Number Theory Seminar
Fall 2011 Archive:
Schedule of talks
Liang Xiao (University of Chicago): On the parity conjecture for Selmer groups of modular forms.
Abstract: The parity conjecture is a weak version of the Birch and Swinnerton-Dyer (BSD)
Conjecture, or more generally, the Beilinson-Bloch-Kato Conjecture. It is conjectured
that the order of an L-function at the central point has the same parity
as the dimension of the Bloch-Kato Selmer group. I will explain an
approach to this conjecture for modular forms by varying the modular
form in a p-adic family. This is joint work with Kiran Kedlaya and
Jason Ekstrand: Continuous diagonalization.
Abstract: During the 2011 summer REU, Prof. Peters and I, along with three
undergraduate students, looked at problems concerning matrices over the
ring C[a,b]. One of the problems discussed was that of continuous
diagonalization. Grove and Patterson gave a non-constructive proof in
1983 that hermitian matrices of this form with distinct eigenvalues
are continuously diagonalizable. I will present my work extending
this in the C[a,b] case to hermitian matrices with a continuous
minimal polynomial, and provide a constructive proof
which offers some insight into the problem.
Thanksgiving break: no seminar.
November 7, 14
Cliff Bergman: Constraint Satisfaction Problems, Graph Theory, and Universal Algebra.
Abstract: Numerous computational problems fall under the general rubric of "Constraint Satisfaction". A very active area of computer science concerns algorithms for these problems. It turns out that questions about constraint satisfaction problems can be reformulated as questions about directed graphs and also in the language of universal algebra.
In these talks I will give an introduction to this area of research. I'll give a general definition of a CSP, discuss the reformulation in terms of graphs and algebras, and outline the most recent results. No particular background will be assumed.
Report on the AMS Central Section Meeting, Lincoln, Nebraska.
Jianqiang Zhao (Eckerd College): Drinfeld's associator and its generalizations.
Abstract: In this talk I'll summarize the theory of Drinfeld's associator using his quasi-triangular quasi-Hopf algebra approach. Then I will give a detailed description of Enriquez's generalization. As an application I will describe the computation showing that the mixed pentagon relations properly include all the regularized double shuffle relations in the cyclotomic setting.
Bin Zhang (Sichuan University): Polylogarithms and multiple zeta values from
free Rota-Baxter algebras.
Abstract: This is joint work with Li Guo. We explore the relations of multiple
zeta values from the viewpoint of free Rota-Baxter algebras. We show that
the shuffle algebras for polylogarithms and regularized MZVs in the sense of
Ihara, Kaneko and Zagier are both free commutative nonunitary Rota-Baxter
algebras with one generator. We apply these results to show that the full
sets of shuffle relations of polylogarithms and regularized MZVs are derived
by a single series.
October 3, 10
Eli Stines: Atomic pseudo-valuation domains.
Abstract: It has long been known that formal power series rings over fields have very interesting factorization and divisibility properties. For example, these rings have a single prime element (up to an invertible element) and a single maximal ideal. Of particular interest in this talk is the fact that power series rings are atomic pseudo-valuation domains.
This talk begins by discussing power series rings over fields where the leading coefficient of the power series is restricted to a subfield. We use the divisibility structure of these rings to visualize the entire ideal lattice of such a ring. Then, following P. Ribenboim's approach to generalized power series, we construct generalized series rings with restricted 'constant terms' and classify their groups of divisibility.
The discussion culminates in a complete characterization of atomic pseudo-valuation domains. This characterization is done solely in terms of the divisibility structure on the given series ring.
September 19, 26
Anna Romanowska (ISU/Warsaw University of Technology): Generalized convex sets in space with holes.
Abstract: We will discuss an extension of the concept of convex subsets of real
vector spaces to convex subsets of modules over certain subrings of the ring
of reals. Real convex sets may be described geometrically, in the
traditional way, or algebraically, as so-called barycentric algebras, and
characterized by a number of equivalent conditions. We will discuss
generalizations of both the geometric and the algebraic definitions, and will see
which of the equvalent conditions concerning the real case are preserved by
our generalizations. We will discuss some of the consequences of the new
definitions, in particular those concerning the concept of their algebraic
closure. In the case of generalized geometric convex sets, we obtain a
purely algebraic description of the topological closure.
Jonathan Smith: Sylow theory for quasigroups, II.
Abstract: The previous talk (August 29) examined the extension to quasigroups of that part of group Sylow theory which concerns itself with the existence of Sylow subgroups. The current talk primarily discusses issues related to the conjugacy or number of Sylow subquasigroups within the context of the permutation representation theory of (left) quasigroups. In particular, we complete the analysis, from the standpoint of quasigroup Sylow theory, for the three prime divisors of the order (120) of the smallest non-associative simple Moufang loop, related to the elements of norm 1 in the E8 lattice.
Labor Day: no seminar.
Jonathan Smith: Sylow theory for quasigroups, I.
Abstract: In the last few years, three extensions of group theory have emerged: Hopf algebras (quantum groups), association schemes (table algebras, supercharacters, etc.), and quasigroups (Latin squares, loops, etc.). Among these, the theory of quasigroups is the only one that has seen development of a broad range of group-theoretic topics: a character theory (based on association schemes), linear representations (module theory), and permutation representations. Nevertheless, a Sylow theory applicable to general finite quasigroups has hitherto been lacking, despite some recent developments for Moufang loops.
The aim of the current work is to initiate a study of Sylow theory for general quasigroups. The starting point here is the permutation representation proof of Sylow's theorems for (finite) groups, examining the behavior of subsets of a given size under left multiplication by group elements. The orbit of a given subset is described as non-overlapping if its members are disjoint; otherwise, the orbit is said to be overlapping. Each subgroup lies in a non-overlapping orbit, and each non-overlapping orbit contains a subgroup. For subsets whose order is a prime-power divisor of the group order, non-overlapping orbits are guaranteed to exist (in a number that is congruent to 1 modulo the prime in question). For general finite quasigroups, the much more richly varied behavior of overlapping and non-overlapping orbits on subsets of different sizes is the main concern of the talk.
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