We will explore the structure of SOP_2 types (defined by Malliaris and Shelah) and types whose realizations fill cuts in regular ultrapowers of infinite linear orders. We will see that both sorts of types and their distributions can be characterized by certain classes of graphs. This will extend to a new characterization of good ultrafilters. Part 1 will motivate the study of these types in terms of Keisler's order on countable complete first-order theories.

Given a manifold M, Chas and Sullivan construct a Lie bialgebra structure on the homology of the space of (unparametrized) loops using intresections and self-intersections of loops. We give an algebraic description of this structure under Chen's isomorphism identifying loop space homology with cyclic homology. More precisely, we construct a homotopy involutive Lie bialgebra structure on cyclic cochains that depends on the partition function of a Chern-Simons type field theory. Moreover, we discuss the (non-)homotopy invariance of that structure and its relation to the configuration space of two points.

Given a manifold M, Chas and Sullivan construct a Lie bialgebra structure on the homology of the space of (unparametrized) loops using intresections and self-intersections of loops. We give an algebraic description of this structure under Chen's isomorphism identifying loop space homology with cyclic homology. More precisely, we construct a homotopy involutive Lie bialgebra structure on cyclic cochains that depends on the partition function of a Chern-Simons type field theory. Moreover, we discuss the (non-)homotopy invariance of that structure and its relation to the configuration space of two points.

Spinors -- spin 1/2 particles -- come already quantized.  They come in
two species, row spinors and column spinors, which already have the
properties of fermion creation and annihilation operators: you can
multiply a row spinor by a column spinor or vice versa, but you cannot
multiply a row spinor by a row spinor or a column spinor by a column
spinor.  The anticommutation property of spinors can be traced to the
fact that the spinor metric in 4 dimensions is antisymmetric.  This
seminar explores building quantum field theory on a foundation of the
known properties of spinors, rather than on postulated rules.  Spinors
have properties beyond those invoked by qft; for example, the algebra of
outer products of spinors yields (is isomorphic to) the multivector
algebra of strings and branes of all dimensions.  Might spinors point to
a way forward for qft at unification scales?

The Math Club and Diversity Committee are having an event to talk to undergraduates about math research and the various opportunities available to the undergraduate students through CU. Our goal is to widen the pool of applicants to the REU and Experimental Math Lab programs and to create an inclusive and equitable opportunity for our students.

In spring 2020, I am starting a virtual seminar on open conjectures in number theory and arithmetic geometry.  The seminar will provide open access to world class mathematics, with a focus on progress on unsolved problems in NT&AG.  The purpose of the seminar is to provide a viable way for researchers to learn about cutting-edge research in NT&AG without the expense and environmental impact of travel.  Another aim of the seminar is to advance understanding on the most exciting open problems in this field.

Please spread the news!

More information about this seminar is on this webpage:
https://sites.google.com/view/vantageseminar

If you would like to join the e-mail list for this seminar, please go to this google group and apply for membership:
https://groups.google.com/forum/#!forum/vantageseminar

This seminar will be run through blue jeans technology and can support 100 connections.  Please consider reserving a room with a projector or a video conferencing room if there are several people at your institution who are interested in watching.  The talks will typically occur on the first and third Tuesdays at 1 pm ET = 12 CT = 11 MT = 10 am PT.


The first topic of the seminar is CLASS GROUPS OF NUMBER FIELDS.  The first talks of the seminar will focus on this survey paper.

On a conjecture for ?-torsion in class groups of number fields: from the perspective of moments - https://arxiv.org/abs/1902.02008

Jan 21: Lillian Pierce. On some questions in number theory, from the perspective of moments

Feb 4: Melanie Matchett-Wood.  Conjectures for number field counting

Feb18: Caroline Turnage-Butterbaugh. Moments of zeta and the vertical distribution of its zeros

March 3: David Zureick-Brown.  Moduli spaces and arithmetic statistics

If you have any suggestions or advice about this seminar, please send me an email. 

Best wishes, Rachel

Commutative algebra and its related field of algebraic geometry are extremely vast fields full of rich theory and daunting notation. However, if we restrict our attention to the simplest varieties (pun intended) of objects, many concepts and problems become completely combinatorial in nature. In particular, we will explore the so-called Stanley-Reisner duality between monomial ideals in a polynomial ring and simplicial complexes, discuss algebraic characterizations of topological properties and vice versa, and explore connections to graph theory.