UIUC Dept. of Mathematics Seminar Calendar

      July 2009             August 2009           September 2009   
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
           1  2  3  4                      1          1  2  3  4  5
  5  6  7  8  9 10 11    2  3  4  5  6  7  8    6  7  8  9 10 11 12
 12 13 14 15 16 17 18    9 10 11 12 13 14 15   13 14 15 16 17 18 19
 19 20 21 22 23 24 25   16 17 18 19 20 21 22   20 21 22 23 24 25 26
 26 27 28 29 30 31      23 24 25 26 27 28 29   27 28 29 30         
                        30 31

Tuesday, August 25, 2009

Number Theory Seminar
1:00 pm Tuesday, August 25, 2009

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Submitted by ahlgren.

No talks this week.
Abstract: The Seminar will begin on Tuesday, September 1. Contact Scott Ahlgren or Jeremy Rouse to volunteer a talk.

Thursday, August 27, 2009

Friday, August 28, 2009

Model Theory and Descriptive Set Theory
4:00 pm in 345 Altgeld Hall, Friday, August 28, 2009

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Submitted by w-henson.

van den Dries / Henson / Solecki (UIUC Math)
Sums and Products -- organizational meeting
Abstract: This semester we will focus our Friday logic seminars on a new development of general mathematical interest, involving descriptive set theory, combinatorics, number theory, and model theory. At this organizational meeting, a plan for the semester will be worked out. The first paper we'll read is "Borel subrings of the reals" by G.A. Edgar and Chris Miller. A main result of this paper is that if E is a proper subring of the field of real numbers that is also a Borel set, then the Hausdorff dimension of E is 0. Indeed, they show that for all n, the set E^n has Hausdorff dimension 0 as a subset of R^n. Further work to which our seminar will aim involves papers by Jean Bourgain, Terry Tao, and others, and especially a preprint by Udi Hrushovski. All interested people are warmly invited to participate.

Tuesday, September 1, 2009

Wednesday, September 2, 2009

Special Topology, Group Theory, and Number Theory Seminar
3:00 pm in 345 Altgeld Hall, Wednesday, September 2, 2009

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Submitted by nmd.

Jordan Ellenberg (University of Wisconsin, Madison)
Congruence subgroups of mapping class groups (especially braid groups.)
Abstract: Abstract: There is a notion of "congruence subgroup" of a mapping class group which appears to have certain properties in common with congruence subgroups of arithmetic groups. We will discuss several open problems in the theory of congruence subgroups, some progress, and applications: for instance, the proof of a "strong congruence subgroup property" for the mapping class group of a once-punctured elliptic curve answers an open question from dynamics about Teichmuller curves in moduli spaces of curves (joint work with D.B. McReynolds). In number theory, theorems about stabilization of cohomology of congruence subgroups yield results towards the Cohen-Lenstra heuristics for ideal class groups over function fields over finite fields. (joint work with A. Venkatesh, C. Westerland.)

Thursday, September 3, 2009

Tuesday, September 8, 2009

Thursday, September 10, 2009

Friday, September 11, 2009

Tuesday, September 15, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, September 15, 2009

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Submitted by ahlgren.

Harold Diamond (UIUC Math)
A biased coin calculation with a biased number theoretic outcome
Abstract: The topic of the talk is the probability that one will be ahead after n tosses of a coin whose probability of heads is 3/5 and that of tails is 2/5. To compensate, each occurrence of heads wins $2 and each tails costs $3. While the expectation is 0, it turns out that there are some interesting patterns; there is a (small) probability of being ahead after n tosses if n is congruent to 1, 3, or 5 mod 5, and being behind in the other cases, and also there is an allover bias toward being ahead. The methods being used involve generating functions, a little Fourier series, and a saddle point type calculation. The details will be given in reasonable detail.

Thursday, September 17, 2009

Tuesday, September 22, 2009

Thursday, September 24, 2009

Tuesday, September 29, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, September 29, 2009

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Submitted by jarouse.

Kenneth Stolarsky (Department of Mathematics, University of Illinois)
Zeros of Polynomials in Analysis and Number Theory
Abstract: If an analytic (esp. meromorphic) function is "well-approximated" by a polynomial near the origin, we may expect its zeros to be "well-approximated" by the zeros of that polynomial near the origin. We also generally expect this to break down outside of the vicinity of the origin. But what are the transitional details? Robert Jentzsch lived just long enough to start a chain reaction of surprising discoveries starting from this question. Rational approximation theory, statistical functions, the Riemann zeta function and the number Pi*e have all shown up as actors on this stage. We shall mostly survey, but hope to sketch a few proofs. The only prerequisite is some basic complex analysis.

Thursday, October 1, 2009

Tuesday, October 6, 2009

Thursday, October 8, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Thursday, October 8, 2009

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Submitted by jarouse.

Bruce Berndt (UIUC Math)
Ramanujan's Lost Notebook
Abstract: In the spring of 1976, while searching through papers of the late G. N. Watson at Trinity College, Cambridge, George Andrews found a sheaf of 138 pages in the handwriting of Srinivasa Ramanujan, generally regarded as India's greatest mathematician. In view of the fame of Ramanujan's earlier notebooks, Andrews naturally called these papers Ramanujan's "lost notebook." This work, comprising about 650 results with no proofs, arises from the last year of Ramanujan's life and represents some of his deepest work. First, we provide a history of the lost notebook. Second, a general description of the topics found in the lost notebook will be provided. For some of the topics, such as ranks and cranks of partitions, we offer some details. The third portion of the lecture will be devoted to a more detailed discussion of one of the topics prominently addressed in the lost notebook, namely continued fractions.

Tuesday, October 13, 2009

Thursday, October 15, 2009

Monday, October 19, 2009

Math 499: Introduction to Graduate Mathematics
4:00 pm in 245 Altgeld Hall, Monday, October 19, 2009

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Submitted by seminar.

Kenneth Stolarsky (Department of Mathematics, University of Illinois)
Polynomials in Number Theory and Analysis
Abstract: In algebra and analysis one needs eigenvalues of matrices, and in number theory one needs the set of all algebraic numbers, so roots of polynomials, with or without integer coefficients, are of central importance. There is much more to the study of polynomials than the fundamental theorem of algebra. Various easily stated problems (even for polynomials with coefficients not constrained to be integers) remain open. In particular, there are many ways to measure the size of a polynomial, and these lead to interesting results and open problems about the largest or smallest polynomial satisfying a given condition. Polynomials can enter into problems whose formulation does not mention polynomials. One example is finding universally optimal distributions of points on spheres. Another is the open problem of determining the mod 1 distribution of x^n for a given x > 1 (e.g. x = 3/2). This has led to the concept of a PV number.

Tuesday, October 20, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, October 20, 2009

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Submitted by jarouse.

Bruce Reznick (UIUC Math)
Sums of two cubes
Abstract: This will include a discussion of some very old formulas of Viete (1591) and Euler-Binet on equal sums of two cubes of rational numbers, polynomials and rational functions over C. A simple necessary and sufficient condition will be given for a polynomial to be a sum of two cubes of polynomials; for example, xy(x^4-y^4) has six essentially different representations as a sum of two cubes of quadratic forms and x^6 + y^6 has four such representations. There are some analogues to the addition of points on elliptic curves. The speaker will present a lot of questions which he cannot answer.

Thursday, October 22, 2009

Tuesday, October 27, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, October 27, 2009

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Submitted by ahlgren.

Harold Diamond (UIUC Math)
Weak logarithmic density estimates for Beurling generalized numbers
Abstract: An infinite collection of real numbers $p_1, p_2, ... $ satisfying $1 < p_1 \le p_2 \le ...$ is called a sequence $P$ of Beurling generalized (g-) primes, and the semigroup $N_P$ it generates under multiplication is called the associated Beurling g-integers. We say that $N_P$ satisfies an upper (resp. lower) logarithmic density condition if $(1/\log x) \sum_{n_i \le x} 1/n_i$ is bounded above (resp. below). We show that these conditions are connected with the behavior of the g-zeta function near its pole. The case of logarithmic density is the famous tauberian theorem of Hardy, Littlewood, and Karamata.

Thursday, October 29, 2009

Tuesday, November 3, 2009

Thursday, November 5, 2009

Tuesday, November 10, 2009

Thursday, November 12, 2009

Tuesday, November 17, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Tuesday, November 17, 2009

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Submitted by jarouse.

Youness Lamzouri (The Institute for Advanced Study)
On the Distribution of large values of L-functions at the edge of the critical strip
Abstract: In this talk we will construct a class of probabilistic random Euler products to study large values of various families of $L$-functions at the edge of the critical strip. In particular this class includes the random models constructed recently by A. Granville and K. Soundararajan to study large values of the Riemann zeta function and Dirichlet $L$-functions on the $1$-line. Among new applications, we study families of symmetric power $L$-functions of holomorphic cusp forms in the level aspect (assuming the automorphy of these $L$-functions) at $s=1$, functions in the Selberg class (in the height aspect), and the family of $L$-functions of quadratic twists of a fixed $GL(m)/{\Bbb Q}$-automorphic cusp form at $s=1$.

Thursday, November 19, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld Hall, Thursday, November 19, 2009

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Submitted by jarouse.

Andrew Shallue (Illinois Wesleyan)
Enumerating class group structures of real quadratic number fields
Abstract: There are many classic conjectures surrounding the structure of the ideal class group of real quadratic number fields. Among them are the Cohen-Lenstra heuristics, which give precise information about the expected structure of such class groups. Supporting such conjectures provides motivation to tabulate class numbers of quadratic number fields. I will discuss the present progress of a project to enumerate the structure of all class groups for discriminants of real quadratic number fields up to 10^11, and to do so without relying on the Extended Riemann Hypothesis. In essence, this has two main components. First, there are algorithms for computing the structure of an abelian group given group operations as a black box. Second, these group operations must be instantiated. This is nontrivial in the case of the ideal class group of a real quadratic number field. In addition to first computing the regulator, elements of the group do not have a unique representative, making identity testing a difficult proposition. We will see how these difficulties are overcome, and discuss exciting new developments in algorithms for finding the structure of a generic group.