Newspaper (El Pais, Spain) articles August 28, 1988 and March 19, 1989.

Papers starting 1985

Papers starting 1995

Papers starting 2005

Papers starting 2015

Papers starting 2020

Research seminar series at Portland State University (2005, 2006, 2010, 2011)

Research seminar series at Rockefeller University (1996, 1997, 2008, 2009)

A language page.

Short Bio and Description of Research

Pescara 2019 summer school and conference on
Mathematical Modeling of Complex Systems

In the summer of 2019, I participated in a summer school and conference in
Mathematical Modeling in Pescara, Italy. I gave a mini-course on the theory
of directed graphs in the summer school: Part 1, part2, part 3, and part 4.
I also served as guest editor of the Proceedings of the conference. A description
of the meeting and links to the papers presented can be found here:
https://www.euro-acad.eu/news?id=50

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Lecture Notes on Number Theory

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Teaching FALL 2020:

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IMPORTANT 1: All answers on home works and exams must be justified,  even if  that is not evident from
                              the phrasing of the question. Answers without justification will receive partial credit at best.
IMPORTANT 2: Before turning in exams or HW's, write your first plus last name in the top right corner
                              of each sheet you turn in (even if you staple them together)!

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MTH 449/549, Number Theory

Here is a flyer, the syllabus, and three months of free lecture notes  for a newly designed course in number theory.
This a sequence of two courses MTH 449/549 and MTH 410/510 (fall 2020 and winter 2021). The
material is intended to give a general overview of all branches of number theory with an emphasis on
more geometric proofs. In due course, we hope to follow this up with a final course
discussing some of the many applications of number theory.

ASSIGNMENTS:
              Tues, Oct 06:           Ch 1.
              Tues, Oct 13:           Ch 2. 
              Tues, Oct 20:           Ch 2.
              Tues, Oct 27:           Ch 3.
              Tues,                               

EXAMS PLANNED (may be modified in class):  Exam 1 due: Thurs, Nov 05, 14:00 hrs.
                                  

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MTH 621, Advanced Differential Equations

Here is a flyer  and the syllabus for my advanced course in dynamical systems MTH 621/2/3. The emphasis this year
will be on surveying the wide range of this branch of mathematics and its applications, rather than
probing deep theorems. We will use a great -- and very accessible -- text written by the eminent
mathematician S. Sternberg plus some  other resources to be determined during the course.
This is three term sequence. In Fall and Winter, we will go through the entire book by Sternberg.
Spring of 2021 will most likely be organized as a 'topics' course. This will in part depend on students'
interests. Among the possibilities are topics in Ergodic Theory, Number Theory, Quantum Mechanics,
and so forth.
To all dynamics students I strongly recommend MTH 610, Directed Networks, as an excellent
companion course to this sequence. It will be taught in Spring 2021. A description can found below.

ASSIGNMENTS:

Tues, Oct 06:  Study Chapter 1. Look at the Julia Set for Newton's method applied to z^3-1: 
                                                           https://en.wikipedia.org/wiki/Julia_set#/media/File:Julia-set_N_z3-1.png
                                                       And do one plus 2 of the following:
                                                       1) Perform a Newton's method in 3 dimensions.
                                                       2) Optimize Sternberg's proof of the Newton's method (quadratic convergence).
                                                       3) Show that the secant method has order golden mean convergence. 
                                                       4) Work out the renormalization arguments on pg 27--29.
                                                           In particular, argue that convergence to 1 suggests a large cluster.
 Tues, Oct 13:  Study Chapter 2, pg 33-56.  Do at least one in great detail:
                                                        1) Plot the attractors of x->mu.x(1-x) as in Figure 2.9, but for mu in [3.5,3.8].
                                                         and numerically determine for which mu the orbits of period 2, 4, 8, 16, and 32 are superstable.
                                                        2) Write a different proof for the related pitchfork bifurcation (what is the relation?).
Tues, Oct 20:  Study Ch 2. Steven Wolfram wrote a nice obituary for Feigenbaum (who passed in June 2019):
                                          https://writings.stephenwolfram.com/2019/07/mitchell-feigenbaum-1944-2019-4-66920160910299067185320382/
                                                        Study these notes.  Do at least one of:
                                                        1) Numerically determine Feigenbaum's constant (as outlined in Sternberg, see also:       
                                                            https://pages.physics.wisc.edu/~snc/papers/coppersmith%20am%20j%20phys%201998.pdf). 
                                                        2) Set g(y) = 1+ay^2+by^4 and try to solve eqn (2.13) for a and b.
                                                        3) Give a full proof for the Sarkovski theorem. It is mandatory to include explicit
                                                            references to all your sources for this home work.
Tues, Oct 27: Study Ch 3 and Ch 4. Do 2 out of the following:
                                                        1) Do this HW.
                                                        2) Draw (numerically) the Mandelbrot set.
                                                        3) Show that conjugation of  f : R -> R by a diffeo does not change the 
                                                            eigenvalues at a periodic fixed point.                                                        
                                                        4) Fig 3.6 Sternberg: How long does F^n(x) stay in the ``bottleneck'' as fn of the 
                                                            distance of F to the diagonal.
Tues, Oct 30: Study Ch 4. Do the following HW's:
                                                        1) Prove that closed nested intervals in R have non-empty intersection.
                                                        2) Show that if the map i on pg 100 is not 1-1, then there is an interval J  and a K>0 such that:
                                                            for all n>0, Q^n is monotone and  | DQ^n (x_n) | < K for some x_n in J.
                                                        3) Give an example of a map whose periodic orbits are dense, but is not chaotic.
                                                        4) Give an example of a map with sensitive dependence, but which is not chaotic.
                                                        5) Give an example of a map which is top. trans., but which is not chaotic.
                                                        6) Prove Thm: If f and g have attracting fixed points at the origin, then f and g are locally conjugate.
                                                        7) Prove Thm 4.6.1 for c < -2.

EXAMS PLANNED (may be modified in class):  Exam1: Due Tue Oct 27, 17:15hrs. Turn in 1 problem for each of Ch 1, 2, and 3.
                                  

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Other Teaching 2020-2021:
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Spring 2021:  DIRECTED NETWORKS ::
Here is a flyer containing references to freely available lecture notes for this newly designed course  
describing theory and applications of directed graphs (MTH 610, spring 2021). This is a very current area with
applications from data science to flocking, finance, chemical networks and others. In spite of its current
relevance, good textbooks are still very rare. So we developed our own course, and are writing a textbook
based on this course.

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General Announcements for Students:
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In most of my classes you will be either strongly encouraged,
or even obliged to turn in your HW in *.pdf format based on LATEX.
Here is a website where LATEX is explained: 
http://www.ctan.org/tex-archive/info/Math_into_LaTeX-4/