Department of

# Mathematics

Seminar Calendar
for Number Theory events the year of Tuesday, February 1, 2005.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2005          February 2005            March 2005
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1          1  2  3  4  5          1  2  3  4  5
2  3  4  5  6  7  8    6  7  8  9 10 11 12    6  7  8  9 10 11 12
9 10 11 12 13 14 15   13 14 15 16 17 18 19   13 14 15 16 17 18 19
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23 24 25 26 27 28 29   27 28                  27 28 29 30 31
30 31


Tuesday, January 18, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, January 18, 2005

#### Cosh-like arithmetic functions having an average value

###### Harold Diamond (UIUC)

Thursday, January 20, 2005

1:00 pm in 241 Altgeld Hall,Thursday, January 20, 2005

#### The correlations of Farey fractions

###### Florin Boca (UIUC)

Abstract: It will be proved that all correlation measures of the sequence of Farey fractions exist. The pair correlation function is explicitly computed and shows an intermediate behaviour between the Poisson and the GUE correlations. This is joint work with A. Zaharescu.

Tuesday, January 25, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, January 25, 2005

#### Integers covered by systems of congruences with distinct moduli.

###### Kevin Ford (UIUC)

Abstract: A covering system is a finite set of arithmetic progressions with distinct moduli >1 whose union is all the integers. Erdos conjectured that there are covering systems whose smallest moduli is arbitrarily large. We consider the following related problem: If N is a large integer and K>1, what is the densest union of arithmetic progressions with distinct moduli in [N,KN]? We show that if K is a bit smaller than a power of N, then the desity of the union of progressions cannot be much more than 1-1/K. In particular, a covering system with distinct moduli in [N,KN] cannot exist. This is joint work with Michael Filaseta, Sergei Konyagin, Carl Pomerance and Gang Yu.

Thursday, January 27, 2005

1:00 pm in 241 Altgeld Hall,Thursday, January 27, 2005

#### Root numbers of abelian varieties

###### Maria Sabitova (University of Pennsylvania)

Abstract: We generalize a theorem of D. Rohrlich concerning root numbers of elliptic curves over the field of rational numbers. Our result applies to abelian varieties over number fields. Namely, under certain conditions which naturally extend the conditions used by D. Rohrlich, we show that the root number W(A,\tau) associated to an abelian variety A over a number field F and a complex finite-dimensional irreducible representation \tau of the absolute Galois group of F with real-valued character is equal to 1. In the case where the ground field is Q, we show that our result is consistent with the refined version of the conjecture of Birch and Swinnerton-Dyer.

Tuesday, February 1, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, February 1, 2005

#### Lehmer numbers and Lehmer points

###### Alexandru Zaharescu (UIUC)

Abstract: We discuss the definition of Lehmer numbers and Lehmer points, and present some asymptotic results on the number of such points.

Wednesday, February 2, 2005

4:00 pm in 245 Altgeld Hall,Wednesday, February 2, 2005

#### Loop spaces and Langlands duality

###### David E. Nadler (University of Chicago)

Abstract: Langlands' vision of number theory has had a tremendous impact on the representation theory and topology of loop groups. In particular, deep results about the structure of loop groups involve the Langlands dual group. In this talk, I will describe a project, joint with D. Gaitsgory (U. of Chicago), devoted to the topology of loop spaces of more general varieties with "lots of symmetry". We show that the singularities of the loop space of such a variety may be described in terms of a dual group. The dual group also governs many aspects of the original variety, such as its differential operators and compactifications. (The talk will not assume any prior knowledge.)

Thursday, February 3, 2005

1:00 pm in 241 Altgeld Hall,Thursday, February 3, 2005

#### he Dirichlet series for the reciprocal of the Riemann zeta function

###### Paul Bateman (UIUC)

Abstract: We consider the behavior of the partial sums of the series $\sum \mu(n)n^{-s}$ for real values of s greater than 1. The talk is related to my recent problem in the American Mathematical Monthly, which deals with some simple cases.

Tuesday, February 8, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, February 8, 2005

#### Effective structure theorems for quadratic spaces and their isometries

###### Lenny Fukshansky, (Texas A&M)

Abstract: A classical theorem of Witt states that a bilinear space can be decomposed into an orthogonal sum of hyperbolic planes, singular, and anisotropic components. I will discuss the existence of such a decomposition of bounded height for a symmetric bilinear space over a number field, where all bounds on height are explicit. I will also talk about an effective version of Cartan-Dieudonné theorem on representation of an isometry of a regular symmetric bilinear space as a product of reflections. Finally, if time permits, I will show a special version of Siegel's Lemma for a bilinear space, which provides a small-height orthogonal decomposition into one-dimensional subspaces.

Thursday, February 10, 2005

1:00 pm in 241 Altgeld Hall,Thursday, February 10, 2005

#### Diophantine Approximation with mild divisibility constraints

###### Emre Alkan (UIUC)

Abstract: I will present a survey on the history of Diophantine approximation problems using special subsets of integers and also talk about recent work on this jointly with A. Zaharescu and G. Harman

Tuesday, February 15, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, February 15, 2005

#### Congruences for the coefficients of weakly holomorphic modular forms, I

###### Stephanie Treneer (UIUC)

Abstract: Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomena is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form. In particular, we give congruences for a wide class of partitions functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level. Tuesday's talk will consist of an introduction to the problem, a statement of the main theorems, and a discussion of the two applications. Thursday we will prove the main theorems.

Thursday, February 17, 2005

1:00 pm in 241 Altgeld Hall,Thursday, February 17, 2005

#### Congruences for the coefficients of weakly holomorphic modular forms, II

###### Stephanie Treneer (UIUC)

Abstract: Recent works have used the theory of modular forms to establish linear congruences for the partition function and for traces of singular moduli. We show that this type of phenomena is completely general, by finding similar congruences for the coefficients of any weakly holomorphic modular form. In particular, we give congruences for a wide class of partitions functions and for traces of CM values of arbitrary modular functions on certain congruence subgroups of prime level. Tuesday's talk will consist of an introduction to the problem, a statement of the main theorems, and a discussion of the two applications. Thursday we will prove the main theorems.

Tuesday, February 22, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, February 22, 2005

#### An Extension of the Supercongruence for Apery Numbers

###### Timothy Kilbourn (UIUC)

Abstract: In 1987, Beukers proved a mod p congruence between the coefficients of a certain modular form and the Apery numbers. He conjectured the existence of a supercongruence'' mod p^2; this was proved by Ahlgren and Ono in 2000. In this talk we prove an extension of this congruence mod p^3. The proof involves the modularity of a Calabi-Yau threefold, the Gross-Koblitz formula, the p-adic gamma function, and some interesting combinatorics.

Thursday, February 24, 2005

1:00 pm in 241 Altgeld Hall,Thursday, February 24, 2005

#### Digital expansions, exponential sums and central limit theorems.

###### Michael Drmota (Technical University, Wien)

Abstract: The purpose of this talk is to present recent results on the distribution of the values of the ($q$-ary) sum of digits function $s_q(n)$, where we mainly focus on two problems, the joint distribution $(s_{q_1}(n),s_{q_2}(n))$ of two different expansions and on the distribution of the sum-of-digits function of squares: $s_q(n^2)$.