K-Theory and Quadratic Forms Seminar
Organizers: Dr K. Hutchinson / Dr R. Osburn
Time: every Wednesday at 4pm
Venue: Mathematical Sciences Seminar Room
Wednesday, September 27, 2006Speaker: Dr. Robert Osburn (UCD)
Title: Gaussian hypergeometric functions and supercongruences
Abstract: In 1984, Greene introduced the notion of hypergeometric functions over finite fields. Special values of these functions have recently been of interest as they are related to numbers of F_p points on algebraic varieties and to Fourier coefficients of modular forms. In this talk, we discuss a general modulo p^3 congruence for these functions which yields extensions to recent supercongruences of Ono-Ahlgren, Loh-Rhodes, and of Mortenson. This is joint work with Carsten Schneider (Linz).
Wednesday, October 4, 2006
Speaker: Dr. Kevin Hutchinson (UCD)
Title: Galois Groups, cup products and K-theory
Abstract: A much-studied paper of Sharifi and McCallum in 2003 relates the structure of the Galois group of the maximal p-ramified (p prime) extension of a number field to a cup product in cohomology and, via this, to K-theory and Iwasawa theory. We will give a survey of these ideas and will provide a particularly simple K-theoretic interpretation of part of the cup product which extends some results of Sharifi and McCallum and gives information about the Galois groups of certain number fields.
Wednesday, October 11, 2006
Speaker: Professor David Lewis (UCD)
Title: Similitudes of algebras with involution under odd-degree extensions
Abstract: We discuss the following recent result of P. B. Barquero-Salavert: Let F be a field and L/F be an odd-degree extension. Let (A_1, sigma_1) and (A_2, sigma_2) be two central simple algebras with involution. We investigate in what cases (including when char(F)=2), we have that (A_1, sigma_1) and (A_2, sigma_2) are similar over L implies they are already similar over F. This will have applications to the solution of injectivity problems in nonabelian galois cohomology.
Wednesday, October 18, 2006
Speaker: Professor Chris Smyth (Edinburgh)
Title: Integer symmetric matrices with small eigenvalues
Abstract: In joint work with James McKee, we describe all integer symmetric matrices whose eigenvalues have modulus at most 2. It turns out that they can be essentially described by certain kinds of graphs, which we call charged signed graphs. What is the significance of the bound 2? I'll explain this in the talk. If time permits I'll also describe a potential application for these matrices.
Wednesday, November 8, 2006
Speaker: Professor Graham Everest (UEA)
Title: The Divisibility of Some Divisibility Sequences
Abstract: It is a pity that the Mersenne Prime Conjecture seems to be out of reach at the moment. However, in the late 19th century Zsigmondy and Bang found a way to guarantee new prime factors of terms of this and other related sequences. I will discuss this kind of result in the context of some integer sequences which arise naturally from geometry using the theory of elliptic curves.
Wednesday, November 15, 2006
Speaker: Professor Werner Nahm (DIAS)
Title: K_3(\bar Q) and modular forms
Abstract: A conjectural relation between modular forms and torsion elements of K_3(\bar Q) has been tested by D. Zagier and appears to hold true. The relationship of the conjecture to the representation theory of Yangians will be explained.
Wednesday, November 29, 2006 (please note that the second talk will begin at 5:15pm)
Speaker: Dr. Amit Kulshrestha (Louvain-la-Neuve)
Title: Similarity factors of hermitian forms over number fields
Abstract: Undoubtedly, number fields exhibit many nice properties. Some of these properties can be expressed in terms of Galois cohomology or quadratic forms over fields (e.g. strong approximation property, torsion-free third power of the fundamental ideal of Witt ring, etc.). In this talk we discuss the group of similarity factors of hermitian forms over some special fields (number fields always being typical examples of those fields) and compare the group of similarities with "Hyp" group and "certain norm" groups. The results mentioned in the talk are a part of my PhD Thesis submitted last year.
Speaker: Ms. Mélanie Raczek (Louvain-la-Neuve)
Title: Certain subspaces of order 3 matrices and their automorphisms
Abstract: Let F be a separably closed field of characteristic not equal to 2 or 3. We consider a 3-dimensional subspace V of the trace zero matrices of M_3(F) which is totally isotropic for the trace form (x\mapsto tr(x^2)). We compute the group of automorphisms of (M_3(F),V). Such a computation is interesting with respect to the classification of degree 3 central simple algebras equipped with a subspace V as above.
Wednesday, December 6, 2006 (please note that this seminar will take place in the Teaching Room)
Speaker: Dr. Colin Wilmott (UCD)
Title: Breaking RSA Encryption with a Quantum Computer: Shor's Factoring Algorithm
Abstract: Arguably the most spectacular breakthrough in quantum computation was achieved when Shor presented a quantum algorithm for factoring an n-bit integer. This is a task which is believed to be intractable on a classical computer. I will discuss the means by which Shor's algorithm provides an exponential speed-up over the best known classical algorithm for factoring. This is a joint Number Theory/Claude Shannon Institute seminar.
Wednesday, December 13, 2006 (please note that this seminar will take place in the Teaching Room)
Speaker: Professor Andrew Ranicki (Edinburgh)
Title: Noncommutative localization in algebra and topology
Abstract: The talk will survey the algebraic K- and L-theory of the noncommutative Cohn localizations of rings, and the applications to the topology of non-simply-connected manifolds.
