MAT 402: Seminar in Mathematics (Fractals)

Spring 2017

Christopher Bishop

Professor, Mathematics
SUNY Stony Brook

Office: 4-112 Mathematics Building
Phone: (516)-632-8274
Dept. Phone: (516)-632-8290
FAX: (516)-632-7631

TuTh 10:00-11:20, Math Tower 4-130 (moved from Earth and Space Sciences 177)

Final Exam 11:15am-1:45pm Friday, May 12

We will follow the book "Fractals in Probability and Analysis" by myself and Yuval Peres of Microsoft Research. This is currently being published and I hope that hard copies will be available by the beginning of class. If not, then here is a link to PDF version of the final version .

The course is about various fractal sets including Cantor sets, snowflake curves, no-where differentiable functions, Kakeya sets and random sets, such as Brownian motion. The main idea is the measure the complexity of such fractals in terms a number called "dimension". There are several types of dimension that we will consider: Hausdorff, Minkowski and packing dimensions (and many others occur in the literature that we will not cover). We will discuss how to compute the dimension of various examples and prove some basic theorems about dimension.

You will need to know some basic analysis (metric spaces, open and closed sets, Euclidean spaces,...) and some measure theory will help (as in MAT 342), but I will try to fill any gaps as they arise.

I will start by giving a few lectures on the initial sections, but after a week or two I will make assignments for students to present in class. We may also due preserntations of solutions of problems from the text. After we have covered the "basics", I will ask students to choose topics to present (individually or in small groups) from the later chapters or from (approved) related sources.

Grades will be based on:
(1) oral persentations of text material in class;
(2) written solutions to exercises in the text (these might also be presented in class);
(3) a final 30-40 minute in-class presentation and accompanying 4-5 page written report on a subject of the students choosing (related to the course topic and subject to intructor's approval);

I will assign topics and dates for most of the semester; final presentations will scheduled two per class meeting for the last few weeks of the class and during our final exam period. These will be done on a first come, first served basis, once the student has selected a topic and had it approved by me.

Final presentations can be on some topic in the book that we did not cover as a class, or on some related topic, e.g., giving a summary of paper that is referenced in the book, or describing more recent progress on a topic that we have discussed. The presentation should be accompanied by a written report that is shared with the other students. Feedback can be obtained to make corrections and improvements and a final draft submitted to me by the end of the semester.

Students can earn points by doing exercises from the text and writing up solutions, preferable in LaTex. A point will be given for the first correct solution to a problem submitted to me by email. Problems that already have a solution given in the text are not eligible.

Send the lecturer (C. Bishop) email at: bishop - at -

Link to history of mathematics There are a lot of iteresting articles here. If you know of other math related sites I should link to, let me know.

Exercise Sign Up

Here is a Google document where you can tell me and the others in the class which exercises you plan to turn in. I prefer different peope do different exercises, but I am willing to allow credit to be split between different people, if they collaborate on a problem. If possible, prepare your solution in LaTeX and include the statement and number of the problem. Possibly I will offer separate points for a written solution to a problem and its presentation in class.

Here is a sample LaTex file and here is what the corresponding PDF

Start by seeing if you can produce the PDF on your own by running LaTex on the sample LaTex file.


challenge 1 - finding citations

Other links

Here is a good, brief review of that basic facts from real analysis and measure theory.

A link to MathSciNet (Online Mathematical Reviews). These gives access to reviews of about 3 million mathematical papers and links to some of the actual articles. You will need a NETID and password to access this.

A link to the Stony Brook virtual SINC site . You will need a NETID and password to access this.

A link to BlackBoard . You will need a NETID and password to access this.

POINT SYSTEM: 5 points per assigned presentation (three possible), 5 points for final presentation, 5 points for final written report, 3 points each for written exercise solution or challenge problem. 2 points each for presentation of an exercise solution in class.

GRADES: A = 30 points or better, B= 25-29 points, C = 20-24 points.

FINAL PRESENTATIONS : Send me an email description of what you want to do. If I approve it, then go to the Google documents sheet sheet (same as for exercises) and write your title and date you want to make your presentation. We will do two presentations per meeting (40 minutes each), and we will save the final exam period for any make-ups for missed lectures. A first draft of the written report should be available by your presentation, but final version is due by final exam period. Something in the range 5-10 pages will be fine. If you hand in drafts earlier, I will look at them and make suggestions for the final version. Choose something that is interesting to you and have fun with it. Your presentations should be accessible, i.e., give the general idea and some examples, if appropriate; if there are technical details you can save some of these for the written version.

Lecture Schedule (as of Feb 13)

Tentative topics to cover
Minkowski dimension
Hausdorff dimension
Mass distribution prinicple
Sets defined by digit restrictions
Billingley's lemma
The law of large numbers
Sets defined by digit frequency
Slices of sets
Self-affine sets
The Weierstrass nowhere differentiable function
Besicovitch-Kakeya sets

Tuesday, January 25:
Bishop - Introduction to course, measures and dimensions, examples of fractals, possible topics
Thursday, January 27:
Bishop - Defn of Minkowski dimension
Tuesday, January 31:
Bishop - Defn of Hausdorff dimension
Thursday, February 2 :
Brawley - page 6, lines 1-16
Capobianco-Hogan - Prop 1.2.6 and Example 1.2.7
Genkin - Lemma 1.2.8
Hyland - Example 1.2.9
Tuesday, February 7:
Hyland - continued
Kolins - Example 1.2.10 to end of Section 1.2
Marinelli - Begining of Section 1.3 page 11, line -6
Monzongo Zola - Defn 1.3.1 and Example 1.3.2
Thursday, February 9:
Classes canceled due to an abundance of snow and caution (= snuation)
Tuesday, February 14:
Mongzong Zola - finishes Example 1.3.2 Pacun - Example 1.3.3
Palmeri - Example 1.3.4
Thursday, February 16:
Puszklewicz - Lemma 1.4.1
Bishop - Comments a,b,c following Lemma 1.4.1
Stein - Example 1.4.2 and Defn 1.4.3
Tuesday, February 21:
Prof. Bishop away, class wiil be covered by Prof Schul
Suk - Lemma 1.4.4 Example 1.4.5
Takeuchi - Theorem 1.5.2
Zhang - Example 1.5.1, Defn of \mu on page 23, Lemma 1.5.4
Vayda - Example 1.5.5
Thursday, February 23:
Prof. Bishop away, Guest Lecture by Prof Rempe-Gillen
Special time: 10:30-11:20 am (half-hour later start)
Tuesday, February 28:
Bishop - remarks on the strong LLN
Zhang - continues
Thursday, March 2:
Zhang - continues
Vayda - Example 1.5.5
Brawley - Theorem 1.6.1 and proof
Capobianco-Hogan - Examples 1.6.3 and 1.6.4
Tuesday , March 7:
Genkin - Defn 5.1.1, Lemma 5.1.2
Hyland - Example 5.1.4
Kolins - Defn 5.1.5, Lemma 5.1.6
Marinelli - Defn 5.1.7, Lemma 5.1.8
Thursday, March 9: no class
Tuesday, March 14: SPRING BREAK - no class
Thursday, March 16: SPRING BREAK - no class
Tuesday, March 21:
Monzongo Zola - Review Fourier series, Section 5.2 before Defn 5.2.1
Pacun - Defn 5.1 and to bottom of page 142
Palmeri - page 143 before Lemma 5.2.2, Do Exercise 5.17 -
Puszklewicz - Statement and proof of Lemma 5.2.2
Thursday, March 23:
Stein - Statement and proof of Theorem 5.2.3
Vayda - Final paragraph of Section 5.2, summarize Exercises 5.24 and 5.25
Suk - State and prove 5.3.1
Tuesday, March 28:
Takeuchi - State Theorem 5.3.3, and proof up to line 9, on page 148
Zhang - Finish proof of Theorem 5.3.3
Brawley - Theorem 9.1.1
Thursday, March 30:
Capobianco-Hogan - Lemma 9.1.2
Genkin - Theorem 9.1.3
Tuesday, April 4:
Hyland - Theorem 9.1.4
Kolins - Section 9.2, page 276
Marinelli - Section 9.2, page 277
Thursday, April 6: final presentations
Monzongo Zola - Section 4.1, before Theorem 4.1.1
Stein - Theorem 4.1.1
Takeuchi - State Theorem 4.2.1 and Defn 4.2.2
Tuesday, April 11: final presentations
Suk - Proof of Theorem 4.2.1 on pages 122 and 123
Zhang - finish proof on page 124
Vayda - Defn 4.3.1 and Theorem 4.3 in special case of power functions.
Thursday, April 13: Bishop - random sets and harmonic measure
Tuesday, April 18: SCGP CG Geometry Workshop Workshop schedule , Workshop webpage , Workshop registration (free, for headcount only) ,
Thursday, April 20: SCGP CG Geometry Workshop
Tuesday, April 25: Genkin (Exponential map is chaotic),
Thursday, April 27: Brawley (packing dimension), Kolins (Mandelbrot set/ Julia sets)
Tuesday, May 2: Suk (transcendental Julia sets), Takeuchi (finite field Kakeya sets)
Thursday, May 4: Pacun (topological dimension), Puszklewicz (Klienian groups)
Monday, May 8, reading day, 10am-2pm: Zhang (Mandelbrot and Julia sets), Marinelli (invairance of domain), Palmeri (Brownian motion), Capobianco-Hogan (diffusion limited aggregation)
Friday, May 12, final exam perion, 11:15am-1:45pm: Monzonga Zola (non-measurable sets), Stein (self-affine sets), Hyland (TBA)

Ideas for final reports

Anything from the book that we don't cover in class, something suggested in the Notes or Exercises sections, some topic that comes up in class. Almost anything related to fractals or dimension would be OK, as long as I approve it

Some random ideas: Topological dimension, intersections of translates of Cantor sets, Fourier dimension, self-similar sets, self-affine sets, packing dimension, the Appolonian gaskit, Poincare sets, microsets, the projection theorem, random Cantor sets, Brownian motion, discrete Hausdorff dimension, Borel sets, analytic sets, Baire category, Julia sets, the Mandelbrot set, limit sets of Klienian groups, Schramm-Lowener evolution (SLE), simple random walks, calculus and differential equations on fractals, the open set conditions, code to draw some fractals, compute the dimesion of some interesting set not in the book.

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