Spring 2018

**CUNY Graduate Center**

**Room 6417**

**Fridays 2:00-3:30pm**

**Organized by:**

**Joel David Hamkins**

**Roman Kossak**

**Russell Miller**

**Philipp Rothmaler**

**Hans Schoutens**

**February 2**

**No seminar** because of MathFest 2018 conference at the Graduate Center.

**February 9**

**Andrew Brooke-Taylor**, University of Leeds

**Products of CW complexes**

CW complexes are the topological spaces of choice for algebraic topology, but the product (as topological spaces) of two CW complexes need not be a CW complex. In the 1940s and 50s Whitehead and Milnor gave sufficient conditions for the product to be a CW complex, and in the 70s and 80s Liu and Tanaka gave characterisations of those pairs of CW complexes whose product is a CW complex, under the assumption of set-theoretic axioms such as CH. In this talk I will present a new characterisation of the pairs of CW complexes whose product is a CW complex, valid in any model of set theory (ie, without any such extra set-theoretic assumptions). Whilst I have stripped the set theory away from the assumptions on the universe, the characterisation is with reference to a cardinal that may not be familiar to non-set theorists: the bounding number.

**February 16**

**Andrei Morozov**, Sobolev Institute of Mathematics

**Infinite time Blum-Shub-Smale machines: a computability for analysis**

We study a concept of computability over the reals based on Blum-Shub-Smale machines working in infinite time (ITBM). We give some characterizations of this computability, prove some its properties, and discuss its adequacy to classical analysis. This is joint work with Peter Koepke.

**February 23**

**Dimitris Tsementzis**, Rutgers University

**Univalent Foundations and Set Theory**

The Univalent Foundations is a proposed foundation for mathematics that takes as primitive a notion of space (rather than a notion of set). I will introduce the basic concepts of the Univalent Foundations from first principles, and give an overview of a class of formal systems, called Homotopy Type Theories, that can be used to formalize these basic notions. I will then describe how to construct a model of set theory in the Univalent Foundations, explain the consequences of the construction, and discuss several open problems. In the process, I will demonstrate the implementation of everything I talk about in a proof assistant.

**March 2**

**Hans Schoutens**, CUNY

**Building non-standard polynomials from models of PA**

Model-theorists, when they see a class of well-known, ‘standard’ structures, want to know what the ‘non-standard’ models look like, that is to say, those structures that have the same first-order theory as the standard structures–already a challenge might be to find an appropriate language! I am interested in non-standard polynomial rings over a field in a finite number of variables–in the language of rings with finitely many constants for those variables–, and I lectured on this topic last Fall, showing among other things, that the multivariate case can be reduced to the one variable case. Moreover, the set of powers of this variable is (an interpretation of) a model of PA, i.e., a non-standard model of the natural numbers. In this talk, I want to explore the converse: given a model of PA and a field in which it embeds, can we build a non-standard polynomial ring (in one variable) interpreting the given model of PA (in the above described manner). At present, I only now how to do this if the model of PA is stably embedded inside the field (note that the field will in general not be ordered), but I do not know how strong an assumption this actually is. I will review some of the theory of non-standard models from last time, and then explain how to construct such a model from a model of PA in three steps: closing under the ring operations, then under composition, and finally using de l’Hopital’s formula!

**March 9**

**Jan Trlifaj**, Charles University

**Faith’s Problem on R-Projectivity Is Independent of ZFC**

In Algebra II, Carl Faith asked for a characterization of the rings over which the Dual Baer Criterion holds. Such rings were called right testing. Sandomierski proved that each right perfect ring is right testing. Puninski et al. have recently proved for a number of non-right perfect rings that they are not right testing, and noted the consistency with ZFC of the statement `each right testing ring is right perfect’ (the proof used Shelah’s uniformization).

We will present these results together with a new one, showing that the existence of a right testing, but not right perfect, ring is also consistent with ZFC (the proof uses Jensen-functions). Thus the answer to Faith’s question above is independent of ZFC.

**March 16**

**Seth Harris**, Drew University

**On-Line Algorithms and Reverse Mathematics**

Consider a two-player game (played by Alice and Bob) in which Alice asks a sequence (a) and Bob responds with a sequence (b) with no knowledge of Alice’s future requests. A problem P is solvable by an on-line algorithm if Bob has a winning strategy in this game, where Bob wins the game if (\bar{a}, \bar{b}) constitutes a solution to P. For example, if we take P to be a graph coloring problem, Alice plays by adding a new vertex and edges connecting it to previous vertices; Bob chooses a color for that vertex. The graph is on-line colorable if Bob has a winning strategy in this game.

Given a problem P, the corresponding sequential problem SeqP asserts the existence of an infinite sequence of solutions to P.

We will show that the reverse-mathematical strength of SeqP is directly related to the on-line solvability of P, and we will exactly characterize which sequential problems are solvable in RCA_0, WKL_0, and ACA_0. This is joint work with François Dorais.

**March 23**

**Joel David Hamkins**, CUNY

**Nonamalgamation in the generic multiverse**

Consider a countable model of set theory $M$ in the context of all its successive forcing extensions and grounds. This generic multiverse has long been known to exhibit instances of nonamalgamation: one can have two extensions $M[c]$ and $M[d]$, both adding merely a generic Cohen real, which have no further extension in common. In this talk, I shall describe new joint work that illuminates the extent of non-amalgamation: every finite partial order (and more) embeds into the generic multiverse over any given model in a way that preserves amalgamability and non-amalgamability. The proof uses the set-theoretic blockchain argument, which has affinities with constructions in computability theory in the Turing degrees. Other arguments, which also resemble counterparts in computability theory, show that the generic multiverse exhibits the exact pair phenonemon for increasing chains. This is joint work with Miha Habič, myself, Lukas Daniel Klausner and Jonathan Verner.

**March 30**

**No seminar** because of spring break

**April 6**

**No seminar** because of spring break

**April 13**

**Henry Towsner**, University of Pennsylvania

**TBA**

**April 20**

**Dave Marker**, University of Illinois-Chicago

**TBA**

**April 27**

** Start time: 12:30pm**

**Hirotaka Kikyo**, Kobe University

**TBA**

**May 11**

**Jan Reimann**, Penn State University

**TBA**