MATHEMATICS COLLOQUIUM AND SEMINARS
FORTHCOMING TALKS
COLLOQUIUM * MONDAY 5 JUNE 2006 * 1:00 PM * E6A Room 133
SPEAKER: Kim Lund Larsen (Visiting Fellow, Macquarie University)
TITLE: The class number problem for imaginary quadratic fields
ABSTRACT:
h(-d), the class number imaginary quadratic field with discriminant d can be expressed in terms of binary quadratic forms with discriminant d. Gauss studied these forms and calculated a comprehensive list of h(-d) for small values of d. He could however only conjecture that his list was complete of a given value of h (the class number problem), and that the h(-d) goes to infinity as d goes to infinity (Gauss' class number conjecture).
The first proof of Gauss' class number conjecture is quite peculiar and shows how inspirational the Riemann Conjecture can be.
The main contribution in solving the class number problem is a theorem from 1976 due to Goldfeld. The problem was solved in 1985-86 when Gross and Zagier found an elliptic curve, whose associated L-series has a zero at 1 of order at least 3.
Goldfelds theorem illustrates how powerful elliptic curves can be in number theory. It also shows how important conjectures such as the Birch and Swinnington-Dyer Conjecture and Shimura-Taniyama conjecture (now an important theorem due to Wiles) can be as inspiration.
The intended endpoint of the talk is to elaborate a bit on a simplified version of Goldfelds theorem due to Oesterlé.
PREVIOUS TALKS IN 2006
COLLOQUIUM * MONDAY 10 APRIL 2006 * 1:00 PM * E6A Room 133
SPEAKER: Winfried Kohnen (University of Heidelberg)
TITLE: Representation numbers of positive integers by positive definite integral quadratic forms
ABSTRACT:
We will report on recent joint work with O. Imamoglu, in which we give new finite explicit formulas for the number of representations of of a positive integer as a sum of s integral squares, for each s divisible by 8. For this we use the theory of modular forms. No pre-knowledge of modular forms is necessary for the talk.
COLLOQUIUM * MONDAY 1 MAY 2006 * 1:00 PM * E6A Room 133
SPEAKER: John Corbett (Macquarie University)
TITLE: Real number continua and understanding quantum mechanics
ABSTRACT:
Following Gauss and Riemann, the mathematical treatment of the problem of space leaves the topological function of labelling the points to the coordinate system, which uses the standard real number continuum, and describes the metrical function with the metric tensor. Applications to classical physics have shown that the metric tensor depends upon the physical situation. I will argue that applying this analysis to quantum mechanics shows that the real number continuum depends upon the physical situation. For example, the problem of non-locality arises because we are using the wrong real number continuum. This raises the possibility that the laws of physics should be invariant under changes of continua.
Using topos theory, we can construct non-standard real numbers, the Dedekind reals on a topological space $X$, $\RsubD(X)$, which is the sheaf of continuous real-valued functions on $X$. The connection with quantum mechanics is made by taking $X$ to be the Schwartz state space $\EsubS$ of linear functionals on an O*-algebra of physical quantities. We call these real numbers qr-numbers. The physical quantities always have qr-number values which satisfy Hamiltonian-type differential equations of motion. If there is time I will indicate how standard quantum mechanical formulae for measurement and dynamical evolution arise as local approximations in this approach.
This is joint work that has been done with Thomas Durt, VUB, Belgium.
TALKS IN 2005
NUMBER THEORY SEMINAR * FRIDAY 18 MARCH 2005 * 11:00 AM * E6A Room 357
SPEAKER: Paula Cohen (Texas A&M University)
TITLE: Special functions, special points and transcendence
ABSTRACT:
Inspired by Hilbert's 7th problem, Siegel (1932) and Schneider (1937) obtained the first significant results about the transcendence of periods of doubly periodic functions and of values of modular functions at certain algebraic points. Siegel formulated similar problems for G-functions, a particular case of which is the classical hypergeometric function. The modern development of this circle of ideas is the focus of our lecture. For example, we discuss results of our own and others on the transcendence of special values of functions defined on Shimura varieties and the surprising role, first noticed by Wolfart, played by non-arithmetic groups acting on the complex ball. A distinguished role in these results is played by special (or complex multiplication) points. This leads to considering how these points are distributed. Results in this direction will also be discussed. The lecture will be self-contained and accessible to a general audience.
MATHEMATICS COLLOQUIUM * FRIDAY 18 MARCH 2005 * 1:00 PM * E7B Room 100
SPEAKER: Marvin Tretkoff (Texas A&M University)
TITLE: The classical periods of abelian integrals
ABSTRACT:
The periods of abelian integrals appear in complex function theory, number theory, algebraic geometry and mathematical physics. Beginning with an example from elementary calculus, we will give an informal discussion of abelian integrals and their periods. There will be no rigorous proofs. Towards the end of the lecture, we hope to illustrate our own contribution to the subject by determining the periods associated to the Fermat curve.
MATHEMATICS COLLOQUIUM * MONDAY 21 MARCH 2005 * 1:00 PM * C5A Room 226
SPEAKER: Arthur Benjamin (Harvey Mudd College / on sabbatical at UNSW)
TITLE: Counting on determinants
ABSTRACT:
We demonstrate how determinants solve many interesting combinatorial problems, and how many interesting theorems about determinants can be viewed combinatorially. Applications to Pascal's Triangle, Fibonacci numbers and Catalan numbers will also be given. We end with a brand new combinatorial proof of Vandermonde's determinant.
This talk is based on joint work with Naiomi Cameron of Occidental College and Gregory Dresden of Washington and Lee University.
NUMBER THEORY SEMINAR * MONDAY 21 MARCH 2005 * 3:00 PM * E7A Room 333
SPEAKER: Arthur Benjamin (Harvey Mudd College / on sabbatical at UNSW)
TITLE: Number theory and combinatorics
ABSTRACT:
In this lecture, I give combinatorial proofs of all of your favorite theorems from number theory, including:
and more!
MATHEMATICS COLLOQUIUM * MONDAY 4 APRIL 2005 * 1:00 PM * E6A Room 131
SPEAKER: Thomas Durt (Department of Physics, Free University of Brussels)
TITLE: Applications of the generalised Pauli group in prime, prime power and non-prime dimensions
ABSTRACT:
It is known that finite fields with N elements exist only when N is a prime or a prime power.
When the dimension N of a finite dimensional Hilbert space is a prime power, we can associate to each basis state of the Hilbert space an element of a finite or Galois field, and construct a finite group of unitary transformations, the generalised Pauli group or discrete Heisenberg-Weyl group. Its elements can be expressed, in terms of the Galois field operations (addition and multiplication). When the dimension is a prime p, these operations reduce to the operations modulo p.
This group presents numerous applications in quantum information science, e.g. tomography, dense coding, teleportation, error correction and so on.
For instance the bases that diagonalize the generalised Pauli operators also generalize the X, Y, and Z qubit bases in the sense that they are mutually unbiased or maximally conjugate. The aim of our talk is to give a general survey of these techniques and to present recently obtained results in connection with two problems:
MATHEMATICS COLLOQUIUM * TUESDAY 12 APRIL 2005 * 1:00 PM * E6A Room 131
SPEAKER: Ben Odgers (University of Bristol)
TITLE: Moments and log moments of characteristic polynomials of random matrices, and a possible connection with families of L-functions
ABSTRACT:
I have been looking at the work of Conrey-Farmer-Keating-Rubinstein-Snaith which gives evidence for a link between the moments of the characteristic polynomials of random matrices, and the moments of particular families of L-functions. In this talk I will give details of my efforts to extend their work away from "the critical point", in particular finding an interpolation between results from various matrix ensembles.
MATHEMATICS COLLOQUIUM * MONDAY 2 MAY 2005 * 1:00 PM * E5A Room 119
SPEAKER: Tuomas Hytönen (Helsinki University of Technology)
TITLE: Littlewood-Paley-Stein theory for semigroups in UMD spaces
ABSTRACT:
Littlewood-Paley theory is centered around different "square-functions" (which roughly means $L^2$ norm type integrals with respect to a parameter) and two-sided estimates for them in $L^p$ spaces. The usefulness of these estimates is to provide new equivalent norms for the $L^p$ spaces, so that the boundedness properties of various operators of interest become more transparent. In his 1970 monograph, Stein extended the classical theory involving the Poisson semigroup to a more general context, exploiting deep connections with stochastics and especially the martingale theory.
In this talk, we study a further generalization of these results to the Bochner spaces $L^p_X$ of $X$-valued $p$-integrable functions, where $X$ is a possibly infinite dimensional Banach space, which may not be a Hilbert space in general. In such an extension, the role of probability theory becomes even more crucial, perhaps also more natural, than before. The "right" vector-valued version of the classical square-functions involves stochastic integration with respect to a Brownian motion. Ideas from the geometry of Banach spaces further contribute to a mixture of methods from different fields of mathematics.
MATHEMATICS COLLOQUIUM * MONDAY 16 MAY 2005 * 1:00 PM * E5A Room 119
SPEAKER: Paul Leopardi (University of New South Wales)
TITLE: Partitions of the unit sphere into regions of equal area and small diameter
ABSTRACT:
In a paper of 1974, Kenneth Stolarsky asserted that for each $N>1$ the unit sphere $S^d$ in $R^{d+1}$ can be partitioned into $N$ regions of equal area, each of diameter at most $O(N^{-1/d})$.
In 2002, Uriel Feige and Gideon Schechtman essentially proved this assertion.
We show that Feige and Schtechtman's proof gives an upper bound on the smallest maximum diameter of an equal area partition of the unit sphere.
We also review an alternative construction, called the recursive zonal equal area partition, based on work by Rahmanov, Saff and Zhou, which yields regions which have a particularly simple description, and which has a provable diameter bound of the correct order for $S^d$ up to to at least $d=8$.
MATHEMATICS COLLOQUIUM * MONDAY 6 JUNE 2005 * 1:00 PM * C5A Room 232
SPEAKER: Douglas Rogers
TITLE: News from the dissecting table: more chips off an old block
ABSTRACT:
The talk gathers together further observations about the Pythagorean proposition, drawing on diverse cultural traditions. We begin by catching Pythagoras in the cross-wires of two congruent right triangles positioned orthogonally. We then move to the dissecting table, and conclude with the emergence of Pergial's dissection demonstration.
NUMBER THEORY SEMINAR * MONDAY 1 AUGUST 2005 * 3:00 PM * E7A Room 333
SPEAKER: Bill Hart (between University of Leiden and University of Illinois)
TITLE: Extending Weber's explicit class field theory
ABSTRACT:
About 100 years ago, Heinrich Weber dramatically enriched the world of class field theory ... by inventing it! However, not only did he develop much of the early theory, he also made much of his work explicit. It is this explicit class field theory which has come back into vogue lately with the advent of computers.
Particularly important to Weber were his functions $f$, $f_1$ and $f_2$. He proved numerous beautiful identities for these functions, developed methods of obtaining modular equations for them and then used these to construct class invariants.
Bill Hart will speak about progress in extending all these results to other Weber-like functions and even extending Weber's own results.
But there have been some suprises along the way! In the search for a true generalization of Weber's work, Bill has discovered an intriguing identity in two variables for the Dedekind eta function, which no one seems to have seen before and which has so far defied proof.
Bill will also discuss some of the related research that he has been involved in over the last year and a half, since he moved on from the Macquarie Mathematics Department.
COLLOQUIUM * MONDAY 15 AUGUST 2005 * 1:00 PM * E6A Room 133
SPEAKER: Florian Luca (Universidad Nacional Autonoma de Mexico)
TITLE: Diophantine m-tuples
ABSTRACT:
Let $n$ be a nonzero integer. A set with the property $D(n)$ is a set of nonzero integers $A={a_1,...,a_m}$ such that $a_i \cdot a_{j+n}$ is a square for all $i$ and $j$. What is of interest in general is to find upper bounds on $m$, the size of a set with the property $D(n)$. In my talk, I will survey various known results about this problem and report on a few new ones. For example, one of the new results is that if $n$ is a prime, then $m<3\cdot2^{168}$.
This work is joint with Andrej Dujella.
NUMBER THEORY SEMINAR * MONDAY 15 AUGUST 2005 * 3:00 PM * E7A Room 333
SPEAKER: Florian Luca (Universidad Nacional Autonoma de Mexico)
TITLE: On the number of ordered factorizations of a positive integer
ABSTRACT:
Let $m(n)$ be the function which counts the number of ordered factorizations of $n$ in factors larger than $1$. In 1941, P. Erdos claimed that there exist two constants $c_1$ and $c_2$ such that the inequality
holds for all sufficiently large positive integers $n$ while the inequality
holds for infinitely many positive integers $n$, where $\rho=1.72864...$ is the real solution to $\zeta(\rho)=2$. In my talk, I will sketch a proof of these inequalities with $c_1=1/\rho-\epsilon$ and $c_2=\rho/(\rho^2-1)+\epsilon$, where $\epsilon>0$ is arbitrary. I will also survey some other properties of the function $m(n)$.
This is joint work with M. Klazar.
COLLOQUIUM * FRIDAY 9 SEPTEMBER 2005 * 1:00 PM * E5A Room 119
SPEAKER: John Leach (University of Reading)
TITLE: The evolution of travelling wave-fronts in a hyperbolic Fisher model
ABSTRACT:
In this presentation we develop, via the Needham-Leach (NL) method, the large time asymptotic structure of the solution to a hyperbolic Fisher model. Reference is also made to the NL methodology in general and how it can be applied to other classes of nonlinear partial differential equations (such as the Korteweg-de Vries-Burgers (KdVB) equation and the Fisher-Kolmogorov equation).
COLLOQUIUM * MONDAY 17 OCTOBER 2005 * 1:00 PM * E7B Room 164
SPEAKER: Svitlana Mayboroda (Ohio State University)
TITLE: Regularity of Green potentials on non-smooth domains and solution of the Chang-Krantz-Stein conjecture
ABSTRACT:
The so-called shift theorem for the Laplacian aims to identify those functional-analytic settings in which the solution of the equation Delta u=f (with homogeneous Dirichlet and Neumann boundary conditions) is two units smoother that the datum f. When the domain in which this PDE is considered is smooth, such a result is valid on all the classical smoothness spaces, of Hardy-Besov-Sobolev-Triebel-Lizorkin type, without extra restrictions on the indices involved (integrability and smoothness). On the other hand, counterexamples due to Dahlberg and Jerison-Kenig paint a dramatically different picture when the domain in question is allowed to have an irregular boundary.
In this talk (joint work with M. Mitrea) I will discuss some recent progress in this direction and present a complete, sharp answer to the question of the regularity of Green potentials in Lipschitz domains. As a corollary, this yields a solution of the conjecture made by D.-C. Chang, S. Krantz and E. Stein in the 90's to the effect that two derivatives on the Green potential is a bounded mappings in the context of local Hardy spaces Hp for p<1 sufficiently close to 1.
COLLOQUIUM * TUESDAY 18 OCTOBER 2005 * 1:00 PM * E7B Room 164
SPEAKER: Yuly Billig (University of Ottawa)
TITLE: Vertex algebras and their applications in mathematics and physics
ABSTRACT:
In this talk we will review the definition and examples of vertex algebras. These algebras were introduced by Borcherds in his work on the Monstrous Moonshine, for which he was awarded a Fields Medal. Vertex algebras have applications in several areas of mathematics and physics. These objects play a crucial role in constructing conformal field theories, they are connected to the theory of modular forms, and have applications to the soliton partial differential equations. Vertex algebras provide efficient tools for constructing representations of infinite-dimensional Lie algebras. I will illustrate this with some of the methods used in my work on representations of toroidal Lie algebras.
COLLOQUIUM * TUESDAY 25 OCTOBER 2005 * 2:00 PM * W6B Room 336
SPEAKER: Bruce Berndt (University of Illinois at Urbana-Champaign) (AMS Mahler Lecturer 2005)
TITLE: Mystery tales from India
ABSTRACT:
After giving a short history of Ramanujan's lost notebook and a brief description of its contents, we focus attention on approximately five entries. These are chosen because they are surprising, strange, unusual, fascinating, or interesting. One of the entries has connections to the famous "circle problem" in number theory. Another involves a divergent continued fraction with three distinct limit points for the partial quotients. A third is an unusual identity for the deriviative of a quotient of certain hypergeometric series that converge for |z|=1 only, i.e., not for |z|<1 or |z|>1.
COLLOQUIUM * THURSDAY 10 NOVEMBER 2005 * 1:00 PM * C5A Room 226
SPEAKER: Henrik Grundling (University of New South Wales)
TITLE: Generalising group algebras away from local compactness
ABSTRACT:
We generalise the concept of a group algebra to other algebraic objects with bounded Hilbert space representation theory - call the generalised group algebra a "host algebra." The main property of a C*-algebra to be a host algebra, is that its representation theory is isomorphic (in the sense of the Gelfand-Raikov theorem) to a specified subset of representations of the algebraic object. We obtain both existence and uniqueness theorems for host algebras. Abstractly, this solves the problem of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra. If time permits, we will also show how to make contact with the more usual point of view of convolution algebras.