Tuesday, August 28: Bruce Berndt, Cranks -- Really the Final Problem. The purpose of this talk is threefold: (1) Provide a survey of Ramanujan's work on cranks in his lost notebook. (2) Ask questions about why Ramanujan made certain calculations and what were his motivations for doing so. (3) Present a (convincing?) argument that the last topic examined by Ramanujan before he died was cranks.

Thursday, August 30: Andrew Schultz, Hilbert 90 for Biquadratic Extensions. Hilbert's Satz 90 is a classic result for cyclic Galois extensions which has been generalized both cohomologically and K-theoretically. In this talk we give a generalization of Hilbert 90 which has the same flavor as the original, but which applies to finite abelian extensions instead of just cyclic ones. This talk only assumes a basic familiarity with Galois theory, so it should be accessible to almost everyone.

Tuesday, September 4: Jeremy Rouse, Iterated endomorphisms of abelian algebraic groups. Let x = cos^(-1)(3/5) and let a(n) = 1 + cos(n x). What can be said about the set of prime numbers that divide the numerators of the a(n)? The answer to this question involves algebraic number theory, Galois theory, and the Chebotarev density theorem.

Thursday, September 6: Heini Halberstam, A conjecture in prime number theory. See the attached file.

Tuesday, September 11: Jonah Sinick, A short and elementary proof of a good upper bound for the partition function. As is well known, Hardy and Ramanujan proved that the number of partitions of n, p(n), obeys the asymptotic relation p(n) ~ (A/n)e^(C*sqrt(n)) where C = pi*sqrt(2/3) and A is a constant. Carl Ludwig Siegel found a much shorter proof of the weaker statement that p(n) < e^(C*sqrt(n)) for the same value of C as in the asymptotic formula. Wladimir Pribitkin has recently used Siegel's method to prove an upper bound closer to the actual asymptotic, showing that p(n) < (1/n^(3/4))*e^(C*sqrt(n)). In this talk I will present Pribitkin's proof.

Thursday September 13: Kevin Ford, Thin additive bases. Consider a set of positive integers S such that every positive integer from 1 to N is the sum of two elements of S. How small can such a set be? We discuss what has been done to answer this question and discuss related problems about continuous functions, polynomials and Sidon sets.

Thursday September 20: Isaac Goldbring, Elliptic Curves and Bezout Domains. Motivated by a question of model theory, we ask the following question: Let k denote the field of algebraic numbers and let S be a multiplicatively closed subset of k[t]. When is the integral closure of S^{-1}k[t] in the algebraic closure of k(t) a Bezout domain, that is a domain such that every finitely generated ideal is principal? We prove a necessary condition on S for this to hold. The condition involves elliptic curves over k and we provide a few examples of an S for which the condition fails. I will end the talk with several open questions. This is joint work with Marc Masdeu.

Tuesday September 25: Harold Diamond, Approximating the Laplace transform of the Gaussian function. The Laplace transform of the Gaussian function is an interesting and important entire function. Prof. Halberstam and I encountered it in our sieve research. I shall describe a few approximations of this function that are useful in various ranges.

Thursday September 27: Khang Tran, The transcendence of pi. The ancient Greeks had tried to construct a square with area equal to that of a given circle using only ruler and compasses. Lindemann succeeded in generalizing Hermite’s methods to prove that pi is transcendental. This implies that the construction above is impossible. The work of Lindemann was simplified by Weierstrass in 1885, and further simplified by Hilbert, Hurwitz and Gordan in 1983. This talk will review the transcendence of e and go over Lindemann’s work on the transcendence of pi.

Tuesday October 9: Scott Ahglren, Rank generating functions as weakly holomorphic modular forms. In the 1950s Atkin and Swinnerton Dyer produced many complicated identities involving the ranks of partitions (some of these confirmed conjectures made by Dyson). Recent work of Ono, Bringman, and Rhoades has explained some of these identities in the context of weakly holomorphic Maass forms. We produce infinite families of phenomena which explain the remaining identities as special cases. All necessary background will be given in the talk (which is joint work with S. Treneer).

Tuesday October 16: Jonah Sinick, Cubic reciprocity and x^2 + 27y^2. Euler conjectured and Gauss proved the fact that a prime p > 3 can be written in the form p = x^2 + 27y^2 if and only if p = 1(mod 3) and z^3 = 2 (mod p) for some z. The proof is intertwined with cubic reciprocity. This result is one of the results that motivated the development of the study of higher reciprocity laws. I'll prove cubic reciprocity and the theorem about primes of the form x^2 + 27y^2.

Thursday October 18: Kenneth Stolarsky, Some interactions of metric geometry with number theory, especially Diophantine approximation. We discuss some ideas from metric geometry (including a game that can be played on a compact, connected metric space) that shed light on or suggest problems about number theory, especially the approximation of irrationals by rationals (Diophantine approximation). We shall mostly avoid topics from the Minkowskian geometry of numbers. Much of this talk should be comprehensible to a wide mathematical audience.

Tuesday October 23: Bruce Berndt, First V. Ramaswamy Aiyer Centennial Lecture Ramanujan's Series for $1/\pi$: A Survey. In his famous paper, Modular equations and approximations to $\pi$, Ramanujan stated without proof 17 hypergeometric-like infinite series representations for $1/\pi$. The first mathematician to provide a general method for proving such formulas was S.~Chowla, who worked out one of Ramanujan's series representations. Using not dissimilar methods, Jonathan and Peter Borwein proved all 17 formulas in 1987. We give a survey of the developments made in proving these and many other similar series for $1/\pi$ since Ramanujan's epic paper. Included will be work of R.~William Gosper, Jr., Gregory and David Chudnovsky, Jesus Guillera, and several others. In particular, in collaboration with Nayandeep Baruah and Heng Huat Chan, we show that, in contrast to the methods employed by all other authors, Ramanujan's original, but previously neglected, ideas yield perhaps the most efficient and fruitful approach.
This lecture is given in memory of V. Ramaswamy Aiyer, Founder of the Indian Mathematical Society in 1907.

Thursday October 25: Amanda Folsom, Duality involving the mock theta function f(q) and analytic properties of Kloosterman-Selberg zeta functions. We show that the coefficients of Ramanujan's mock theta function $f(q)$ are the first nontrivial coefficients of a canonical sequence of modular forms. This fact follows from a duality which equates coefficients of the holomorphic projections of certain weight 1/2 Maass forms with coefficients of certain weight 3/2 modular forms. This work depends on the theory of Poincar\'e series, and a modification of an argument of Goldfeld and Sarnak on Kloosterman-Selberg zeta functions.

Thursday November 1: Rob Rhoades, Eulerian Series as Modular Forms. In 1988 Hickerson proved the celebrated "mock theta conjectures", a collection of ten identities from Ramanujan's "lost notebook" which express certain modular forms as linear combinations of mock theta functions. Using the modern perspective of Weak Maass forms we show that this phenomenon is generic. Specifically, we use this general theory to construct several infinite families of modular forms which are linear combinations of Eulerian series.

Tuesday November 6: Renate Scheidler, Construction of Hyperelliptic Function Fields of High Three-Rank. A hyperelliptic function field is a field of the form k(x,y) where k is a finite field of odd characteristic and y^2 = D(x) with D(x) a square-free polynomial with coefficients in k. If D has even degree, or if D has odd degree and the leading coefficient of D is a non-square in k, then the Jacobian of the hyperelliptic curve y^2 = D(x) is essentially isomorphic to the ideal class group of the ring k[x,y]. This is the finite Abelian group of fractional ideals of k[x,y] modulo principal fractional ideals. Although generically, the 3-Sylow subgroup of this ideal class group is small (and frequently trivial), it is possible to generate hyperelliptic function fields -- even infinite families of such fields -- whose 3-rank is unusually large.

Thursday November 8: Ivan Horozov, Non-Commutative Two-Dimensional Modular Symbols. Modular symbols are integrals of modular forms over a region connecting cusp points. A few years ago Manin has defined non-commutative modular symbols for finite index subgroups of SL(2,Z). His method is based on iterated integrals over a path as defined by Chen. Manin iterates modular forms and the generating series of all such integrals, which he calls non-commutative modular symbol, has interesting properties. My construction consist of two parts. First, I define iterated integrals over a (real) two-dimensional region so, that all such iterated integrals naturally form a Hopf algebra. Then I apply this construction to Hilbert modular surfaces. Using this new type of iteration, I construct non-commutative two-dimensional modular symbols, which are group-like elements in the Hopf algebra.

Tuesday November 13: Bruce Reznick, On the non-monotonicity of the sequence { | Im(z^n) | }. We prove the somewhat amusing fact that if z is a non-real complex number, then the title sequence is never monotone. Much can be said about its behavior. For example, if w = 1 + 2 i, then the set of integers n for which | Im(w^{n+1}) | > | Im(w^n) | has density exactly 1/4. If w = 20 + 7 i, the density is arctan(1/32)/pi. This talk is based on a paper published in 1999; previous versions of the talk presented examples with w = 19 + 87 i and w = 19 + 98 i.

Thurday November 15: Mr. Cevallos, Euler and the Zeta function. We discuss the extent to which Euler realized that the Zeta function satisfied a functional equation. In the course of doing this we shall explain/review some fundamental properties of the Zeta function.

Tuesday November 27: Rafe Jones, Arboreal Galois representations and arithmetic dynamics. The preimages of a point under an iterated map form a tree in a natural way. When the point and map are defined over a global field $K$, the absolute Galois group of $K$ preserves the tree structure, giving an "arboreal" Galois representation. The image of this representation encodes information about a surprising variety of arithmetical and dynamical problems. In this talk I will give concrete descriptions of some of these problems, and discuss how they can be translated into the setting of arboreal representations. Then I will give some theorems resolving the problems. Along the way I will point out several conjectures and open questions, and give a brief introduction to the field of arithmetic dynamics.

Thursday November 29: Paul Pollack, The distribution of irreducible polynomials over finite fields. After reviewing some of the basic analogies between the ring of rational integers and the ring of univariate polynomials over a finite field, we look at some recent results on the distribution of irreducible polynomials over a finite field. Much of the theory of irreducible polynomials runs parallel to classical prime number theory, but there are some surprises. Our emphasis in this talk will be on analogues of the twin prime conjecture as well as Schinzel's more general Hypothesis H.

Tuesday December 4: Scott Parsell, Quantative issues for diophantine inequalities. Using a classical method of Davenport and Heilbronn, one can show that the values taken by an irrational indefinite form at integer points are dense in the real line, provided that the number of variables is sufficiently large in terms of the degree. In other words, an inequality of the shape $|f(x_1, \dots, x_s)-\mu| < \varepsilon$ has integer solutions for every real number $\mu$ and every positive number $\varepsilon$. It is natural to ask for quantitative versions of this result, in which one counts solutions lying in a box and/or replaces $\varepsilon$ by an explicit function of the box size. We discuss the limitations of the classical method in achieving such refinements and the extent to which the recent Bentkus-G\"otze-Freeman technology makes these possible. We also describe some new work (joint with T. Wooley) on the set of exceptional $\mu$ for which an inequality in relatively few variables fails to have a solution.

Thursday December 6: Atul Dixit, Ramanujan's circular summation of theta functions. In his Lost Notebook, Ramanujan claimed that the circular summation of n^th powers of the symmetric theta function f(a,b) satisfies a factorization of the form f(a,b)*F_n(ab), where F_n(x)=1+2nx^{(n-1)/2}+...., n>=3. Ramanujan also recorded elegant identities for F_n(ab) when n=2, 3, 4, 5 and 7 in terms of his theta functions phi(q), psi(q) and f(-q). Here we discuss the recent proof of the circular summation formula by Heng Huat Chan, Zhi-Guo Liu and Say Tiong Ng which employs elliptic function theory.