaliação:
Método Aulas teóricas e de exercícios.
Critério Listas = 50 %
Prova 1 = 25 %
Prova 2 = 25 %
Bibliografia:
1. K. Houston, How to Think Like a Mathematician: A Companion to Undergraduate Mathematics, Cambridge University Press, 2009.
2. M. Hutchings, Introduction to mathematical arguments, notes, p. 1 - 27.
3. Michael Spivak, Calculus, (we will pick some topics from the first 2 chapters only, namely pages 3--35), 4'th edition, 2008.
4. L. Lovász, J. Pelikán, K. Vesztergombi, Discrete Mathematics, Brazilian Mathematical Society, 2006.
5. D. J. Velleman, How to Prove It: A Structured Approach, 2nd ed., Cambridge University Press, 2006.
6. Yoshiharu Kohayakawa, An exercise list, https://www.ime.usp.br/~yoshi/2018i/mac105/html/exx/Exercicios.pdf
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MAC 6915, at IME/USP, the University of São Paulo
2 de Marco - Junho, 2020
Lista 1, aqui
Objectives:
We will study Fourier transforms of polytopes, Fourier-Laplace transforms of cones, and see 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), as well as some more applied topics.
Evaluation:
Exercises: 80%
Prova: 20%
Bibliografia:
1. Matthias Beck and Sinai Robins, Computing the continuous discretely: integer point enumeration in polytopes, 2'nd edition, Springer, 2015.
2. Sinai Robins, Fourier analysis on polytopes, and the geometry of numbers, draft.
A preliminary arxive version may be found here: https://arxiv.org/abs/2104.06407
3. 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
Agosto - Dezembro, 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 diagonalizávei.
Programa:
1. Espaços vetoriais: definição, subespaços, dependência linear, bases, dimensão.
2. Cálculo matricial, determinantes, sistemas lineares.
3. Transformações lineares e matrizes, núcleo, imagem, posto.
4. 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 = 33 %
Prova 2 = 33 %
Prova 3 (ultima prova) = 33 %
Bibliografia:
1. Gilbert Strang, , 4'th edition, 2016
http://math.mit.edu/~gs/linearalgebra/
2. Howard Anton and Chris Rorres, Wiley Publications, 11'th edition, 2014
https://danboak.files.wordpress.com/2017/01/howard-anton-chris-rorres-elementary-linear-algebra-applications-version-11th-edition.pdf
3. M. Barone Jr., ÁLGEBRA LINEAR, 3 ed., IME-USP, São Paulo, 1988
4. Debora 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 (and some additional ones as the semester progresses) :
Dan Spielman's course notes, located at
1. (2009) http://www.cs.yale.edu/homes/spielman/561/2009/index.html
2. (2015) http://www.cs.yale.edu/homes/spielman/561/
Algebraic Graph Theory, Norman Biggs, Cambridge University Press, 2nd edition, 1993.
Algebraic Graph Theory, Chris Godsil, Gordon Royle, Graduate Texts in Mathematics, Springer-Verlag, 2001.
An Introduction to the Theory of Graph Spectra, Dragos Svetkovic, Peter Rowlinson, Slobodan Simic, Cambridge University Press, 2010.
Matthias Beck, Sinai Robins, Computing the continuous discretely: integer point enumeration in polytopes, 2nd edition, Springer-Verlag, 2016.
We will pick various sub-topics from the sources above, with some flexibility that may depend on the students' interests.
GRADING:
Homework = 30 %
1 Midterm = 30 %
Final exam = 40 %
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Graduate course, at IME/USP, the University of São Paulo
March - June, 2018
This is a graduate course which consists of an introduction to the analytic theory of numbers. Key topics in the area of Number Theory include advanced Linear Algebra, the finite Fourier Transform, Dirichlet L-functions, its functional equations and continuation, Dirichlet's theorem concerning primes in arithmetic progressions, group characters for finite abelian groups, theta functions, and possible applications.
Homework assignments:
HW 1. (due March 21)
In Apostol's book (1) below, please do the following exercises:
Chapter 3, number 4a 4b, 5a, 5b.
Also, prove that
$\sum_{k=0}^{n-1} \floor x + k/n \floor = \floor nx \floor$,
for all real numbers x, and positive integers n.
GRADING:
Homework = 30 %
1 Midterm = 30 %
Final exam = 40 %
1. Tom Apostol, Introduction to analytic Number Theory, Springer UTM series, 1998.
2. Tom Apostol, Modular functions and Dirichlet series in Number Theory, Springer GTM series, 2'nd edition, 2012.
3. Matthias Beck and Sinai Robins, Computing the continuous discretely: integer point enumeration in polytopes, 2'nd edition, 2015.
4. Edmund Hlawka, Johannes Schoissengeier, Rudolf Taschner, Geometric and analytic number theory, Springer Science & Business Media, 2012.
5. Ram Murty, Problems in analytic number theory, Springer GTM series, 2007.
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(at IME/USP, the University of São Paulo)
August - December 2017
Teacher Responsible: Sinai Robins
Objectives: To study topics in the Theory of Numbers,
With an emphasis on their algorithms and applications.
Rationale: Many algorithms developed for Number Theory have proved to be important both in mathematical research and in many applications. The subject is undergoing intense research and development.
Some topics we will cover:
The gcd algorithm, modular arithmetic, Fast exponentiation, the 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, Primality testing.
RSA, factoring algorithms. Quadratic residues, Gauss' law of quadratic reciprocity.
A geometric introduction to continued fractions, Pell's equation, factoring using continued fractions.
The LLL algorithm for finding a short vector in a lattice, a natural attempt to generalize the gcd algorithm to higher dimensions.
The main reference will be the book by Bressoud and Wagon, below. This text uses Mathematica calculations to facilitate the subject.
Sage will also be used on practical exercises along the course.
Bibliography
1. David Bressoud and Stan Wagon, Computational Number Theory, Wiley publishers, 2000.
2. Murray Bremner, Lattice Basis Reduction: an introduction to the LLL algorithm
And its applications, CRC Press, 2011.
3. John Harris, Karen Kohl, and John Perry, Peering into advanced mathematics through Sage-colored glasses, open-source text freely available online, based on sage calculations.
4. Tom Apostol, Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer-Verlag publishers, 1976.
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Computational Geometry
(at IME/USP, the University of São Paulo)
August - December 2016
We will use various references for our course, including mostly books but also research papers, and mainly picking some topics from the following sources:
Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars, Computational Geometry, 2008, 3rd revised edition, Springer-Verlag.
Martin Aigner, Gunter M. Ziegler, Proofs from the Book, 2010, 4th edition, Springer-Verlag.
Matthias Beck, Sinai Robins, Computing the continuous discretely: integer point enumeration in polytopes, 2016, 2nd edition, Springer-Verlag.
Satyan Devadoss, Joseph O'Rourke, Discrete and Computational Geometry, 2011, Princeton University Press.
Richard Stanley, Enumerative Combinatorics, 2012, 2nd edition, Cambridge University Press.
GRADING
Homework = 30 %
1 Midterm = 30 %
Final exam = 40 %
For a great introduction to rational functions of one variable and their precise relations to linear recurrence sequences, see chapter 4 of Richard Stanley's book:
Enumerative Combinatorics, 2nd edition.
Here are some interesting animations involving graphs:
https://upload.wikimedia.org/wikipedia/commons/7/71/Moebius-ladder-16-animated.svg
Homework
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Algebraic Graph Theory
(at IME/USP, the University of São Paulo)
August - December 2016.
References:
Algebraic Graph Theory, Norman Biggs, Cambridge University Press, 2nd edition, 1993.
Algebraic Graph Theory, Chris Godsil, Gordon Royle, Graduate Texts in Mathematics, Springer-Verlag, 2001.
An Introduction to the Theory of Graph Spectra, Dragos Svetkovic, Peter Rowlinson, Slobodan Simic, Cambridge University Press, 2010.
Matthias Beck, Sinai Robins, Computing the continuous discretely: integer point enumeration in polytopes, 2nd edition, Springer-Verlag, 2016.
We will pick various sub-topics from the sources above, with some flexibility that may depend on the students' interests.
GRADING
Homework = 30 %
1 Midterm = 30 %
Final exam = 40 %
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Some previous course that I've taught:
Brown University, 2014, Fall Semester: Harmonic analysis on polytopes, a graduate course.
Nanyang Technological University, Singapore, 2014, Semester 2: Topics in applied and discrete mathematics, an undergraduate/graduate seminar.
NTU, Singapore, 2012-2013: It’s a discreetly discrete world (A large undergrad course taught 3 times).
NTU, Singapore, 2012: Multivariable calculus with Linear Algebra, for CN Yang Scholars.
NTU, Singapore, 2011: It's a discreetly discrete world (301 students)
NTU, Singapore, 2011: Complex analysis (62 students)
NTU, Singapore 2010: It's a discreetly discrete world (347 students)
NTU, Singapore, 2010, Semester 2: Discrete Methods, a 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, Nanyang Technological University (158 students).
2005: MSRI / BIRS two-week summer course in Banff, Canada, with Matthias Beck (40 graduate students took the course).
Temple University, Number Theory (Ph.d. level), Differential Equations
Temple University, Combinatorial geometry, from my book with Matthias Beck: “Computing the Continuous Discretely”, Springer UTM series, 2006.
Temple University, Number Theory (undergraduate), Cryptography and information theory for IST majors (undergraduate)
Temple University, Discrete Geometry, combinatorial topology/knots.
Temple University, Calculus I, II, III, real Analysis.
Temple University, Analytic Number Theory, Algebraic Number Theory
Temple University, For All Practical Purposes (FAPP for non-majors).
UCSD: Calculus for the Life Sciences I, II, III, Spring of 1995-96.
UNC: Calculus I, II, III, Abstract Algebra (Masters level), Analytic number theory.
UNC: Math for Elementary School Teachers.
UNC: History of Mathematics.
UNC: Real Variables I, II, Math for the Liberal Arts, Linear Algebra.
UNC: Seminar in Number Theory, Topics in Finite Mathematics, Seminar on Mathematical Discovery using Polya's books.
Some Videos:
Google headquarters (googleplex), "Bay Area Discrete Math Day XII: integer linear programming", 2007.
Teaching Masterclasses for kids:
Royal Institution Masterclasses, 2009 December Program, Singapore
Books:
(Joint with Matthias Beck), published in the Springer Undergraduates Texts series:
"Computing the continuous discretely: integer point enumeration in polytopes", Springer 2008, Undergraduate Texts in Mathematics. [Get the pdf, and see reviews of our book, here]
Fourier analysis on polytopes, and the geometry of numbers, part I: a friendly introduction, 2024, by the American Mathematical Society book series "the student mathematical library":
https://bookstore.ams.org/view?ProductCode=STML/107
Undergraduate dissertation students, in NTU/ Singapore :
Czarina Ann Marifosque, 2008
Nhat Le Quang , 2009
Stephanie Gabriela, 2010
Danh Nguyen Luu (Danny), 2011
Zhang Yichi, 2012