Seminar Calendar
for number theory events the next 4 month of Monday, January 14, 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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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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Submitted by jarouse.
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.