UIUC Dept. of Mathematics Seminar Calendar

     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
 16 17 18 19 20 21 22   20 21 22 23 24 25 26   20 21 22 23 24 25 26
 23 24 25 26 27 28 29   27 28                  27 28 29 30 31      
 30 31

Tuesday, January 18, 2005

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

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Submitted by boylan.

Harold Diamond (UIUC)
Cosh-like arithmetic functions having an average value

Thursday, January 20, 2005

Tuesday, January 25, 2005

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

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Submitted by boylan.

Kevin Ford (UIUC)
Integers covered by systems of congruences with distinct moduli.
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

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

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Submitted by boylan.

Maria Sabitova (University of Pennsylvania)
Root numbers of abelian varieties
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

Wednesday, February 2, 2005

Mathematics Colloquium: Special Lecture 2004-05
4:00 pm in 245 Altgeld Hall, Wednesday, February 2, 2005

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Submitted by seminar.

David E. Nadler (University of Chicago)
Loop spaces and Langlands duality
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

Tuesday, February 8, 2005

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

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Submitted by boylan.

Lenny Fukshansky, (Texas A&M)
Effective structure theorems for quadratic spaces and their isometries
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

Tuesday, February 15, 2005

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

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Submitted by boylan.

Stephanie Treneer (UIUC)
Congruences for the coefficients of weakly holomorphic modular forms, I
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

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

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Submitted by boylan.

Stephanie Treneer (UIUC)
Congruences for the coefficients of weakly holomorphic modular forms, II
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

Thursday, February 24, 2005

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

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Submitted by boylan.

Michael Drmota (Technical University, Wien)
Digital expansions, exponential sums and central limit theorems.
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

Thursday, March 3, 2005

Tuesday, March 8, 2005

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

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Submitted by boylan.

Ben Howard (University of Chicago)
The Iwasawa theoretic Gross-Zagier theorem
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

Tuesday, March 15, 2005

Thursday, March 17, 2005

Tuesday, March 29, 2005

Thursday, March 31, 2005

Monday, April 4, 2005

Tuesday, April 5, 2005

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

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Submitted by boylan.

Ken Stolarsky (UIUC)
A curious confluence of three problems in the analytic theory of polynomials.
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

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

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Submitted by boylan.

Ben Brubaker (Stanford University)
The form of Fourier coefficients of Eisenstein Series
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

Thursday, April 14, 2005

Tuesday, April 19, 2005

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

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Submitted by nevins.

Kiran Kedlaya (MIT)
Some p-adic differential equations in algebraic geometry and number theory
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

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

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Submitted by boylan.

Andrew Booker (University of Michigan)
Converse theorems and Artin's conjecture
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

Thursday, April 28, 2005

Tuesday, May 3, 2005

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

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Submitted by boylan.

Kathrin Bringmann (University of Wisconsin)
Traces of Singular Moduli on Hilbert Modular Surfaces.
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

Thursday, July 21, 2005

Tuesday, August 30, 2005

Thursday, September 1, 2005

Tuesday, September 6, 2005

Wednesday, September 7, 2005

Thursday, September 8, 2005

Thursday, September 15, 2005

Tuesday, September 20, 2005

Thursday, September 22, 2005

Tuesday, September 27, 2005

Thursday, September 29, 2005

Tuesday, October 4, 2005

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

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Submitted by ford.

Florin Boca (UIUC Math)
The Pascal triangle with memory
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

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

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Submitted by ford.

Robert Tichy (Technische Universitat Graz, Austria)
Normal numbers and discrepancy
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.

Thursday, October 13, 2005

Tuesday, October 18, 2005

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

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Submitted by ford.

Kevin Ford (UIUC Math)
A Carmichael-type conjecture for Carmichael's function
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

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

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Submitted by ford.

Michael Lacey (Georgia Tech.)
Roth's Theorem in two dimensions and an argument of Shkredov
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.

Thursday, October 27, 2005

Thursday, November 3, 2005

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

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Submitted by szuta.

Marc Masdeu Sabate (UIUC Math)
Computing roots of certain polynomials
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

Thursday, November 10, 2005

Tuesday, November 15, 2005

Thursday, November 17, 2005

Tuesday, November 29, 2005

Thursday, December 1, 2005

Tuesday, December 6, 2005

Friday, December 9, 2005

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

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Submitted by ahlgren.

Carl Pomerance (Dartmouth College)
Irreducible radical Galois extensions
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 $M(n)$.