UIUC Dept. of Mathematics Seminar Calendar

     October 2009          November 2009          December 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    1  2  3  4  5  6  7          1  2  3  4  5
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 18 19 20 21 22 23 24   22 23 24 25 26 27 28   20 21 22 23 24 25 26
 25 26 27 28 29 30 31   29 30                  27 28 29 30 31

Tuesday, January 20, 2009

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

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Submitted by jarouse.

Jeremy Rouse (UIUC Math)
Primes Dividing Recurrence Sequences
Abstract: The properties and distribution of prime numbers have fascinated humanity for several millenia. In this talk, we will discuss which primes divide certain recurrence sequences, and connections with Galois theory and elliptic curves.

Thursday, January 22, 2009

Number Theory Seminar
1:00 pm in Altgeld Hall, Thursday, January 22, 2009

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Submitted by jsinick2.

Jonah Sinick (UIUC math)
The Elliptic Curve y^2 = x^3 + x and Abelian Extensions of Q(i)
Abstract: It is well known that every abelian extension of the rational numbers Q is a subfield of a cyclotomic extension of the rational numbers Q. Kronecker's Jugendtraum ("dream of youth") is that of finding a similarly explicit description of the abelian extensions of other number fields K. Kronecker's Jugendtraum can be realized for Q(i) as follows: endow the elliptic curve y^2 = x^3 + x with the usual group law and let C[n] be the set of x and y coordinates of points of order dividing n. Then every abelian extension of Q(i) is contained in Q(i)(C[n]) for some n. In this talk I will discuss the main points of the argument given in Silverman and Tate's *Rational Points on Elliptic Curves* that Q(i)(C[n]) is in fact an abelian extension of Q(i) for every n.

Tuesday, January 27, 2009

Thursday, January 29, 2009

Tuesday, February 3, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld, Tuesday, February 3, 2009

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Submitted by pppollac.

Byungchan Kim (Department of Mathematics, University of Illinois)
On the subpartitions of the ordinary partitions
Abstract: Let $a_1 \ge a_2 \ge \cdots \ge a_&ob;\ell&cb;$ be an ordinary partition. A subpartition with gap $d$ of an ordinary partition is defined as the longest sequence satisfying $a_1 > a_2 > \cdots > a_s$ and $a_s > a_&ob;s+1&cb;$, where $a_i - a_j \ge d$ for all $i < j \le s$. This is a generalization of the Rogers-Ramanujan subpartition which was introduced by L. Kolitsch. In this talk, we will doscuss various properties of the subpartition and as an application, we will give a combinatorial proof of two entries, which are in Ramanujan's lost notebook.

Thursday, February 5, 2009

Thursday, February 12, 2009

Tuesday, February 17, 2009

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

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Submitted by pppollac.

Atul Dixit (Department of Mathematics, University of Illinois)
Analogues of a transformation formula of Ramanujan and a theorem of Ramanujan and Hardy-Littlewood
Abstract: In a manuscript in 'The Lost Notebook and Other Unpublished Papers' is present a beautiful transformation formula of Ramanujan involving the Gamma and Riemann zeta functions which can be proved using an identity in Ramanujan's paper 'New expressions for Riemann's functions $\xi(s)$ and $\Xi(t)$'. In the first part of this talk, I will derive two analogues of this formula. Both involve infinite series consisting of Hurwitz zeta functions and their relation to a certain integral containing Riemann $\Xi$ function. In the latter part, I will discuss another integral involving $\zeta(s)$ which yields, under certain assumptions, an interesting theorem of Ramanujan and Hardy-Littlewood about a certain transformation formula which involves infinite series consisting of Mobius function and is related to the complex zeros of the Riemann zeta function.

Thursday, February 19, 2009

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

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Submitted by pppollac.

Paul Pollack [email] (Illinois at Urbana-Champaign)
Rational cubic reciprocity
Abstract: Many first courses in number theory culminate in a proof of Gauss's law of quadratic reciprocity. Usually very little is said about other reciprocity laws. This is understandable, since to even state the cubic and quartic reciprocity laws in their usual form, one needs to expand one's domain beyond the rational integers. In this talk I will describe a `rational' cubic reciprocity law due to Jacobi, in a recent elegant reformulation of Z.-H. Sun. Here `rational' means that it requires for its statement nothing beyond the rational integers. The statement of this law (though perhaps not its proof) would be right at home in an honors introductory number theory course.

Tuesday, February 24, 2009

Thursday, February 26, 2009

Tuesday, March 3, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld, Tuesday, March 3, 2009

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Submitted by pppollac.

Michael Dewar (Department of Mathematics, University of Illinois)
Simple congruences in the coefficients of modular forms
Abstract: It is well-known that the partition function has the congruence p(5n+4) = 0 mod 5. Although there are also simple congruences like this for the primes 7 and 11, Ahlgren and Boylan have proven there are no others. In this talk we show how Ramanujan's theta operator can be used to prove the non-existence of simple congruences (except at small primes) in many modular forms which vanish only at the cusps. Applications include overpartitions, crank differences, and 2-colored F-partitions. I owe a great debt to Byungchan Kim for suggesting these applications.

Thursday, March 5, 2009

Tuesday, March 10, 2009

Thursday, March 12, 2009

Tuesday, March 17, 2009

Thursday, March 19, 2009

Tuesday, March 31, 2009

Tuesday, April 14, 2009

Tuesday, April 21, 2009

Thursday, April 23, 2009

Tuesday, April 28, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld, Tuesday, April 28, 2009

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Submitted by pppollac.

Richard McIntosh (Department of Mathematics, University of Illinois)
Lambert Series as Theta Functions
Abstract: A Lambert series is a sum of the type \sum_{n=1}^\infty a_n q^n/(1-q^n). If a_n=1 when n is congruent to k modulo m and a_n=0 otherwise, then this Lambert series, denoted by L_{k,m}(q), is the generating function for the number of positive divisors of n that are congruent to k modulo m. Dirichlet's representation theorem for integral binary quadratic forms expresses a theta series in terms of divisor-sums, and hence, as a Lambert series. I am interested in the inverse problem, that is, which Lambert series can be expressed in terms of theta functions. The asymptotic expansion of L_{k,m}(q) as q-> 1 leads to the conjecture that L_{k,m}(q)-L_{m-k,m}(q) can be expressed as a finite sum of theta quotients times powers of q. An outline of the proof will be presented.

Thursday, April 30, 2009

Tuesday, May 5, 2009

Number Theory Seminar
1:00 pm in 241 Altgeld, Tuesday, May 5, 2009

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Submitted by pppollac.

Bruce Reznick (Department of Mathematics, University of Illinois)
Recent results in 19th century algebra
Abstract: The length of a form p of degree d is the minimum number of linear forms so that a linear combination of their d-th powers sum to p. Two familiar theorems about quadratic forms is that the length cannot decrease if the base field is enlarged, and, in the real case, Sylvester's Law of Inertia. We show that the length can depends on the field when d > 2 in a fundamental way and that the Law of Inertia holds for binary quartic forms, but fails for binary sextic forms. We will also present a mysterious identity involving quadratics taken to the 14-th power.

Tuesday, August 25, 2009

Thursday, August 27, 2009

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

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.