Algebra and Number Theory Seminar

Fall 2004

Thursdays, 11:15-12:05, 320 Whitmore

September 9           Luis Gallardo, Brest

Waring's problem for polynomials

Note: Special Time and Place, 4:00-5:00, 201 Thomas.

September 16         Joe Hundley, PSU

A new Rankin-Selberg construction for GSO(8)

Let $\pi$ be an automorphic representation of a split reductive algebraic group $G$ over a number field $F$.  Langlands has associated to $G$ a complex Lie group $^LG$, and defined, for every finite dimensional representation $r$ of $^LG$, an $L$-function $L(\pi,r,s),$ which is conjectured to have analytic properties similar to those of the Riemann zeta function and/or Dirichlet $L$-functions, and contain interesting information about the representation $\pi.$  In the Rankin-Selberg method, analytic properties of $L$-functions are proved by expressing the $L$-functions in terms of integrals containing Eisenstein series, whose analytic properties are known.  We will consider a new integral on the similitude orthogonal group GSO(8), in which the representations $r$ that arise are the standard and half-spin representations of SO(8,C), and use it to relate poles of $L$-functions with the existence of a non-trivial functorial lift to the exceptional group $G_2.$

September 23          Alexander Borisov, PSU

Two elementary results concerning rational polynomials

The two elementary results in the title are part of my recent work. Both have to do with the wonderful fact that in polynomials with integer coefficients both the coefficients and the powers are taken from essentially the same set. Most of the talk will be accessible to most graduate students.

The first result is the following theorem.

Theorem. A polynomial with rational coefficients divides a derivative of the polynomial which is split over rationals if and only if all of its irrational roots are real and simple.

I will explain the motivation as well as the proof of this result, which resembles the famous three point theorem of Belyj. The second result is related to the classification of the set of solutions of the system of functional equations satisfied by the so-called quantum integers with respect to multiplication.

September 30          Robert Vaughan, PSU

Report of the Oberwolfach meeting on the Riemann zeta and allied functions

October 7                David Terhune, PSU

Evaluations of a class of double L-values

An anlaytic proof for the `convolution' -type double L-values is given.  Along the way, Dirichlet character analogues of generalized single and double polylogarithms are defined, and the monodromies of these new functions play a pivotal role.

October 14              Scott Parsell, Butler U.

Paucity Problems

In analogy with Fermat's Last Theorem, one typically expects that a diophantine equation in few variables (relative to the degree) will have few, if any, non-trivial solutions.   When counting the number of solutions in a box, one can frequently show that the number of non-trivial solutions grows more slowly than the number of trivial ones as the size of the box tends to infinity, and in this case we say that the equation (or system of equations) exhibits a "paucity" of non-trivial solutions.  We will present several examples of this phenomenon in symmetric systems and discuss some recent results in this direction.

October 21              Bill Hoffman, LSU

Modular forms on noncongruence subgroups and Atkin-Swinnerton-Dyer relations

This is a joint work with Liqun Fang, Ben Linowitz, Andrew Rupinski, and Helena Verrill. We give new examples of modular forms on noncongruence subgroups whose $l$-adic representations are modular and whose expansion coefficients satisfy Atkin-Swinnerton-Dyer congruences.

October 28              Cam Stewart, Waterloo U.

On intervals with few prime numbers

In this talk we shall survey the known results on large gaps between consecutive prime numbers and we shall then discuss recent work of Maier and the  speaker on long intervals with fewer than the expected number of primes.

October 29              Vladimir Retakh, Rutgers

Note: Special Day, Friday, 11:15-12:05, 320 Whitmore.    

Universal algebra of pseudo-roots of non-commutative polynomialsa and non-commutative  symmetric functions

A generic monic polynomial $P(t)\in R[t]$ where $R$ is a (non-commutative) division ring admits many different facotrizations into a product of linear polynomials $t-a$.  Elements $a$ are called pseudo-roots of $P(t)$ and may be considered as generators of a quadratic algebra.  This algebra has a deep structure.  I will talk about this structure anbd its relations to other problems.

November 4            Dale Brownawell, PSU

Independence of derivatives of Carlitz periods

Carlitz defined an exponential function in the setting of function fields F_q(t) over finite fields F_q.  In particular, its periods are all multiples of a fundamental period \pi_q which is the analogue of the usual 2{\pi}i, whose transcendence was shown by Carlitz's student Wade.  Recently Laurent Denis has investigated the algebraic independence of the derivatives of \pi_q.  Alf van der Poorten and I have taken up this investigation.

November 11          Katherine Hurley, PSU

Vertex Operator Algebras and Spherical Harmonics

This talk introduces the axioms for a vertex operator algebra and gives the examples of Heisenberg and Lattice VOAs.  It then discusses the use of harmonic polynomials (spherical harmonics) to construct highest-weight vectors for the Virasoro Lie algebra and determine their graded traces on the vertex operator algebra.

November 18          Robert Vaughan, PSU

Diophantine Approximation on Planar Curves

Let $\cal C$ be a non--degenerate planar curve and for a real, positive decreasing function $\psi$ let $\cal C(\psi)$ denote the set of simultaneously $\psi$--approximable points lying on $\cal C$. In joint work with Beresnevich, Dickinson and Velani, we show that $\cal C$ is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on $\cal C$ of $\cal C(\psi)$ is full. We also obtain the Hausdorff measure analogue of the divergent Khintchine type result.  In the case that $\cal C$ is the unit circle the convergence counterparts of the divergent results  are  also obtained.  Furthermore, for functions $\psi$ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of $\cal C(\psi)$.  These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

December 2             Ulrike Vorhauer, U. of Michigan

Comparative prime number theory

Chebyshev conjectured that $\pi(x;4,3) \ge \pi(x;4,1)$ for all  $x\ge 3$.  Although $x$ are now known for which this fails, and it is known that the difference $\pi(x;4,3)-\pi(x;4,1)$ changes sign infinitely many times, for the general modulus it is still not known that $\pi(x;q,a) - \pi(x;q,b)$ changes signs for arbitrarily large $x$ when $(a,q) = (b,q) = 1$.  We survey the existing knowledge, and add a new result in this area.

December 9             George Andrews, Penn State

Partitions with Short Sequences and Ramanujan's Mock Theta Functions

In a recent paper "Integrals, Partitions and Cellular Automata" in the Transactions of the American Mathematical Society, Holroyd, Liggett and Romik evaluated an intriguing definite integral and applied it to a variety of probability models.  The application to integer partitions concerned partitions in which no sequence of consecutive integers of length k appears (k=2,3,...).  The authors note that in one instance, a proof of their result can also be based on a little known partition theorem of P.A. MacMahon.  Our object in this talk will be to introduce these ideas and to develop the study of such partitions from a purely combinatorial, q-series point of view.  Surprisingly one of Ramanujan's mysterious mock theta functions arises.