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

Papers starting 1985

Papers starting 1995

Papers starting 2005

Papers starting 2015

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

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

A language page.

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Teaching Fall 2018:
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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 434/534, Set Theory

Here you can find the syllabus.
All assignments, home works, and exams are announced in class.

ASSIGNMENTS:    Tues, Oct 02: 1.1: all. 1.2: 1--9a. 1.3: unstarred questions 1--8.
                                Tues, Oct 09: 1.3: 9--11; 1.4: 1--14, 18.
                                Tues, Oct 16: 1.5: 1--4; 2.1 & 2.2 all unstarred. Also do exercise in this note.
                                Tues, Oct 23: 2.3: unstarred. Extra: use Zorn to prove that every vector space has a basis (see pg 123).
                                                            Extra: 1) Let Si be the set of points on the circle of radius i. Show that |S1|=|S2|.
                                                                      2) Show that in the reals: |[0,1]|=|(0,1)|.
                                Tues, Oct 30: 2.4 and 2.5: unstarred. Extra:
                                                     1) Show that cancelation does not hold for cardinals:  aa=a does not imply a=1
                                                                                                                                      a+a=a does not imply a=0.
                                                     2) Prove that A_0A_0=A_0 and A_0+A_0=A_0 without using 2.4 or 2.5. (A_0 is aleph0).
                                                     3) Prove that for cardinals (a+b)+c=a+(b+c) and ab=ba.
                                                     4) Prove that the set of roots of polynomials with integer coefficients is countable.
                                Tues, Nov 06: 2.6: unstarred problems plus problem 3.
                                Tues, Nov 13: 3.1: unstarred problems.
                                Tues, Nov 20: 3.2: unstarred problem PLUS:
                                                      1) Complete the proof that =< gives a well-ordering on the ordinals on pg 55, just above Thm 24.

                                                    
EXAMS:                  Midterm1:    Thurs, Oct 18, in-class. Chapter 1.
                                Midterm2:    Thurs, Nov 08, in-class. Chapter 2.

                          Final:        Tues, Dec 04, 17:30-19:20.

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Here you can find the syllabus.
All assignments, home works, and exams are announced in class.

ASSIGNMENTS:    Tues, Oct 02:  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 1/2 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 09:  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 16:  Study Ch 2. Study these notes.  Do at least one of:
                                                        1) Numerically determine Feigenbaum's constant (as outlined in Sternberg).
                                                        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 23: 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 conjugat.
                                   Tues, Nov 06: Study Ch 5. Do the following HWs.
                                                        1) Complete the missing details of the proof on pg 109.
                                                        2) Prove (or disprove) the ballot theorem and its application on pg 118 if probability of voting P is not 1/2.
                                                        3) Does Lemma 5.4.3 still hold in that case?
                                                        4) Verify statements about the simulation on pg 121.
                                                        Optional: Why is the Von Neumann thm better suited for numerical work than the Birkhoff  thm?
                                                                       Why are translations not considered as finite dimensional isometries?
                                   Tues, Nov 13: Study Ch 5. Study Ch 6.
                                                         1) Show that the quantity C/(t(1-t))  is invariant under \phi_a  (pg 124) (Use Jaynes' paper, pg 20.)
                                                             (Note: it does not integrate to 1, so it is actually not a density but something called an improper
                                                             density.)
                                                         2) Show that the space X on pg 138, is a complete metric space.
                                   Tues, Nov 20: Study Ch 7.
                                                         1) Apply the contraction in the bottom of pg 141 to  the diff. eqn   x'=3x   on   [0,1]   and   x(0)=2.
                                                         2) Find an open set for Moran's condition for the middle 3rd Cantor set and one for Sierp. gasket.
                                                         3) Write a program to generate the Sierpinski gasket. Use the program to draw other fractals in R^2.
                                                         4) In R^2 we are given the iterated function system T_i(x) = 0.5 (x + d_i), where d_i are the 4
                                                             corners of the unit square. Prove that the unique compact invariant set is the unit square.
                                                         5) Optional: finish the sketch of the proof completeness on page 145.
                                                         6) Optional: in addition (see question above), show that if X is compact, then H(X) is compact.

EXAMS:                  To be discussed in class. (Change in Final: Dec 3, 10.15-12.05 in CH 254)

General Announcement 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/