Seminar Calendar
for Number Theory events the year of Tuesday, December 16, 2008.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery. 
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Tuesday, January 15, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, January 15, 2008
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Edray Goins (Purdue Math)
Does There Exist an Elliptic Curve $E/Q$ with Mordell-Weil Group $\Z_2 \times \Z_8 \times \Z^4$?
Abstract: An elliptic curve $E$ possessing a rational point is an arithmetic-algebraic object: It is simultaneously a nonsingular projective curve with an affine equation $Y^2 = X^3 + AX + B$, which allows one to perform arithmetic on its points; and a finitely generated abelian group $E(\Q) \simeq E(\Q)_{tors} \times \Z^r$, which allows one to apply results from abstract algebra. The abstract nature of its rank $r$ can be made explicit by searching for rational points $(X,Y)$. In this talk, we give some history on the problem of determining properties of $r$, explain its importance by discussing the conjecture of Birch and Swinnerton-Dyer, and analyze various approaches to finding curves of large rank.

Thursday, January 17, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, January 17, 2008
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Mehmet Haluk Sengun (UW-Madison Math)
Galois Representations of Small Quadratic Fields
Abstract: For a quadratic field K, we investigate continuous mod p representations of the absolute Galois group of K that are unramified away from p and infinity. We prove that for certain (K,p), there are no such irreducible representations. We also list some imaginary quadratic fields for which such irreducible representations exist. As an application, we look at elliptic curves with good reduction away from 2 over quadratic fields.

Tuesday, January 22, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, January 22, 2008
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Submitted by jarouse.
Kevin Ford (UIUC Math)
Random partitions and random factorizations
Abstract: We discuss the problem of how the large prime factors of a typical integer are distributed, in particular the distribution of the largest prime factor. These distributions are usually written in terms of a somewhat complicated object, the Dickman function. We show that one can obtain the distribution in a simpler way by considering a certain random partition of the unit interval.

Thursday, January 24, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, January 24, 2008
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Submitted by jarouse.
Jonah Sinick (UIUC Math)
Every natural number is the sum of a bounded number of primes
Abstract: In the 1930's Lev Schnirelman proved that there exists a c > 0 such that every natural number can be written as the sum of at most c primes. Since then significantly stronger results have been obtained using different methods, however, his proof is still striking in that the result obtained is quite strong relative to the simplicity of the method. In this talk I will outline his proof and describe related subsequent results.

Tuesday, January 29, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, January 29, 2008
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Kenneth Stolarsky (UIUC Math)
Sets with few products have many sums
Abstract: We examine the problem of showing that if A is a finite set of distinct positive real numbers then the sum and product sets A+A and A*A cannot both be small in cardinality. We follow a rather surprising method that uses Euler's relation for connected planar graphs.

Thursday, January 31, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, January 31, 2008
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Bruce Berndt (UIUC Math)
Modular relations, functional equations, and equivalent identities
Abstract: It is well known that the classical theta transformation formula and functional equation of the Riemann zeta function are equivalent. We focus on other identities that are equivalent to general modular relations and functional equations.

Tuesday, February 5, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, February 5, 2008
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Kevin Ford (UIUC Math)
Pratt trees and random fragmentations
Abstract: Below a prime p write the prime factors q of p-1, below each q write the prime factors of q-1, and so forth. The resulting structure we call the Pratt tree for p, named after V. Pratt, who used this tree to construct a certificate of primality for p. Our interest is in the depth of this tree (length of the longest path down the tree), which we call D(p). We present upper and lower bounds on D(p) which are valid for almost all p. Based on the behavior of a random fragmentation process, we give a heuristic argument that the normal order of D(p) is e loglog p, where e=2.718281828... is Euler's constant.

Thursday, February 7, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, February 7, 2008
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Jeremy Rouse (UIUC Math)
Lehmer's conjecture
Abstract: This will be an expository talk about the conjecture of Lehmer on the non-vanishing of Ramanujan's tau(n) function and the mathematics that is related to it.

Tuesday, February 12, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, February 12, 2008
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Paul Jenkins (UCLA Math)
Integral traces of singular values of Maass forms
Abstract: In his influential paper, Zagier proved that the generating functions for traces of singular moduli associated to polynomials in $j(\tau)$ are weight 3/2 modular forms on $\Gamma_0(4)$. At the end of his paper, he suggested a method for generalizing these results to higher weights. One such generalization was given by Bringmann and Ono, who give an identity for the traces associated with certain Maass forms in terms of the Fourier coefficients of certain half integral weight Poincare series. However, it does not seem to be known when these traces are integral or even rational. We give an identity for the traces associated to an arbitrary weakly holomorphic form $f$ of negative integral weight on $SL_2(Z)$ in terms of the coefficients of specific weakly holomorphic forms of half integral weight on $\Gamma_0(4)$ in Kohnen's plus space. If the coefficients of $f$ are integral, then these traces are integral as well. We use this correspondence to obtain a negative weight analogue of the classical Shintani lift.

Wednesday, February 13, 2008

Math 499: Introduction to Graduate Mathematics
4:00 pm   in 245 Altgeld Hall,  Wednesday, February 13, 2008
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Bruce Berndt (Department of Mathematics, University of Illinois)
Number Theory - The Queen of Mathematics

Thursday, February 14, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, February 14, 2008
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Leon McCulloh (UIUC Math)
Stickelberger evolution--an alternate branch
Abstract: The classical Stickelberger relations of Kummer/Jacobi were reinterpreted by Iwasawa as an ideal J in the integral group ring ZC where C is the Galois group of the cyclotomic field K of pth roots of unity over Q. The classical relations then say that J annihilates the ideal class group Cl(K). Iwasawa further showed that the classical formula for the first factor of the class number of K could be interpreted as saying that the first factor was the minus (skew symmetric) part of the index of J in ZC. Generalizing this to higher cyclotomic fields of p-power roots of 1 led to his interpretation of the Kubota-Leopoldt p-adic L-functions as limits of Stickelberger elements and evolved into what is now known as Iwasawa Theory. Trying to extend Iwasawa's class number formula to an integral group ring ZG (instead of a cyclotomic field) has led through several intermediate steps to a Stickelberger module (not ideal) in ZG itself arising from the character theory of the group G (not necessarily abelian).

Tuesday, February 19, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, February 19, 2008
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Andrew Schultz (UIUC Math)
Absolute Galois groups via module theory
Abstract: Absolute Galois groups encode a great deal of information about their corresponding fields, yet their structure isn't understood in great generality. In this talk we'll discuss how module structures of certain Galois cohomology groups can be used to determine conditions that absolute Galois groups must (or must not) satisfy. In particular we'll see how the appearance of certain Galois groups over a field F forces the appearance of (larger) Galois groups over F. We'll also give a few explicit relator shapes that cannot appear in absolute Galois groups over any field.

Thursday, February 21, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, February 21, 2008
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Paul Bateman (UIUC Math)
Famous Conjectures in Number Theory
Abstract: This is a reprise of a talk given in this seminar on September 11, 2001 (of all days). That talk consisted of two parts, namely ``Ten Conjectures Settled Between 1940 and 2001'' and ``Ten Conjectures Not Settled by September 2001.'' Needless to say, we will attempt to bring things up to date; specifically we will discuss an important conjecture which was been settled since 2001.

Tuesday, February 26, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, February 26, 2008
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Eunmi Kim (UIUC Math)
The Erdos-Turan theorem
Abstract: We shall give a proof of the Erdos-Turan theorem on distribution of real numbers modulo 1. This theorem can be viewed as a more quantitative version of the famous theorem of H. Weyl on uniform distribution. The version of the proof presented will use the remarkable Beurling-Selberg functions.

Thursday, February 28, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, February 28, 2008
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Kevin Ford (UIUC Math)
Local injectivity of Carmichael's function
Abstract: Let lambda(n) be Carmiachael's function, the largest order of an element in the multiplicative group of reduced residues modulo n. It has been conjectured that for any n there is another number m with lambda(m)=lambda(n), analogous to the famous (and unsolved) Carmichael conjecture for Euler's function phi(n). We ''almost'' prove the conjecture, in the sense that the proof hinges on a finite but impractical computation. The proof uses properties of Pratt trees, and is joint work with Florian Luca.

Tuesday, March 4, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, March 4, 2008
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Dohoon Choi (Korea Aerospace University)
Divisibility of traces of CM values of modular functions
Abstract: CM values of modular functions play important roles in number thoery. For example, the Hilbert class field of an imaginary quadratic field is generated by a CM value of a modular function. Recently, Ahlgren and Ono (when the modular group is \Gamma(1)) and Treneer (when the modular group is \Gamma^*_0(p) for any prime p) studied the divisibility of traces of CM values of weakly holomorphic modular functions. In this talk, we study the extention of the results to weakly holomorphic modular functions on \Gamma^*_0(N) for any positive integer N. This is a joint work with Jeon, Kang and Kim.

Thursday, March 6, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, March 6, 2008
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Byungchan Kim (UIUC Math)
Combinatorial proofs of certain identities involving partial theta functions
Abstract: Recently, G.E. Andrews and S.O. Warnaar proved the interesting identity for the product of two partial theta functions. In their paper, two identities involving partial theta functions played an important role. We will give a combinatorial proofs for them. If time permits, we will also give a combinatorial proof for the product identity.

Tuesday, March 11, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, March 11, 2008
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Harold Diamond (UIUC Math)
Properties of the Dickman function
Abstract: The Dickman function arises in approximations of the number of integers in some interval [1, x] which have all their prime factors in some interval [2, y]. After reviewing this relation, we shall establish some bounds for the Dickman function and a related function called xi. The rho bounds are a special case of results in the sieve book of Halberstam, Galway, and myself.

Thursday, March 13, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, March 13, 2008
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Jonah Sinick (UIUC Math)
Quaternion algebras, the Hasse-Minkowski theorem and quadratic reciprocity
Abstract: A quaternion algebra is a generalization of Hamilton's quaternions with the real numbers replaced by an arbitrary field K and for which we replace the condition i^2 = j^2 = -1 with the condition that i^2 = a and j^2 = b, where a and b are arbitrary nonzero elements of K. We describe the classification theorem for quaternion algebras over a number field K and explain how its proof is related to the Hasse-Minkowski theorem and quadratic reciprocity. This material has bearing on the topology of hyperbolic 3-manifolds.

Tuesday, March 25, 2008

Joint Number Theory/Differential Geometry Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, March 25, 2008
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Submitted by nmd.
Nathan Dunfield (UIUC)
Increasing the number of fibered faces of arithmetic hyperbolic 3-manifolds.
Abstract: I will exhibit a closed hyperbolic 3-manifold which satisfies a very strong form of Thurston's Virtual Fibration Conjecture. In particular, this manifold has finite covers which fiber over the circle in arbitrarily many fundamentally distinct ways. More precisely, it has a tower of finite covers where the number of fibered faces of the Thurston norm ball goes to infinity, in fact faster than any power of the logarithm of the degree of the cover. The example manifold M is arithmetic, and the proof uses detailed number-theoretic information, at the level of the Hecke eigenvalues, to drive a geometric argument based on Fried's dynamical characterization of the fibered faces. The origin of the basic fibration of M over the circle is the modular elliptic curve E=X_0(49), which admits multiplication by the ring of integers of Q[sqrt(-7)]. We first base change the holomorphic differential on E to a cusp form on GL(2) over K=Q[sqrt(-3)], and then transfer over to a quaternion algebra D/K ramified only at the primes above 7; the fundamental group of M is a quotient of the principal congruence subgroup of level 7 of the multiplicative group of a maximal order of D. This is joint work with Dinakar Ramakrishnan.

Thursday, March 27, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, March 27, 2008
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Ling Long (Iowa State)
Finite index subgroups of the modular group and their modular forms
Abstract: The modular group which consists of all 2-by-2 integral matrices with determinant 1 is one of the most famous and important discrete groups. Modular forms are spectacular functions whose symmetries can be essentially described by the modular group or its subgroups. The theory of congruence modular forms has been one of the central topics in number theory for over one century. It is well-known that the Hecke theory for noncongruence modular forms, which outnumber congruence ones, is missing. However, investigations in the past 40 years revealed many wonderful properties satisfied by noncongruence modular forms. In this talk, we will first introduce "KFarey", a computational package for finite index subgroups of the modular group. Then, we will review the developments of noncongruence modular forms. Finally, we will discuss the following properties of noncongruence modular forms: unbounded denominator property, Atkin and Swinnerton-Dyer congruences, and modularity.

Tuesday, April 1, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, April 1, 2008
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Scott Ahlgren (UIUC Math)
Vanishing of Fourier coefficientsof modular forms at infinity
Abstract: I will describe some recent work with Jeremy Rouse and Nadia Masri which gives upper bounds for the order of vanishing of a modular form at infinity. I will also discuss some nice applications.

Thursday, April 3, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, April 3, 2008
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Karl Dilcher (Dalhousie University)
Divisibility properties of some classes of binomial sums
Abstract: In this talk I will present congruence and divisibility properties of two different classes of combinatorial sums. The first class involves products of powers of two binomial coefficients; we will see that even though in general there is no evaluation in closed form, the sums behave in certain respects like single binomial coefficients. This is evident through a result similar to Wolstenholme's theorem, and through the fact that under certain conditions the sums are divisible by all primes in specific intervals. The second class of combinatorial sums is the alternating version of a well-known sum that was used in the theory of Bernoulli numbers. This new sum is evaluated modulo an odd prime, and as an application it is shown that the n-th Bernoulli polynomial cannot have multiple roots.

Tuesday, April 8, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, April 8, 2008
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Ghaith Hiary (UMinn Math)
Fast methods to compute the Riemann zeta function
Abstract: The Riemann zeta function on the critical line can be computed using a straightforward application of the Riemann-Siegel formula, Sch\"onhage's method, or Heath-Brown's method. The complexities of these methods have exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this talk, two new fast and potentially practical methods to compute zeta are presented. One method relies on an algorithm to compute quadratic exponential (theta) sums. Its complexity has exponent 1/3. The second method employs an algorithm to compute cubic exponential sums. Its complexity has exponent 4/13 (approximately, 0.307). If time permits, I will also present the results of recent computations (with Andrew Odlyzko) of moments and other statistics of zeta. The computations were done for a set of 20*10^9 zeros near the zero 10^23, as well as for lower sets.

Thursday, April 10, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, April 10, 2008
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Michael Dewar (UIUC Math)
Overpartitions and Maass forms
Abstract: An overpartition of $n$ is a partition in which the first occurrence of a part may be overlined. The overpartition rank generating function for $n$ lying in an arithmetic progression is not quite modular, but it is the holomorphic part of a Maass form. By computing the nonholomorphic part explicitly, we find linear combinations of the rank generating functions which are modular. Overpartitions are just one of many combinatorial objects which may be studied by looking at their shadows.

Friday, April 11, 2008

Group Theory Seminar (special meeting)
2:00 pm   in Altgeld Hall 141,  Friday, April 11, 2008
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Submitted by kapovitc.
Nigel Boston (University of Wisconsin -Madison)
Random p-groups
Abstract: What is the probability that r relators chosen at random from a g-generator free group present a particular group? We make sense of and answer this question for p-groups. This leads to some mysterious mass formulae and an application to number theory.

Tuesday, April 15, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, April 15, 2008
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Kevin Ford (UIUC Math)
Report on the Analytic Number Theory workshop (Oberwolfach, March 2008)
Abstract: We summarize some of the results that were announced at the Oberwolfach meeting.

Wednesday, April 16, 2008

Math 499: Introduction to Graduate Mathematics
4:00 pm   in 245 Altgeld Hall,  Wednesday, April 16, 2008
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Alexandru Zaharescu (Department of Mathematics, University of Illinois)
Exponential sums and their role in number theory

Thursday, April 17, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, April 17, 2008
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Byoung Du Kim (Northwestern Math)
Iwasawa theory of elliptic curves for supersingular primes
Abstract: Studying the Selmer groups of elliptic curves for a supersingular prime is difficult. It turned out we should instead use the plus/minus Selmer groups defined by Kobayashi. In this talk, we will see the plus/minus Selmer group theory for supersingular primes is very analogous to the Selmer group theoery for ordinary primes, and as an application, we will prove the parity conjecture of elliptic curves for supersingular primes among other things. We will report some other recent progress as well.

Tuesday, April 22, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, April 22, 2008
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Khang Tran (UIUC Math)
Shapiro Conjecture
Abstract: An exponential polynomial is an entire function of the form f(z) = a_1 exp(alpha_1 z) + ... + a_n exp(alpha_n z) where the a_i and alpha_i are complex numbers. Let E be the ring of exponential polynomials. The Shapiro conjecture is that given any two exponential polynomials with infinitely many roots in common, then there exists an exponential polynomial h with infinitely many roots such that h | f and h | g in the ring E. The talk will discuss some steps related to settling this conjecture.

Wednesday, April 23, 2008

Math 499: Introduction to Graduate Mathematics
4:00 pm   in 245 Altgeld Hall,  Wednesday, April 23, 2008
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A.J. Hildebrand (Department of Mathematics, University of Illinois)
Number Theory and Computing at Illinois

Thursday, April 24, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, April 24, 2008
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András Sárközy (Eötvös Loránd Tudományegyetem)
Equations in finite fields with restricted solution sets
Abstract: A survey of 5 papers will be given. In the first paper I proved that if p is a prime number and A, B, C, D are "large" subsets of F_p, then the equation a+b = cd can be solved with a, b, c, d belonging to A, B, C and D, resp. In the second paper I proved a similar result with ab+1 in place of a+b. The proofs in these papers are based on character sum estimates. In two joint papers with Katalin Gyarmati we extended these results in various directions. In the first paper we gave new character sum estimates, while in the second one we studied more general algebraic equations in finite fields with restricted solution sets. In a joint paper with Peter Csikvari and Katalin Gyarmati we studied similar equations in other structures.

Tuesday, April 29, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, April 29, 2008
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Hei-Chi Chan (University of Illinois, Springfield Math)
From Ramanujan's cubic continued fraction to an analog of Ramanujan's "Most Beautiful Identity"
Abstract: In this talk, I will discuss an analog of Ramanujan's "Most Beautiful Identity" that is derived from the Ramanujan's cubic continued fraction. I will also discuss certain applications of this identity that involve the congruence properties of a certain partition function.

Thursday, May 1, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, May 1, 2008
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Nadia Masri (UIUC Math)
Higher Weierstrass Points on X_0(p) and Supersingular j-Invariants
Abstract: I will discuss the arithmetic properties of higher Weierstrass points associated to certain meromorphic k/2-differentials on modular curves, and the relationship, for any even k, between reductions mod p of the collection of these points on X_0(p) and the supersingular locus in characteristic p.

Tuesday, August 26, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, August 26, 2008
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Harold Diamond (Department of Mathematics, University of Illinois)
Oscillation in Mertin's PNT

Thursday, August 28, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, August 28, 2008
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Robert Maier (University of Arizona)
Nonlinear DE's for Modular Forms

Tuesday, September 9, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, September 9, 2008
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Dimitris Koukoulopoulos (Department of Mathematics, University of Illinois)
Localized factorizations of integers
Abstract: Consider the set of numbers up to x which are divided by a product of k integers with each of the factors lying in a prescribed interval. We obtain the order of magnitude of the cardinality of this set and discuss applications to multiplication tables and sums of Farey fractions.

Thursday, September 11, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, September 11, 2008
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Krishnaswami Alladi (University of Florida)
Partitions with non-repeating odd parts and q-hypergeometric identities
Abstract: Much is known about partitions with non-repeating even parts, a primary reason being the famous identity of Lebesgue. In contrast, partitions with non-repeating odd parts have not attracted much attention. We show that by studying these partitions using 2-modular Ferrer's graphs, we get a unified and efficient treatment of several important q-hypergeometric identities such as those of Sylvester, Lebesgue, Roger-Fine, Gollnitz, and others.

Tuesday, September 16, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, September 16, 2008
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Ken Stolarsky (Department of Mathematics, University of Illinois)
Extremality, Symmetry, and Chebyshev-like polynomials
Abstract: If polynomials are extremal in some important way, must they have nice algebraic properties? If n points are arranged on a sphere to minimize a "reasonable" potential, must they be arranged symmetrically? The first problem leads us into results, some recent, concerning Chebyshev-like polynomials and their discriminants. The second provides additional motivation. Our survey will also touch upon various generalizations and analogues of the Chebyshev polynomials, and connections to number theory. We shall include descriptions of some open problems.

Thursday, September 18, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, September 18, 2008
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Pace Nielsen (University of Iowa)
A Covering System with Minimum Modulus 40
Abstract: A covering system is a set of congruence classes, with distinct moduli, for which each integer lives in at least one of the congruence classes. It is an open question of Paul Erdos whether covering systems exist where the minimum modulus can be chosen arbitrarily large. We introduce the readers to some new ways of picturing covering systems, and sketch the proof that there is a covering system with smallest modulus equal to 40.

Analysis Seminar
2:00 pm   in 243 Altgeld Hall,  Thursday, September 18, 2008
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Ken Stolarsky (Department of Mathematics, University of Illinois)
Analytic and algebraic properties of Chebyshev-like polynomials
Abstract: We shall survey various analytic and functional properties of Chebyshev polynomials and see how they extend to wider classes of polynomials. Some particular attention will be paid to the discriminants and zero distributions of such polynomials. If time permits, some open problems and connections with number theory will be given. Please note that there will be substantial overlap with the talk given on Tuesday in the number theory seminar.

Tuesday, September 23, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Tuesday, September 23, 2008
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Jonah Sinick (Department of Mathematics, University of Illinois)
Ramanujan Congruences for a Family of Eta Quotients
Abstract: Given a sequence of integers c(n), define a Ramanujan congruence for c(n) to be a congruence of the form c(Ln + a) = 0 (mod L) where L is prime. Last spring Hei-Chi Chan showed us that the coefficients of the generating function 1/[(1 - q^n)(1 - q^(2n))] obey a Ramanujan congruence (mod 3) and asked if the coefficients obey other Ramanujan congruences. We answer his question in the negative and indicate how to give a complete characterization of Ramanujan congruences for the coefficients of any member of a class of generating functions similar to 1/[(1 - q^n)(1 - q^(2n))]. The talk will be accessible to nonspecialists.

Thursday, September 25, 2008

Number Theory Seminar
1:00 pm   in 241 Altgeld Hall,  Thursday, September 25, 2008
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Ken Ono (University of Wisconsin)
Infinite products and the Han-Nekrasov-Okounkov Theorem
Abstract: This lecture will survey recent results on hook products in the general theory of partitions. In short, we shall sketch the proof of a conjecture of Han, and we shall give a sample of recent results at the interface of the theory of modular forms and work of Nekrasov and Okounkov.

Mathematics Colloquium
4:00 pm   in 245 Altgeld Hall,  Thursday, September 25, 2008
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Ken Ono (University of Wisconsin, Madison)
Freeman Dyson's challenge for the future: The mock theta functions
Abstract: In his last letter to Hardy, Ramanujan defined 17 peculiar functions which are now referred to as his mock theta functions. Although these mysterious functions have been investigated by many mathematicians over the years, many of their most basic properties remain unknown. At the 1987 Ramanujan Centenary Conference at the University of Illinois, Freeman Dyson proclaimed:

"Mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure... This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include not only theta-functions but mock theta-functions."

Here we describe the solution to Dyson's "challenge for the future", and we give a brief indication of some the applications of this theory to number theory.


Tuesday, September 30, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall ,  Tuesday, September 30, 2008
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Submitted by berndt.
Mat Rogers (Department of Mathematics, University of Illinois)
L-series, hypergeometric functions, and computation conjectures
Abstract: In recent years, computational searches have lead to the discovery of many surprising formulas relating special values of L-series of elliptic curves to rational hypergeometric functions. These identities can often be thought of as analogues of classical formulas for $\pi$. I will discuss the proofs of several relations, and mention connections to lattice sums and Mahler measure.

Thursday, October 2, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall,  Thursday, October 2, 2008
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Atul Dixit (Department of Mathematics, University of Illinois)
A Transformation Formula involving the Gamma and Riemann Zeta Functions in Ramanujan's Lost Notebook
Abstract: In 'The Lost Notebook and Other Unpublished Papers' of Ramanujan are present some manuscripts of Ramanujan in the handwriting of G. N. Watson which are 'copied from loose papers'. We present a proof of a beautiful formula of Ramanujan in one of these manuscripts, namely a transformation formula involving the Gamma function and Riemann Zeta function. This formula elegantly yields a modular relation. This is joint work with Bruce C. Berndt.

Tuesday, October 7, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall,  Tuesday, October 7, 2008
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Jeremy Rouse (Department of Mathematics, University of Illinois)
Modular forms with coefficients supported on finitely many square classes (mod l)
Abstract: We will discuss half-integer weight modular forms whose Fourier coefficients are supported on finitely many square classes modulo a prime $\ell$, and their applications to values of $L$-functions and values of the partition function.

Thursday, October 9, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall,  Thursday, October 9, 2008
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Joseph Vandehey (Department of Mathematics, University of Illinois)
Containment in (s,t)-core Partitions
Abstract: A partition is (s,t)-core if none of the hook numbers in its Ferrers Diagram is divisible by either s or t. A partition is said to be contained in another if superimposing the Ferrers Diagram of the first over the second gives us the second again. We will sketch a proof of Olsson and Stanton's conjecture that there exists a maximal (s,t)-core partition under partition containment and show how this method can be generalized to (t1,t2,...,tn)-core partitions.

Tuesday, October 14, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall,  Tuesday, October 14, 2008
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Paul Pollack (Department of Mathematics, University of Illinois)
The distribution of sociable numbers
Abstract: Let s(n) denote the sum of the proper divisors of n, so that s(n) = sigma(n)-n. Interest in the behavior of this arithmetic function can be traced back thousands of years to the early interest in perfect numbers. Let s_k denote the kth iterate of s. We call a number n *sociable* if s_k(n) = n for some k >= 1; the least such k is then referred to as the *order* of n. Thus the sociable numbers of order 1 are precisely the perfect numbers and those of order 2 are precisely the amicable numbers. In this talk we describe some recent results on the distribution of sociable numbers. This is joint work with M. Kobayashi and C. Pomerance, both from Dartmouth College.

Thursday, October 16, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall,  Thursday, October 16, 2008
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Jonah Sinick (Department of Mathematics, University of Illinois)
Finite volume hyperbolic 3-manifolds and their associated number fields
Abstract: A surprising context in which number theory emerges is that of the study of hyperbolic 3-manifolds. After I describe hyperbolic 3-manifolds, their isometries and Kleinian groups I will discuss how there's a natural way in which to associate a number field K and a quaternion algebra over K to a finite volume hyperbolic 3-manifold and give an example. This talk will serve as background material to a series of talks that I hope to give in the future concerning results about hyperbolic 3-manifolds that have been proved using number theoretic methods.

Thursday, October 23, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall ,  Thursday, October 23, 2008
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Supawadee Prugsapitak (Department of Mathematics, University of Illinois)
A converse of the Erdos-Fuchs theorem
Abstract: The Tarry-Escott problem asks to find two different sets of integers $A = \{a_1,\dots,a_n\}$ and $B = \{b_1,\dots,b_n\}$ so that $\sum a_i^j = \sum b_i^j$ for $1 \le j \le k$. We call $k$ the degree of a solution. In any solution $n \ge k+1$; if $n=k+1$, the solution is called ideal. In this talk, we discuss the Tarry-Escott problem over the Gaussian integers and characterize ideal solutions of degree 2. A. Choudry has proved that there is no non-trivial solution over the integers to: $$ u^j + v^j + w^j = x^j + y^j + z^j,\quad j= 1,2,5. $$ If there is time, we will give the solution to this system over the Gaussian integers.

Tuesday, October 28, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall ,  Tuesday, October 28, 2008
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Jonah Sinick (Department of Mathematics, University of Illinois)
An Introduction to Arithmetic Hyperbolic 3-Manifolds & Applications
Abstract: Though all elements of S = {finite volume hyperbolic 3-manifolds} have number theoretic invariants, there's a subclass A of S called the "arithmetic" hyperbolic 3-manifolds which are more intimately tied to number theory. In this talk I will give several characterizations of the arithmetic hyperbolic 3-manifolds and state a formula due to A. Borel which relates the values of the Dedekind zeta function of a number fields of specified type at s = 2 to the volume of an arithmetic hyperbolic 3-manifold.

Graph Theory and Combinatorics
3:00 pm   in 241 Altgeld Hall,  Tuesday, October 28, 2008
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Peter Horak (University of Washington)
Graph theory as an integral part of mathematics
Abstract: Nobody doubts the significance of applications of graph theory to other sciences (e.g. electrical engineering, psychology, economics) or to real life problems. In this talk, we concentrate on a different type of application; the possibility of using graph theory in other branches of mathematics. To demonstrate this kind of application, graph-theoretic proofs of well-known theorems in set theory, number theory, analysis, etc. will be presented.

Tuesday, November 11, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall ,  Tuesday, November 11, 2008
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Maria Sabitova (Department of Mathematics, University of Illinois)
Root numbers of algebraic varieties
Abstract: Root numbers are important fundamental invariants that arise in connection with several influential conjectures of number theory and representation theory, such as Birch--Swinnerton-Dyer conjecture, conjectural functional equations for L-functions, and the Langlands program. In this talk I will give a brief overview of the history of the subject and then discuss the case of root numbers attached to abelian varieties.

Wednesday, November 19, 2008

Math 499: Introduction to Graduate Mathematics
4:00 pm   in 245 Altgeld Hall,  Wednesday, November 19, 2008
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Kenneth Stolarsky (Department of Mathematics, University of Illinois)
Polynomials in Number Theory and Analysis
Abstract: In number theory one needs the set of all algebraic numbers, so the roots of polynomials with integer coefficients are of central importance. There is much more to the study of polynomials than just 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. Finally, polynomials can enter into problems whose formulation does not mention polynomials. An example is the open problem of determining the mod 1 distribution of xn for a given x > 1 (e.g. x = 3/2). This has led to the concept of a PV number.

Thursday, November 20, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall ,  Thursday, November 20, 2008
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Jason Sneed (Department of Mathematics, University of Illinois)
Quasi-Prime Number Races
Abstract: In 1853, Chebyshev noted that there were always at least as many primes $p \leq x$ congruent to $3 \pmod{4}$ as there were primes congruent to $1\pmod{4}$ for relatively small $x$ values. In this discussion, we consider "quasi-primes", the product of two primes that are not necessarily distinct, and examine the number of quasi-primes $pq \leq x$ congruent to $3 \pmod{4}$ and $1\pmod{4}$. We then find a "percentage" of the time that there are more quasi-primes up to $x$ congruent to $1\pmod{4}$ as $x$ tends to infinity.

Tuesday, December 2, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall ,  Tuesday, December 2, 2008
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Geremias Polanco (Department of Mathematics, University of Illinois)
Beatty Sequences, Characteristic Words and an Unusual Continued Fraction
Abstract: If a, b are irrational numbers such that 1/a + 1/b = 1, A={Floor[a*n] for n= 1,2,3,...}, and B={Floor[b*n] for n=1,2,3,...}, then, every positive integer is an element of A or B and no integer belong to both, i.e, A and B form a partition of the integers. A and B are called Complementary Beatty Sequences. In this talk we present well known properties of the so called characteristic words, give a proof for this theorem about Beatty Sequence and apply properties of characteristic words to construct a number whose continued fraction expansion has quotients equal to 2 raised to the power of the Fibonacci Numbers.

Tuesday, December 9, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall ,  Tuesday, December 9, 2008
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Eunmi Kim (Department of Mathematics, University of Illinois)
Sum of distances between points on the unit circle
Abstract: We will discuss distributions of N points p_1, p_2, ..., p_N on the unit circle which maximize ∑|p_i-p_j|^s for s>0.

Thursday, December 11, 2008

Number Theory
1:00 pm   in 241 Altgeld Hall,  Thursday, December 11, 2008
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Kevin Ford (UIUC math)
Generalized Wieferich primes
Abstract: Primes p for which p^2 divides 2^p-1 are called Wieferich primes and are quite rare. More generally, if b>1, one can ask about primes p with p^2 dividing b^p-1. We discuss the problem of bounding the least number b for which p^2 does not divide b^p-1. Tools used are bounds for smooth numbers and exponential sums. This is joint work with Jean Bourgain, Sergei Konyagin and Igor Shparlinski.