Number Theory Seminar
Fall 2006 Archive:
Mehmet Dagli: Applications of algebra to quantum control theory.
Ling Long: Finite index subgroups of PSL(2 ,Z) and their modular functions.
Abstract: The modular group PSL(2,Z), consisting of elements in SL(2,Z) modulo the scalar matrices, is a free group generated by two elements. The modular group is of fundamental importance in number theory, geometry, and algebra. The talk will present some basic properties of finite index subgroups of PSL(2,Z), their actions on the meromorphic functions on the upper half complex plane, and their invariants called modular functions. We will also mention some constructions on certain finite index subgroups which cannot be described by congruence relations and the arithmetic properties of their modular functions. .
Thanksgiving break: no seminar.
Faculty meeting: no seminar.
Chris Kurth: Construction of noncongruence subgroups.
Siu-Hung Ng: On arithmetic properties of Frobenius-Schur indicators II
Abstract: In this talk, we will continue to discuss more arithmetic consequences of Frobenius-Schur indicators for a spherical fusion category. We will mention some details for the proof of a higher indicator formula for modular tensor categories, and highlight some relations to the associated representations of SL(2,Z). The talk is intended to be accessible to graduate students.
Siu-Hung Ng: On arithmetic properties of Frobenius-Schur indicators I
Abstract: In this talk, we will introduce the higher Frobenius-Schur indicators for a spherical fusion category which generalize the classical higher indicators for the representations of a finite group. We will discuss the Galois group action on a sequence of higher indicators, and some arithmetic consequences such as a version of Fermat's Little Theorem, and an analog of Cauchy's Theorem for certain spherical fusion categories. The talk is intended to be accessible to graduate students.
A recursive decomposition of the unitary operators on qubits.
Abstract: Decompositions of the unitary group U (2N) are useful tools in quantum information theory as they allow to decompose unitary evolutions into local evolutions and evolutions causing entanglement. We propose a recursive procedure to obtain a decomposition of any unitary evolution on N qubits. This decomposition systematically uses Cartan decompositions of the classical Lie algebras and induces a graded structure on the Lie
algebra u(2N). We present the linear algebra tools involved in the actual calculation of the factors of the decomposition, and present several examples of applications.
The continuing story of zeta.
Abstract: The Riemann zeta function ζ(s) is defined for all s > 1 by the
summation of 1/(ns) from n = 1 to infinity.
The same definition works for all complex numbers s with real part Re (s) > 1.
The importance of ζ(s) for number theory, the distribution of
primes and other asymptotic counting problems stems in part from the
possibility of extending it to a function on the whole complex plane,
with only a simple pole at s = 1.
There are many ways to obtain this result. The high road, Riemann's
own, uses contour integration at an early stage, and
leads directly to the famous functional equation linking the value
ζ(s) to ζ(1 - s). Other methods are known (Chapter 2 of
Titchmarsh's book lists seven) but a toll
seems inevitable on any route ending with the functional equation.
There are less sophisticated ways of extending ζ(s) without
proving the functional equation.
We aim to give a particularly simple proof, obtaining the values of
ζ(s) at negative integers in the process in terms of Bernoulli
We will prove everything we need and provide some motivation, in
particular a neat link between Bernoulli numbers and binomial
September 25, October 2
Key One Chung:
Weak coalgebras and their applications.
Abstract: Coalgebras have been studied to model various structures in theoretical computer science, including automata, transition systems, and object oriented systems. Likewise, there are natural ways to turn some mathematical objects such as graphs and topological spaces into coalgebras. One advantage of the coalgebraic point of view in mathematical objects is the use of extensive results from the theory of coalgebras. When we express these mathematical objects in coalgebraic language, the standard coalgebraic notion of a homomorphism is too strict. In this talk, we propose a relaxation of the condition for the definition of a homomorphism. As one example of the advantages that accrue, we demonstrate the finite completeness and non-cocompleteness of the category of all coalgebras for the finite powerset functor. This results can naturally be applied to the category of locally finite graphs including loops.
September 11, 18
Anna Romanowska (Warsaw University of Technology):
Algebraic intervals, convex sets, and fuzzy logic.
Abstract: Real convex sets can be presented algebraically with binary operations given by weighted means, the weights taken from the open unit interval in the real numbers. The class of convex sets is a
quasivariety (defined by certain implications) and generates the variety (defined by identities) of so-called barycentric algebras. Both these classes have a well developed theory. However, in the specification of convex sets and barycentric algebras, the open unit interval itself has not hitherto been axiomatized. The talk will discuss the problem of
axiomatization, and propose one possible solution. We extend the
open unit interval of operations to the closed one, and consider
barycentric algebras as two-sorted algebras, one sort
corresponding to the set of elements of a traditional barycentric
algebra, and the second corresponding to a certain algebra of
fuzzy logic, a so-called LP-algebra. The LP-algebras yield an algebraic description of the closed unit interval, and suggest interesting extensions of the class of barycentric algebras. The new structures encompass two-sorted
counterparts of barycentric algebras over any ordered field,
and also include other algebras providing links to such
notions as Boolean affine spaces, the B-sets of Bergman and Stokes,
and ``if-then-else'' algebras.
Labor Day: no seminar.
Jonathan Smith: The algebra of 3-nets and quasigroup homotopies.
Abstract: 3-nets are known as a combinatorial construction. To differential geometers, the comparable concept is a 3-web. These constructions are given an algebraic interpretation in terms of quasigroup homotopies.
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