Department of

Mathematics


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

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     January 2005          February 2005            March 2005     
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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: $s_q(n^2)$.
Interestingly there are strong relations to "classical analytic number theory". For example, one can use Baker's theorem on linear forms (and exponential sums) to prove that the pairs $(s_{q_1}(n),s_{q_2}(n))$, $n

Tuesday, March 1, 2005

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

On some very old theorems of Erdos

Heini Halberstam (UIUC)

Thursday, March 3, 2005

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

The Dirichlet series for the reciprocal of the Riemann zeta function

Paul Bateman (UIUC)

Abstract: For real values of s greater than 1, we compare the partial sums of the series $\sum \mu(n)n^{-s}$ with its sum $1/\zeta(s)$, where $\zeta$ denotes the Riemann zeta-function. Specifically, we ask: For what values of N does the difference $$\sum_{n=1}^N \mu(n)n^{-s} - 1/\zeta(s) $$ have a fixed sign for all s in $(1,\infty)$. The talk is related to my recent problem in the American Mathematical Monthly, which deals with some simple cases.

Tuesday, March 8, 2005

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

The Iwasawa theoretic Gross-Zagier theorem

Ben Howard (University of Chicago)

Abstract: The Gross-Zagier theorem relates the central value of the derivative of the L-function of an elliptic curve to the Neron-Tate height of a special point, a Heegner point, in the Mordell-Weil group. Combining this result with work of Kolyvagin gives some of the strongest known results in the direction of the Birch and Swinnerton-Dyer conjecture. In this talk we will explain an Iwasawa theoretic analogue (conjectured by Mazur and Rubin) of the Gross-Zagier theorem. The result relates towers of Heegner points in a Z_p-extension to the derivative of a two-variable p-adic L-function.

Thursday, March 10, 2005

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

Computing with the Jacobian of a Hyperelliptic Curve

Paul van Wamelen (Louisiana State University)

Abstract: We will define the algebraic and analytic Jacobians of a hyperelliptic curve and explain how to go from the one to the other. We will demonstrate functionalities in the computer algebra system MAGMA for working with these objects.

Tuesday, March 15, 2005

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

Special values of L-functions.

A. Raghuram (University of Iowa)

Abstract: This talk will be an introduction to Deligne's conjectures on the special values of the symmetric power L-functions associated to a holomorphic cuspform. The latter half of the talk will be an account of some work in progress, which is joint work with Freydoon Shahidi, toward the special values of the symmetric fourth power L-functions.

Thursday, March 17, 2005

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

Ramanujan's 40 Identities for the Rogers-Ramanujan Functions.

Bruce Berndt (UIUC)

Abstract: Ramanujan's list of 40 identities was found in the Oxford University Library by Bryan Birch and published in 1975. The list is in the handwriting of G. N. Watson who privately held the manuscript for many years and evidently lost Ramanujan's original list. We provide a survey of the methods used over the years to prove the identities, with emphasis on recent work. This is joint work with Geumlan Choi, Youn-Seo Choi, Heekyoung Hahn, Boon Pin Yeap, Ae Ja Yee, Hamza Yesilyurt, and Jinhee Yi.

Tuesday, March 29, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, March 29, 2005

A computational version of a theorem of Polya

Bruce Reznick (UIUC)

Abstract: In 1928, Polya proved that if p is a homogeneous polynomial in n variables which is strictly positive on the unit simplex, then for sufficiently large N, all the coefficients of (x1+...+xn)^N * p are non-negative. Vicki Powers and I proved in 2001 an explicit estimate on N, which is sharp in at least one special case. This theorem is becoming popular in certain circles of mathematical optimization. All proofs are elementary.

Thursday, March 31, 2005

1:00 pm in 241 Altgeld Hall,Thursday, March 31, 2005

The Dirichlet series of a certain arithmetic function associated with the continued fraction algorithm.

Florin Boca (UIUC)

Monday, April 4, 2005

4:00 pm in 245 Altgeld Hall,Monday, April 4, 2005

Exponential sums and their role in number theory

Alexandru Zaharescu (Department of Mathematics, UIUC)

Tuesday, April 5, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, April 5, 2005

A curious confluence of three problems in the analytic theory of polynomials.

Ken Stolarsky (UIUC)

Abstract: How a polynomial behaves at critical points (e.g., max-min theory) and how the symmetry of a polynomial relates to the distribution of its zeros are two major problem areas in the study of polynomials. A more esoteric problem is to what extent a polynomial and some of its derivatives can all be composite (say without having multiple zeros). Certain multivariable polynomials over the rationals shall be displayed that seem to be remarkable (at least to me) with respect to each of these three problem areas. They are somehow related to Shabat polynomials.

Thursday, April 7, 2005

1:00 pm in 241 Altgeld Hall,Thursday, April 7, 2005

The form of Fourier coefficients of Eisenstein Series

Ben Brubaker (Stanford University)

Abstract: We will begin with a review of ordinary spectral Eisenstein series on GL(2) and its Fourier coefficients and then describe a new class of Eisenstein series associated to covers of semi-simple Lie groups (called metaplectic groups since Weil originally worked over the symplectic group Sp(2n)). The Fourier (or more properly Whittaker) coefficients of these Eisenstein series have extremely rich arithmetic information, e.g. they can contain Dirichlet L-functions or interesting combinations of Gauss sums. Hence analytic information about the Eisenstein series can be translated into information about the special values of L-functions, etc. We'll present some recent examples of this and give a combinatorial approach to dealing with these objects that avoids much of the complication with computing the Fourier-Whittaker coefficients directly.

Tuesday, April 12, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, April 12, 2005

On the difference between a number and its inverse modulo n.

Kevin Ford (UIUC)

Abstract: We study the distribution of the function M(n) = max(|a-b| : 0 < a, b < n and ab=1 (mod n)) and its connection to divisor problems. In particular, we examine what can be said (i) for infinitely many n, (ii) for infinitely many prime n, and (iii) for almost all n.

Thursday, April 14, 2005

1:00 pm in 241 Altgeld Hall,Thursday, April 14, 2005

Ten minute talks

Various faculty and graduate students (UIUC)

Abstract: Volunteers from the regular seminar group will offer short talks in the area of general number theory, roughly ten minutes in length.

Tuesday, April 19, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, April 19, 2005

A Table of Values in Ramanujan's Lost Notebook

O-Yeat Chan (UIUC)

Abstract: On pages 179 and 180 of his Lost Notebook, Ramanujan lists a table of values related to the crank generating function. In this talk, we shall use the circle method to prove the completeness of this table, and, as a bonus, prove 2 conjectures of Andrews and Lewis regarding cranks modulo 3 and 4.

3:00 pm in 243 Altgeld Hall,Tuesday, April 19, 2005

Some p-adic differential equations in algebraic geometry and number theory

Kiran Kedlaya (MIT)

Abstract: It has been known for over a century that certain special differential equations (e.g., Picard-Fuchs equations) occupy a distinguished role in algebraic geometry, by regulating the variation of periods (integrals of algebraic differential forms). It emerged from the work of Dwork on zeta functions of varieties over finite fields that similar differential equations (in fact, often the very *same* equations) play a similar role in controlling zeta functions! Since then, much work has gone into trying to explain Dwork's insights by constructing a form of de Rham cohomology for varieties over finite fields; this project continues to the present. I'll describe some recent progress in this area, and perhaps point out some surprising offshoots: some consequences in the theory of p-adic Galois representations (which lead to advances in the theory of modularity of Galois representations, as initiated by Wiles), and some "practical" applications (e.g., in cryptography).

Thursday, April 21, 2005

1:00 pm in 241 Altgeld Hall,Thursday, April 21, 2005

Converse theorems and Artin's conjecture

Andrew Booker (University of Michigan)

Abstract: Given a finite Galois extension of the rationals and a representation of its Galois group, Artin defined an L-function which he conjectured to have analytic continuation to the complex plane and satisfy a functional equation. All progress to date on this conjecture has come through the related Langlands' program which anticipates that Artin's L-functions agree with those associated to automorphic forms. I will show that if Artin's conjecture is true for a given 2-dimensional representation then so is Langlands' conjecture. The technique used in the proof may be extended to give a Weil-type converse theorem. If time permits, I will discuss generalizations to number fields.

Tuesday, April 26, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, April 26, 2005

Parity of the partition function

Matthew Boylan (UIUC)

Abstract: Let p(n) denote the ordinary parition function. In 1966 Subbarao conjectured that in every arithmetic progression r (mod t) there are infinitely many integers N = r (mod t) with p(N) even, and infinitely many integers M = r (mod t) with p(M) odd. Using classical facts about modular forms mod 2, we prove the conjecture for every arithmetic progression r (mod t) where t is a power of 2. This is joint work with Ken Ono from 2001.

Thursday, April 28, 2005

1:00 pm in 241 Altgeld Hall,Thursday, April 28, 2005

Eisenstein series and representing natural numbers as sum of integer squares.

Ling Long (Iowa State University)

Abstract: In this talk we are going to address two questions. We will give a short proof of Milne's formulae for sums of $4n^2$ and $4n^2+4n$ integer squares using the theory of modular forms, in particular Eisenstein series. This work was done jointly with Yifan Yang. In the second part, we will address a question regarding the zeros of some Eisenstein series used in the first part of the talk.

Tuesday, May 3, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, May 3, 2005

Traces of Singular Moduli on Hilbert Modular Surfaces.

Kathrin Bringmann (University of Wisconsin)

Abstract: Suppose that $p\equiv 1\pmod 4$ is a prime, and that $\Op$ is the ring of integers of $K:=\Q(\sqrt{p})$. A classical result of Hirzebruch and Zagier asserts that certain generating functions for the intersection numbers of Hirzebruch-Zagier divisors on the Hilbert modular surface $(\h\times \h)/\SL_2(\Op)$ are weight $2$ holomorphic modular forms. Using recent work of Bruinier and Funke, we show that the generating functions of traces of singular moduli over these intersection points are often weakly holomorphic weight $2$ modular forms. For the singular moduli of $J_1(z)=j(z)-744$, we explicitly determine these generating functions using classical Weber functions, and we factorize their ``norms" as products of Hilbert class polynomials.

Thursday, May 5, 2005

1:00 pm in 241 Altgeld Hall,Thursday, May 5, 2005

Computing with the Jacobian of a Hyperelliptic Curve.

Paul van Wamelen (Louisiana State University)

Abstract: We will define the algebraic and analytic Jacobians of a hyperelliptic curve and explain how to go from the one to the other. We will demonstrate functionalities in the computer algebra system MAGMA for working with these objects.

Thursday, July 21, 2005

1:00 pm in 241 Altgeld Hall,Thursday, July 21, 2005

Weight-dependent congruence properties of modular forms.

YoungJu Choie (POSTECH)

Abstract: In this talk, we study congruence properties of modular forms in various ways. By proving a weight-dependent congruence property of modular forms, we give some sufficient conditions for a modular form to be non $p-$ordinary in terms of the weights of modular forms. As applications of our main theorem we derive a linear relation among coefficients of new forms. This is joint work with D.H.Choi.

Tuesday, August 30, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, August 30, 2005

Modular Equations, Class Invariants and an Unusual New Identity

Bill Hart (UIUC Math)

Abstract: Bill Hart, a new postdoc in the department (formerly at Leiden University), will give an overview of his work on modular equations for Weber type functions, their application to computing class invariants and the development of an unusual two variable eta function identity which he has discovered.

Thursday, September 1, 2005

1:00 pm in 241 Altgeld Hall,Thursday, September 1, 2005

An Infinite Product for the Lemniscate Constant

Tim Huber (UIUC Math)

Abstract: An infinite product for the lemniscate constant bearing a striking similarity to Viete's product for pi recently appeared in a paper by Aaron Levin. The product is constructed from the curve x^4 + y^4 = 1 in much the same way that Viete's product is constructed from the circle. Using analogues of the trigonometric functions in this context and some basic elliptic function theory, we will derive this charming result and discuss its geometric interpretation.

Tuesday, September 6, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, September 6, 2005

Cyclotomic Numbers and Primitive Idempotents of Minimal Cyclic Codes

Vishwa Dumir (Panjab University)

Abstract: Let Fq be a field with q elements (q, m) = 1, q odd. A cyclic code of length m over Fq can be identified as an ideal in the ring Rm = Fq[x]/. For many values of m, expressions for the primitive idempotents (idempotent generators of a minimal cyclic code) in terms of eigen values of a Matrix involving Cyclotomic Numbers are given.

Wednesday, September 7, 2005

4:00 pm in 245 Altgeld Hall,Wednesday, September 7, 2005

Old and new number theory

Scott Ahlgren (Dept. of Mathematics, UIUC)

Abstract: I'll try to illustrate how some very old and seemingly elementary problems are connected to research in modern number theory.

Thursday, September 8, 2005

1:00 pm in 241 Altgeld Hall,Thursday, September 8, 2005

Two entries on bilateral hypergeometric series and two false entries from Ramanujan's lost notebook

Bruce Berndt (UIUC)

Abstract: We discuss two entries on bilateral hypergeometric series from Ramanujan's lost notebook. One is a formula for the derivative of a quotient of two such series. An introduction to bilateral hypergeometric series will be given. This is joint work with Wenchang Chu. In the last 15 minutes of the lecture, two false entries from the lost notebook will be presented. The speaker will ask the audience if they think that corrected versions exist.

2:00 pm in Altgeld Hall 347,Thursday, September 8, 2005

Organizational Meeting

The voice of the collective (UIUC Math)

Abstract: Algebraic Number Theory seminar is starting up again at the usual time of 2 pm every thursday. Join us for the (hopefully short) organizational meeting. First and second year graduate students are especially encouraged to come.

Thursday, September 15, 2005

2:00 pm in Altgeld Hall,Thursday, September 15, 2005

Postponed

Abstract: The Algebraic Number Seminar has been postponed to next week.

Tuesday, September 20, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, September 20, 2005

Points on curves over finite fields and residue races.

Alexandru Zaharescu (UIUC)

Thursday, September 22, 2005

1:00 pm in 241 Altgeld Hall,Thursday, September 22, 2005

Estimates on the discrepancy of fractions with divisibility constraints.

Emre Alkan (UIUC)

Abstract: Generalizing a result of Niederreiter we will present estimates on the discrepancy of Farey fractions satisfying certain divisibility constraints. In particular, we obtain the correct order of magnitude for the discrepancy.

2:00 pm in Altgeld Hall 241,Thursday, September 22, 2005

On a conductor discriminant formula of McCulloh

Patrick Szuta   [email] (UIUC Math)

Abstract: I will discuss the IJM vol. 40 paper titled "On a conductor discriminant formula of McCulloh." The talk will end with a statement of a conjecture that Micah James and I worked on during the summer. A followup talk will be given by Micah probably next thursday. Algebraic number theory at the math 530 level is probably the only prerequisite.

Tuesday, September 27, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, September 27, 2005

Covering Congruences: Old and New

Donald Gibson (UIUC Math)

Abstract: Romanov asked Erdos: Does there exist an infinite arithmetic progression consisting only of odd numbers, no term of which is the sum of a prime and a power of 2? The subject of covering congruences begins in 1950 when Erdos answered this question. In this talk, we learn that answer and survey some of what has happened in the area since then.

Thursday, September 29, 2005

1:00 pm in 241 Altgeld Hall,Thursday, September 29, 2005

Consecutive values of arithmetic functions

Kevin Ford (UIUC Math)

Abstract: Let f be a real valued, additive arithmetic function. In 1961, Erdos and Schinzel gave necessary and sufficient conditions on f so that for every positive integer h, there is a constant c=c(h) so that the vectors (f(n+1),f(n+2),...,f(n+h)) are dense in [c,infinity]^h. We present their proof and discuss some related problems about the distribution of consecutive values of arithmetic functions.

2:00 pm in Altgeld Hall 241,Thursday, September 29, 2005

On a conductor discriminant formula of McCulloh (Part 2)

Patrick Szuta   [email] (UIUC Math)

Abstract: We continue our discussion from last week.

Tuesday, October 4, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, October 4, 2005

The Pascal triangle with memory

Florin Boca (UIUC Math)

Abstract: One possible (and interesting) way of enumerating the rationals is given by the classical Stern-Brocot sequence. This sequence is closely related with the continued fraction algorithm and can be diagrammatically generated by an algebraic-combinatorial configuration called (according to A. Knauf) the Pascal triangle with memory. In this talk I will examine this configuration as a Bratteli diagram and discuss some of the properties of the AF algebra it defines. This algebra is related with classical classes of C*-algebras such as Effros-Shen AF algebras, rotation algebras, and Temperley-Lieb algebras.

Thursday, October 6, 2005

1:00 pm in Altgeld Hall,Thursday, October 6, 2005

Normal numbers and discrepancy

Robert Tichy (Technische Universitat Graz, Austria)

Abstract: In the first part various constructions of normal numbers are discussed. We focus on quantitative results involving suitable concepts of discrepancy. In the second part several probabilistic methods and theorems are presented. Here some very recent results of Berkes, Philipp and Tichy are included. In the third part some modified concepts of discrepancy are investigated: this complements recent work of Rudnick, Sarnak and Zaharescu on pair correlations and of Mauduit and Sarkoezy on distribution properties of pseudorandom sequences.

2:00 pm in Altgeld Hall 241,Thursday, October 6, 2005

On a conductor discriminant formula of McCulloh (Part 3)

Micah James   [email] (UIUC Math)

Abstract: This goal of this talk is to present two non-commutative examples of the conductor discriminant formula in action. We will begin with a review of the construction of T(E) and a summary of deSmit's results in the commutative case. Note: This is Part 3, so you know that this is really important stuff.

Thursday, October 13, 2005

2:00 pm in Altgeld Hall 241,Thursday, October 13, 2005

Coding theory over finite Frobenius rings

Iwan Duursma   [email] (UIUC Math)

Abstract: Linear codes are classically defined over finite fields. Some of the best known codes are linear only when considered as a module over a finite ring. We show that much of the classical theory holds for such codes provided that the ring is a finite Frobenius ring.

Tuesday, October 18, 2005

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

A Carmichael-type conjecture for Carmichael's function

Kevin Ford (UIUC Math)

Abstract: In 1922, R. D. Carmichael conjectured that for any positive integer n, there is another positive integer m so that phi(n)=phi(m). Here phi is the Euler totient function, and phi(n) can be interpreted as the order of the multiplicative group (Z/nZ)*. This conjecture still has not been proved. A closely related function is Carmichael's function lambda(n), which is the largest order of any element of (Z/nZ)*. On can ask the same question: for any n, is there another integer m with lambda(n)=lambda(m)? We give an affirmative answer, subject to a famous conjecture about primes in arithmetic progressions, and give some indication of how one might relax this hypothesis and (maybe) give a complete proof. The techniques we present are very elementary.

Thursday, October 20, 2005

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

Roth's Theorem in two dimensions and an argument of Shkredov

Michael Lacey (Georgia Tech.)

Abstract: For a finite group G, define a Ramsey number r(G) to be the largest subset A of G \times G that does not contain three points { (x,y), (x+d,y), (x,y+d) }, with d \not= 0. We are primarily interested in the case of G being a finite field F_p ^n, and show that r (F_p ^n ) < p ^{2n} / (log n )^4 We follow an argument of Shkredov, and find several simplifications along the way. The inequality above immediately implies Roth's theorem on arithmetic progressions of length 3, in the finite field setting. We will also discuss a (new) three dimensional variant of this question, related to four term arithmetic progressions. Joint work with Bill McClain.

2:00 pm in Altgeld Hall 241,Thursday, October 20, 2005

MacWilliams Theorems and Coding Theory

Iwan Duursma   [email] (UIUC Math)

Abstract: Per request, Prof. Duursma will prove the MacWilliams theorems and discuss more coding theory.

Thursday, October 27, 2005

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

To Be Announced

Abstract: The seminar has been moved to 1pm for this week only. So, if you show up to 241 Altgeld at 2pm and there are people there, then they're probably on their way out. This means that you're one hour late. Don't be an hour late; show up at 1 pm for Jennifer's talk.

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

On Hyperelliptic Curves with Elliptic Quotients

Jennifer Paulhus (UIUC Math)

Abstract: The Jacobian of small genus hyperelliptic curves often factors into a product of elliptic curves. We can determine properties of the elliptic curves which arise in this way by considering them as quotients of the hyperelliptic curves. In particular, we would like to know: What are their torsion groups? Are they Q-curves? What sort of isogeny relations exist between them? The genus 2 case has been studied by various mathematicians. We discuss the genus 3 and 4 cases.

Thursday, November 3, 2005

2:00 pm in Altgeld Hall,Thursday, November 3, 2005

Computing roots of certain polynomials

Marc Masdeu Sabate (UIUC Math)

Abstract: In 1831, the French Academy rejected a proof given by Galois of the statement "A degree-p irreducible polynomial over the rationals is solvable by radicals if and only if all its roots can be expressed as rational functions over Q of any two of them". This theorem is actually true, and the result is still attributed to Galois. However, it is highly non-constructive, so finding the actual relations among such roots is still an open problem. I will explain how Spearman and Williams found a solution for the first non-trivial example (the dihedral group of 10 elements), and how it can be extended to all dihedral groups of 2p elements for p prime. The next unsolved case is the Frobenius group of 20 elements, so everyone is invited to propose their own algorithm for this case.

Tuesday, November 8, 2005

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

Analytic number theory and elliptic curves

Matt Young (American Institute of Mathematics)

Abstract: The Birch and Swinnerton-Dyer conjecture provides a way for methods of analytic number theory to produce results on ranks of families of elliptic curves by studying properties of their L-functions. In particular, we are interested in questions such as: How many L-functions in a given family vanish (or not) at the central point? What is the average (analytic) rank of the family? In this talk I will discuss recent progress on these problems.

Thursday, November 10, 2005

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

Farey fractions and pair-correlation of torsion points on elliptic curves

Maosheng Xiong (UIUC Math)

Abstract: We compute the pair-correlation of Farey fractions and the sum of Farey fractions over subintervals, which implies the existence of the pair-correlation of torsion points and also the sum of torsion points along elliptic curves over $Q$ (more precisely, along the unbounded real part of such elliptic curves). This is joint work with Emre Alkan and Alexandru Zaharescu.

2:00 pm in Altgeld Hall,Thursday, November 10, 2005

No seminar this week

Abstract: Due to prelims, the seminar for this week has been cancelled.

Tuesday, November 15, 2005

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

The Stern sequence

Bruce Reznick (UIUC Math)

Abstract: This talk is an advertisement for a Spring 2006 Math 595 course devoted to the Stern sequence. Some of the basic properties of the Stern sequence (s(0) =0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1)) will be described, as well as other mathematical objects which can be understood through the Stern sequence. These objects reside in number theory, combinatorics, analysis, geometry and elsewhere.

Thursday, November 17, 2005

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

Nonlinear recurrences with rational generating functions, and certain number triangles

Ken Stolarsky (UIUC Math)

Abstract: There is a notable equivalence between rational generating functions, exponential polynomials, and linear recurrences. We show that some nonlinear recurrences also have rational generating functions, and that they are connected with certain number triangles that have divisibility properties.

Tuesday, November 29, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, November 29, 2005

Ten minute talks

Abstract: Ten minute talks contributed by members of the seminar.

Thursday, December 1, 2005

1:00 pm in 241 Altgeld Hall,Thursday, December 1, 2005

Covering Congruences: An Empirical Quest

Don Gibson (UIUC)

Abstract: In a previous talk, we surveyed some of the history and results associated with covering congruences, a favorite topic of Erdos. In this talk, we examine some computational approaches to the problem of constructing covering congruences.

Tuesday, December 6, 2005

1:00 pm in 241 Altgeld Hall,Tuesday, December 6, 2005

A method of Andrews and Roy in partition theory

Chadwick Gugg (UIUC Math )

Abstract: We discuss Andrews and Roy's vast generalization of an elementary method for proving Ramanujan's congruence modulo 5 for the partition function p(n).

1:00 pm in 241 Altgeld Hall,Tuesday, December 6, 2005

A method of Andrews and Roy in partition theory

Chadwick Gugg (UIUC Math )

Abstract: We discuss Andrews and Roy's vast generalization of an elementary method for proving Ramanujan's congruence modulo 5 for the partition function p(n)

1:00 pm in 241 Altgeld Hall,Tuesday, December 6, 2005

A method of Andrews and Roy in partition theory

Chadwick Gugg (UIUC Math )

Abstract: We discuss Andrews and Roy's vast generalization of an elementary method for proving Ramanujan's congruence modulo 5 for the partition function p(n).

Friday, December 9, 2005

4:00 pm in 243 Altgeld Hall,Friday, December 9, 2005

Irreducible radical Galois extensions

Carl Pomerance (Dartmouth College)

Abstract: An irreducible radical field extension is one of prime degree $p$ which is formed by adjoining a $p$th root of an element. Can a solvable extension of the rationals always be decomposed into a chain of irreducible radical Galois extensions? The answer is clearly no; for example, take the field of 7th roots of unity. To reach this by prime-degree extensions, there must be a degree-3 extension at some point, and for this to be irreducible radical Galois, we will need to have the 3rd roots of unity present. However, if we throw in the 3rd roots of unity, so as to arrive at the 21st roots of unity, we can indeed decompose into a chain of irreducible radical Galois extensions. In general, let $M(n)$ denote the minimal degree of an extension of the rationals which can be reached by such a chain and which contains the $n$th roots of unity. So, in our example, $M(7)=12$. In this talk we will discuss the normal order of $M(n)$.