Analysis Seminar

Speaker: Jon Handy (Michigan)

"The Corona Problem on Complements of Square Cantor Sets"

Time: 15:00

Room: MC 108

Some 45 years after Carleson proved the corona theorem for simply connected domains, the question for general domains still eludes an answer. In this talk a quick sketch of the historical progress in the problem will be given followed by a discussion of the recent progress in the case of the complements of certain square Cantor sets, where the explicitness of the construction of the set allows greater insight. The main ingredients of the proof in this case are a method of solving the d-bar problem introduced in a paper of P. Jones and J. Garnett and some ideas from rational approximation theory.

Analysis Seminar

Speaker: André Boivin (Western)

"On the proof of Mergelyan's theorem IV"

Time: 15:00

Room: MC 108

Analysis Seminar

Speaker: Adam Coffman (Fort Wayne)

"CR Singularities of 4-manifolds in C³"

Time: 14:30

Room: MC 109a

CR singularities of real 4 -submanifolds in {mathbb C}³ - points where the tangent space is a complex hyperplane - are classified by using holomorphic coordinate changes to transform the quadratic coefficients of the real analytic local defining equations into one of a list of normal forms. The quadratic coefficients determine an intersection index, which appears in global enumerative formulas for CR singularities. The geometry, both locally and globally, is a natural generalization of the well-known case of surfaces in {mathbb C}2 and Bishop's elliptic/hyperbolic classification of CR singular points.

Distinguished Lecture

Speaker: Florian Pop (Penn)

"New developments in anabelian geometry I"

Time: 16:00

Room: MC 107

At the beginning of the 1980's, Grothendieck announced his new ideas concerning an "anabelian geometry". His fundamental observation was that algebraic fundamental groups of "complicated" varieties tend to become very rigid objects when endowed with the Galois action of fields of definition of those varieties, and morphisms between such "rigidified objects" tend to originate from geometry and arithmetic. Conjecturally, this would enable us with a completely new tool to attack problems in arithmetic/algebraic geometry, namely by translating them into questions about the category of (etale) fundamental groups via the (etale) fundamental group functor. There are several aspects of these problems, like recovering varieties and points of these from the corresponding fundamental group setting, giving new non-tautological descriptions of absolute Galois groups in purely topological/combinatorial terms, describing the representations of absolute Galois groups, etc. Grothendieck's Esquisse d'un Programme sparked a very intensive research activity. Meanwhile some of the questions have quite satisfactory answers, but some of the most fundamental ones are not answered yet. After an introduction to anabelian geometry, I plan to present some new results, concerning: the birational aspects of the theory, the anabelian geometry of curves, Grothendieck's section conjecture; and to touch upon the problem of describing absolute Galois groups in purely topological/combinatorial terms (thus answering the Ihara Question / Oda-Matsumoto Conjecture).

Distinguished Lecture

Speaker: Florian Pop (Penn)

"New developments in anabelian geometry II""

Time: 15:00

Room: MC 107

At the beginning of the 1980's, Grothendieck announced his new ideas concerning an "anabelian geometry". His fundamental observation was that algebraic fundamental groups of "complicated" varieties tend to become very rigid objects when endowed with the Galois action of fields of definition of those varieties, and morphisms between such "rigidified objects" tend to originate from geometry and arithmetic. Conjecturally, this would enable us with a completely new tool to attack problems in arithmetic/algebraic geometry, namely by translating them into questions about the category of (etale) fundamental groups via the (etale) fundamental group functor. There are several aspects of these problems, like recovering varieties and points of these from the corresponding fundamental group setting, giving new non-tautological descriptions of absolute Galois groups in purely topological/combinatorial terms, describing the representations of absolute Galois groups, etc. Grothendieck's Esquisse d'un Programme sparked a very intensive research activity. Meanwhile some of the questions have quite satisfactory answers, but some of the most fundamental ones are not answered yet. After an introduction to anabelian geometry, I plan to present some new results, concerning: the birational aspects of the theory, the anabelian geometry of curves, Grothendieck's section conjecture; and to touch upon the problem of describing absolute Galois groups in purely topological/combinatorial terms (thus answering the Ihara Question / Oda-Matsumoto Conjecture).

Distinguished Lecture

Speaker: Florian Pop (Penn)

"New developments in anabelian geometry III"

Time: 15:00

Room: MC 107

At the beginning of the 1980's, Grothendieck announced his new ideas concerning an "anabelian geometry". His fundamental observation was that algebraic fundamental groups of "complicated" varieties tend to become very rigid objects when endowed with the Galois action of fields of definition of those varieties, and morphisms between such "rigidified objects" tend to originate from geometry and arithmetic. Conjecturally, this would enable us with a completely new tool to attack problems in arithmetic/algebraic geometry, namely by translating them into questions about the category of (etale) fundamental groups via the (etale) fundamental group functor. There are several aspects of these problems, like recovering varieties and points of these from the corresponding fundamental group setting, giving new non-tautological descriptions of absolute Galois groups in purely topological/combinatorial terms, describing the representations of absolute Galois groups, etc. Grothendieck's Esquisse d'un Programme sparked a very intensive research activity. Meanwhile some of the questions have quite satisfactory answers, but some of the most fundamental ones are not answered yet. After an introduction to anabelian geometry, I plan to present some new results, concerning: the birational aspects of the theory, the anabelian geometry of curves, Grothendieck's section conjecture; and to touch upon the problem of describing absolute Galois groups in purely topological/combinatorial terms (thus answering the Ihara Question / Oda-Matsumoto Conjecture).