An iterated tower is a sequence of number fields that is generated by the iteration of a rational map defined over the base field. These dynamically generated number fields have received a growing attention over the past decade for a variety of reasons; in this talk I will discuss how these towers could lead to the first construction of infinite, yet finitely and tamely ramified, extensions. One way to ensure a successful construction is to find a tower in which the discriminants of the number fields does not grow too fast. I will demonstrate a method for how one might seek to study the growth of these number fields using the Chebyshev polynomials as an example. The iterated extensions arising from the Chebyshev polynomials are interesting in their own right.
Arithmetic properties of iterated towers
Sep. 09, 2014 12:10pm (MATH …
Kempner
Mati Rubin (Ben Gurion University)
X
A locally moving (LM) group for a regular topological space , is a group of homeomorphisms of which has the following property: For every nonempty open set , there is such that: (1) is not the identity function, and (2) for every , .
A group is a locally moving (LM) group, if it is isomorphic to a group as above. Many groups of automorphisms of topological spaces, linear orderings, trees, Boolean algebras, measure algebras and some other types of structures are LM.
Examples: (1) The group of all auto-homeomorphisms of a Euclidean space is LM. (2) The group of all homeomorphisms of a Euclidean space is LM. (3) The group of all automorphisms of the binary tree is LM.
Using a certain theorem about LM groups, one can prove theorems of the following type: If and are topological spaces such that (= the group of all auto-homeomorphisms of ) is isomorphic to , then is homeomorphic to . Similar theorems for linear orderings, trees, Boolean algebras, measure algebras and some other types of structures are also true.
A recent theorem states that: The first order theory of every locally moving group is undecidable. (This solves in much more generality a question of Mark Sapir about the R. Thompson groups). I shall state and explain that "certain theorem" mentioned above. Then I'll speak about consequences of that theorem including the undecidability theorem.
If time allows, I shall mention some open problems.
Given a parabolic subgroup of a Coxeter group, we may hope to describe a set of nice coset representatives for this subgroup. Deodhar's Lemma shows us how to directly compute a set of coset representatives with the property that every prefix of a coset representative is also on the list. Time permitting, we'll discuss an application of Deodhar's Lemma to the 56-dimensional minuscule module for the simple Lie algebra of type E7. No previous knowledge of Coxeter groups will be assumed.
Deodhar's Lemma and an Application to Representation Theory
Sep. 09, 2014 3pm (Math 350)
Algebraic Geometry
Elizabeth Gillaspy (University of Colorado)
X
The goal of this talk is to explain my thesis research and why we care. In the first part of the talk, I will define the objects in the title and explain how the K-theory of groupoid C*-algebras can tell us about other mathematical objects, from dynamical systems to string theory. My hope is that this will be a gentle introduction to the topic(s) at hand; no prior familiarity with groupoids or with C*-algebras will be assumed. In the second part of the talk, I'll discuss the particular question I've investigated about the K-theory of twisted groupoid C*-algebras, why I chose it, and the progress that I've made so far. Time permitting, I will sketch some of the proofs, so this part of the talk will be more technical than the first part.
Arithmetic properties of iterated towers