University Seminars: Logic Across Disciplines Archives

Fall 2017


Title:   An intuitive description of toposes, toposes as a description of intuitionism

Speaker:    Tslil Clingman, Johns Hopkins University

Date and Time: Thursday, November 30,  2017, 2:30pm-3:30pm

Place: Rome Hall (801 22nd Street), Room 771

Abstract:   In this talk we will begin by examining, by way of example, the notion of a topos, a category exhibiting certain structural properties satisfied in particular by the familiar category of sets and functions but many strange and wonderful examples besides. We will then observe how the presence of certain structural properties naturally equips a topos with `internal' and `external' models intuitionistic propositional logic. From here we will explore the `Mitchell-Bènabou' language of a topos -- a framework which will allow us to redevelop much of the usual language of set theory internal to any topos. That is, we will see how we may make sense of forming objects of a topos via comprehensions, quantifiers and many other appropriate tools as in the case of "surjections(X,Y) = {f  Y^X | yxX [f(x)=y]}" for the (object of) surjections from X to Y. That objects defined in this manner are in factthe ones we desired depends further on the semantics of internal language, the so termed Kripke-Joyal semantics of a topos. Time allowing we will develop this more fully and explore the relation this bears to the forcing arguments of Cohen.

Title:   Gödel Index Sets of Computable Structures

Speaker:  Valentina Harizanov, GWU

Date and Time: Thursday, November 16 2017, 02:30pm-03:30pm

Place: Rome Hall 771

Abstract:  For a computable structure, we define its index set to be the set of all Gödel codes for computable isomorphic copies. We will show how to calculate precisely the complexity of the index sets for some familiar algebraic structures. We will further discuss the most recent results in this area.

Title:   Classification and measure for algebraic fields

Speaker:    Russell Miller, City University of New York

Date and Time: Friday, November 10, 2017, 03:00pm-04:00pm

Place: Rome Hall 771

Abstract:  The algebraic fields of characteristic 0 are precisely the subfields of the algebraic closure of the rationals, up to isomorphism.  We describe a way to classify them effectively, via a computable homeomorphism onto Cantor space.  This homeomorphism makes it natural to transfer Lebesgue measure from Cantor space onto the class of these fields, although there is another probability measure on the same class, which seems in some ways more natural than Lebesgue measure.  We will discuss how certain properties of these fields – notably, relative computable categoricity – interact with these measures:  the basic result is that only measure-0-many of these fields fail to be relatively computably categorical.  (The work on computable categoricity is joint with Johanna Franklin.)

Title:   Computable Classification Problem

Speaker:    Valentina Harizanov, GWU

Date and Time: Thursday, November 2, 2017, 02:30pm-03:30pm

Place: Rome Hall 771

Abstract:  The Scott Isomorphism Theorem says that for any countable structure M there is a sentence, in countable infinitary language, the countable models of which are exactly the isomorphic copies of M. Here, we consider a computable structure A and define its index set to be the set of all Gödel codes for computable isomorphic copies of AWe will present evidence for the following thesis. To calculate the precise complexity of the index set of A, we need a good description of A, using computable infinitary language, and once we have an optimal description, the exact complexity within a computability-theoretic hierarchy will match that of the description.

Title:   Trees of orderings

Speaker:   Jennifer Chubb, University of San Francisco

Date and Time: Friday, October 27, 2017, 04:15am-05:15pm

Place: Rome Hall 771

Abstract:  An ordering of an algebraic structure with identity can be often identified with the corresponding set of positive elements.  For a given algebraic structure, we can organize the cones of all the admitted orderings on a tree.  When the structure is computable, the tree can be constructed in an effective way.  Topological properties of this space of orderings can provide insight into algorithmic properties of the orderings, and vice versa.  In this talk, we will see how to construct these trees and what they can tell us.

Title:  Orderings of algebraic structures

Speaker:   Jennifer Chubb, University of San Francisco

Date and Time:  Thursday, October 19, 2017, 2:30pm-3:30pm

Place: Rome 771

Abstract:  A left- or bi- partial ordering of an algebraic structure is a partial ordering of the elements of the structure that is invariant under the structure acting on itself on the left or, respectively, both on the left and on the right. I will discuss algorithmic properties of the orderings admitted by a computable structures and their general properties, and describe some open problems.

Logic-Topology Seminar

Title:   Mathathon II: Search for Interesting Torsion in Khovanov Homology

Speaker:    Jozef Przytycki, GWU
Date and Time: Thursday, October 5, 2017, 2:30-3:30pm

Place: Rome Hall (801 22nd Street), Room 771

Abstract:  We will describe the work of our Mathathon group (Sujoy Mukherjee, Marithania Silvero, Zhao Wang, Seung Yeop Yang), Dec. 2016–Jan. 2017, on torsion in Khovanov homology different from Z_2. Khovanov homology, one of the most important constructions at the end of XX century, has been computed for many links. However, computation is NP-hard and we are limited to generic knots of up to 35 crossings with only some families with larger number of crossings. The experimental data suggest that there is abundance of Z_2-torsion but other torsion seems to be rather rear phenomenon. The first Z_4 torsion appears in 15 crossing torus knot T(4,5), and the first Z_3 and Z_5 torsion in the torus knot T(5,6). Generally, calculations by Bar-Nathan, Shumakovitch, and Lewark suggest Z_p^k torsion in the torus knot T(p^k,p^k+1), p^k >3, but this has not yet been proven. We show, with Mathathoners, the existence of Z_n-torsion, n>3, for some infinite family of knots.  The simplest of them is obtained by deforming the torus knot T(5,7) by a t_2k-moves. We also prove the existence of knots with other torsion, the largest being Z_2^23, so the cyclic group of over 8 millions elements. We combine computer calculations (and struggle with NP hardness) with homological algebra technique.

The talk will be elementary and all needed notions will be defined.