Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, May 2, 2006.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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30


Tuesday, May 2, 2006

12:00 pm in 322 Loomis,Tuesday, May 2, 2006

#### Electric-Magnetic Duality and the Geometric Langlands correspondence following Kapustin and Witten

###### Dave Morrison (Duke University)

Abstract: Kapustin and Witten have recently shown how to relate electric- magnetic duality for Yang-Mills theories in four dimensions, mirror symmetry for sigma-models in two dimensions, and the geometric Langlands program. I will give an informal discussion of some aspects of this work.

12:30 pm in 347 Altgeld Hall ,Tuesday, May 2, 2006

#### Noncommutative Maximal Function

###### Marius Junge (UIUC)

Abstract: We are starting early! I will talk about the noncommutative version of Birkhoff's maximal theorem and applications to differential operators on free groups.

1:00 pm in 345 Altgeld Hall,Tuesday, May 2, 2006

#### Groups and the N.I.P (continued)

###### Krzysztof Krupinski (UIUC MAth)

Abstract: We will conclude our exposition of Section 7 of the Groups, Measure and the NIP paper, continuing where we left of Friday in the model theory seminar.

2:00 pm in 243 Altgeld Hall,Tuesday, May 2, 2006

#### No meeting this week

3:00 pm in 241 Altgeld Hall,Tuesday, May 2, 2006

#### Combinatorial Properties of the Stern Sequence

###### Bruce Reznick (UIUC Math)

Abstract: In this talk, we survey some old and new results about the Stern sequence, a highly underappreciated mathematical object. It is defined by the recurrence s(2n) = s(n) and s(2n+1) = s(n) + s(n+1), with s(0) = 0 and s(1) = 1. It is most easily written by imagining a Pascal triangle with memory, and starting with (1,1). The rows of the resulting"diatomic" array give s(n) for 2r \le n \le 2r+1:

(r=0): 1 1
(r=1): 1 2 1
(r=2): 1 3 2 3 1
(r=3): 1 4 3 5 2 5 3 4 1

Stern himself proved in 1858 that every ordered pair (a,b) of relatively prime positive integers occurs exactly once as the pair (s(n),s(n+1)), and the binary representation of n is encoded by the continued fraction representation of a/b. The maximum values in the rth row is the (r+2)-nd Fibonacci number. The entries in the rth row sum to 3r + 1; the sum of the cubes of the entries is 9*7r-1 (for r > 0). The Stern sequence also has many interesting divisibility properties: s(n) is even iff n is a multiple of 3; the set of n for which s(n) is a multiple of 3 has a simple recursive description. The set of n for which s(n) is a multiple of the prime p has density 1/(p+1). Further, s(n) is the number of binary representations of n-1, if one allows digits from {0,1,2}. If n is odd and m is the integer whose base 2 representation is the reversal of n's, then s(n) = s(m) and s(n+1)s(m+1) = 1 mod (s(n)). The Stern sequence affords a clear understanding of the alluringly-named Minkowski ?-function, which gives a strictly increasing map from [0,1] to itself, taking the rationals to the dyadic rationals, and the quadratic irrationals to the non-dyadic rationals. There are few other mathematical objects which are as generous in their grooviness; as one final example, s(729) = 64.

3:00 pm in 243 Altgeld Hall,Tuesday, May 2, 2006

#### Divisors on Kontsevich Moduli Spaces

###### Izzet Coskun (MIT mathematics)

Abstract: In the last decade the Kontsevich moduli spaces of stable maps have emerged as an invaluable tool for answering questions of algebraic geometry, mathematical physics and combinatorics. In joint work with Joe Harris and Jason Starr, we were able to describe which divisors on the Kontsevich spaces are effective and which are ample. I will describe these results and discuss some applications to rationally connected varieties.

3:00 pm in 443 Altgeld Hall,Tuesday, May 2, 2006

#### A nonstandard proof of the Ergodic Theorem, revisited

###### David Ross (visiting UIUC from Hawaii 05-06)

Abstract: Kamae's nonstandard proof of Birkhoff's Ergodic Theorem used the von Neumann-Maharam structure theory for measure spaces, a common uniformizing trick from nonstandard analysis, and a clever approximation argument. Later standardizations of this proof lost the power of the nonstandard uniformization. I'll show that the structure theory in Kamae's original proof can be replaced by a much simpler appeal to S-measurability, while retaining the simplicity of the rest of the argument.