Lecture Summaries

• Lecture 1: Monday, September 8, 2008.
  I collected the e-mail addresses of everyone in the course. If you did not provide this information, please send me an e-mail message.

  I discussed the course web page, the project ground rules and pointed out there is a page of ideas for projects. On Wednesday, September 10, I hope to identify people who are willing to present their term project presentation on September 26.

  I talked about the kinds of questions we will address in this course. Here are some notes that describe the problems; you should only look at section 1.5.

• Lecture 2: Wednesday, September 10, 2008.
  I continued talking about the kinds of questions we will address in the course. I talked about morphisms, iterated morphisms, and some questions involving them. These questions were already in the Lecture 1 notes given above.

• Lecture 3: Friday, September 12, 2008.
  I proved that the Thue-Morse morphism μ preserves the property of overlap-freeness, and hence that the Thue-Morse word is overlap-free. I proved the existence of squarefree words.

  The notes are here.

  September 15-19: Instructor away at a conference.

• Lecture 4: Monday, September 22, 2008.
  I proved the Restivo-Salemi factorization theorem (see the notes) and that there are only polynomially many overlap-free words of length n.

  For Wednesday, we will discuss Open Problem 1 on the open problems list.

• Lecture 5: Wednesday, September 24, 2008.
  I proved that the lexicographically least overlap-free word over { 0,1 } starting with 1 is t, and the lexicographically least overlap-free word over { 0,1 } starting with 0 is 001001t.

  In our open problem session, we discussed Open Problem 1 on the open problems list. This list has been updated.

• Lecture 6: Friday, September 26, 2008.
  The first of our student presentations: Zhi Xu talked about multidimensional nonrepetitive words. He talked about some results in the classic paper of Bean, Ehrenfeucht, and McNulty, and in this paper of Currie and Simpson. There is a 1988 paper of Carpi that has even stronger results.

• Lecture 7: Monday, September 29, 2008.
  I talked about De Bruijn words (words that contain each block of length n over a k-letter alphabet exactly once). I proved that they exist, and gave an algorithm to generate them. I discussed the connection between these words and a certain directed graph. For more details, see the course notes.

  For Wednesday, we will discuss the open problems numbers 4 and 7 on the open problems list.

• Lecture 8: Wednesday, October 1, 2008.
  We mostly discussed problems 7 and 9 on the open problems list. For problem 7, there were several ideas proposed:
  • I gave a construction that gives an upper bound of 3n - o(n) on the length of the shortest word, basically by exploiting the fact that you can go through all the cyclic shifts of a word cheaply.
  • by Yu Hin Au, who offered a possible algorithm to generate the shortest string containing all the squares of length n (However, it later seemed to fail for n = 5.)
  • by Peter Nelson, who proposed a construction based on the De Bruijn graph, and
  • by Elyot Grant, who offered an argument based on assigning weights to edges of the de Bruijn graph

  Mathieu Guay-Paquet offered some observations on problem 9 (smallest DFA accepting all words having as factors all strings of length n).

• Lecture 9: Friday, October 3, 2008.
  Mathieu Guay-Paquet gave a presentation on de Bruijn words. He covered material on how to generate them, in two different ways: algebraically, using a primitive polynomial over a finite field, and as the concatenation of Lyndon words of length dividing n.

• Lecture 10: Monday, October 6, 2008.
  I talked about decision procedures for checking whether a word generated by a uniform morphism (or a k-automatic sequence) is r-power-free, for any rational number r. See the lecture notes here.

  For Wednesday, please look at problem 4 on the open problems list.

• Lecture 11: Wednesday, October 8, 2008.
  Mathieu Guay-Paquet presented some ideas about problem 4. He defined ψ(a, b) as the lexicographically least overlap-free word starting with a and ending with b. He conjectures that φ^l(k) = ψ(k,k+l), where φ is the morphism discussed here. He observes that ψ(k,k+1) = k ψ(0,k) ψ(0,k) k^-1 (k+1) .

  Peruvemba Ravi suggested an attack on the question of whether φ(a) is a palindrome, if you drop the first and last letter. He observes that φⁱ(φ^j(0)^R)² = φ^j(φⁱ(0)^R)²)^R .

  Eliot Grant discussed his program to find the length l(n) of the shortest binary string containing all the squares of length 2n. He gives the following table:

  n	l(n)
  1	4
  2	12
  3	24
  4	48
  5	82
  6	166
  7	294

• Lecture 12: Friday, October 10, 2008.
  Gary Au discussed generalizations of de Bruin words. His presentation can be found here.

• Monday, October 13, 2008.
  No class (Thanksgiving Holiday).

• Lecture 13: Wednesday, October 15, 2008.
  The Lovász local lemma and its applications. See the Lecture 13 notes here.

• Lecture 14: Friday, October 17, 2008.
  Peter Nelson discussed "squarefull" words, that is, words that have a square (or infinitely many squares) starting at every position. See this paper by Currie and Rampersad.

• Lecture 15: Monday, October 20, 2008.
  More about the Lovász local lemma. The Lecture 13 notes have been revised.

• Lecture 16: Wednesday, October 22, 2008.
  Solutions to Problem Set 1 are available (get username and password from instructor). Problem Set 2 is now available; it is due November 5.

• Lecture 17: Friday, October 24, 2008.
  Hussein Hirjee spoke about avoiding repetitions in music, specifically the music of Per Nørgård.

• Lecture 18: Monday, October 27, 2008.
  Abelian repetitions. The course notes are here.

• Lecture 19: Wednesday, October 29, 2008.
  We discussed open problem 21. Prof. Shallit started by sketching the 1997 Cummings-Smyth "proof" of a lower bound of Ω(n log n) on the problem of determining if a length-n string has an abelian square as a subword. However, it appears this proof is incomplete. We tried various strategies for a lower bound, including reducing from element distinctness.

• Lecture 20: Friday, October 31, 2008.
  Thomas Ang discussed avoiding abelian powers (papers of Pleasants and Keränen). For his slides, go here.

• Lecture 21: Monday, November 3, 2008.
  More about abelian powers; the work of Dekking.

• Lecture 22: Wednesday, November 5, 2008.
  We discussed open problems 23, 24, and 25 from the open problems list. Open problem 26 has been solved. Thomas Ang reported on his progress on open problem 20.

• Lecture 23: Friday, November 7, 2008.
  The solution to open problem 26. Arbitrarily large abelian squares cannot be avoided over an alphabet of size 2. The course notes are here.

• Lecture 24: Monday, November 10, 2008.
  Introduction to Dejean's conjecture. The notes are here. Problem Set 3 is now available; it is due November 24.

• Lecture 25: Wednesday, November 12, 2008.
  I handed back the marked problem set 2. We discussed open problems 27-30. Elyot Grant sketched a possible solution to problem 28; the writeup is here.

  The solutions to Problem Set 2 are now available.

• Lecture 26: Friday, November 14, 2008.
  Hamed Shirazi spoke about Dejean's conjecture.

• Lecture 27: Monday, November 17, 2008.
  Algorithms for finding squares in a string of length n. The notes are here.

• Lecture 28: Wednesday, November 19, 2008.
  Gary Au presented some of his results. For example, he showed that if an infinite word w has the property that in any prefix, the difference between the number of occurrences of the most frequently-occurring letter and the least frequently-occurring letter is bounded, then w has abelian powers of all orders. He noted that this could be generalized to the case where the difference between the Parikh vector of a prefix of length n and nv (where v is a vector with rational entries) is bounded.

  He also discussed the case of using the word 01020103010201204... in the naive algorithm for square testing, in which as you get Ω(n² log n) comparisons.

  Prof. Shallit finished up the discussion of the efficient algorithm for square testing.

• Lecture 29: Friday, November 21, 2008.
  Thomas Stoll will talk about his results on diversity of sequences.

• Lecture 30: Monday, November 24, 2008.
  Results about the complexity of pattern matching. The notes are here.

• Lecture 31: Wednesday, November 26, 2008.
  Counting abelian squares. The notes are here.

• Lecture 32: Friday, November 28, 2008.
  Peruvemba Ravi will talk about the paper of Bell and Goh.

• Lecture 33: Monday, December 1, 2008.
  Guest lecture by Dan Brown on tandem repeats and pattern-matching in biological sequences.