Department of Mathematics

The Graduate Center of The City University of New York

THE TALKS WILL ALL BE DONE ON ZOOM THIS SEMESTER.

THIS WEEKS ZOOM INFORMATION:

Topic: New York City Category Theory Seminar

Time: Wednesdays 07:00 PM Eastern Time (US and Canada)

Join Zoom Meeting

https://us02web.zoom.us/j/81139396554

Meeting ID: 811 3939 6554

Passcode: NYCCTS

Usually our talks take place at

365 Fifth Avenue (at 34th Street) map

(Diagonally across from the Empire State Building)

New York, NY 10016-4309

Wednesdays 7:00 - 8:30 PM.

Usually the talks are in

Room 6417 .

Some of the talks are videoed and available here.

Contact N. Yanofsky to schedule a speaker

or to add a name to the seminar mailing list.

Expanding the domain of definition to extended pseudo metric spaces enables the construction of a realization functor on diagrams of spaces, which has a right adjoint Y |--> S(Y), called the singular functor. The realization of the Vietoris-Rips system V(X) for an ep-metric space X is the space itself. The counit of the adjunction defines a map \eta: V(X) --> S(X), which is a sectionwise weak equivalence - the proof uses simplicial approximation techniques.

This is the context for the Healy-McInnes UMAP construction, which will be discussed if time permits. UMAP is non-traditional: clusters for UMAP are defined by paths through sequences of neighbour pairs, which can be a highly efficient process in practice.

Joint work with Pieter Hofstra.

Using extensive unpublished materials available via Vladimir Voevodsky’s memorial webpage (https://www.math.ias.edu/Voevodsky/) I reconstruct Vladimir’s vision of mathematics and its role in science incuding his original strategy of bridging the gap between the pure and applied mathematics. Finally, I show a relevance of Univalent Foundations to Vladimir’s unachieved project and speculate about a possible role of Univalent Foundations in science.

'discrete mathematics : algebra :: continuous mathematics : coalgebra'

The idea is that objects in continuous math, like real numbers, are often understood via their approximations, and coalgebra gives tools for understanding and working with those objects. Some examples of this are Pavlovic and Escardo's relation of ordinary differential equations with coinduction, and also Freyd's formulation of the unit interval as a final coalgebra. My talk will be an organized survey of several results in this area, including (1) a new proof of Freyd's Theorem, with extensions to fractal sets; (2) other presentations of sets of reals as corecursive algebras and final coalgebras; (3) a coinductive proof of the correctness of policy iteration from Markov decision processes; and (4) final coalgebra presentations of universal Harsanyi type spaces from economics.This talk reports on joint work with several groups in the past 5-10 years, and also some ongoing work.

References:

Paper (and references therein)

Paper (original paper of Baez, Fritz, and Leinster)

[1] J. Funk, P. Hofstra, B. Steinberg. Isotropy and crossed toposes. Theory and Applications of Categories 26, 660-709, 2012.

[2] P. Hofstra, J. Parker, P.J. Scott. Isotropy of algebraic theories. Electronic Notes in Theoretical Computer Science 341, 201-217, 2018.

Far from being 'generalisation for generalisation's sake', the original motivation came from theoretical & practical computer science. The mathematics of card shuffles is commonly used to describe processing in multi-threaded computations. Moving to the infinite case gives a language in which one may talk about potentially non-terminating processes, or servers with an unbounded number of clients, etc.

However, this talk is entirely about algebra & category theory -- just as in the finite case, the mathematics is of interest in its own right, and should be studied as such.

We model shuffles using operads. The intuition behind them of allowing for arbitrary n-ary operations that compose in a hierarchical manner makes them a natural, inevitable choice for describing such processes such as merging multiple packs of cards.

We use very concrete examples, based on endomorphism operads in groupoids of arithmetic operations. The resulting structures are at the same time both simple (i.e. elementary arithmetic operations), and related to deep structures in mathematics and category theory (topologies, tensors, coherence, associahedra, etc.)

We treat this as a feature, not a bug, and use it to describe complex structures in elementary terms. We also aim to give previously unobserved connections between distinct areas of mathematics.

Working with finiteness spaces forces the number of summands in certain calculations to be finite and thus avoid convergence questions. An excellent example of this is how Ribenboim’s theory of generalized power series rings can be naturally interpreted by assigning finiteness monoid structure to his partially ordered monoids. After Ehrhard’s linearization construction is applied, the resulting structures are the rings of Ribenboim’s construction.

There are several possible choices of morphism between finiteness spaces. If one takes structure-preserving partial functions, the resulting category is complete, cocomplete and symmetric monoidal closed. Using partial functions, we are able to model topological groupoids, when we consider composition as a partial function. We can associate to any hemicompact etale Hausdorff groupoid a complete convolution ring. This is in particular the case for the infinite paths groupoid associated to any countable row-finite directed graph.

Globularly generated double categories are minimal solutions to lifting problems of bicategories into double categories along given categories of vertical arrows. Globularly generated double categories form a coreflective sub-2-category of general double categories. This, together with an analysis of the internal structure of globularly generated double categories yields a numerical invariant on general double categories. We call this invariant the vertical length. The vertical length of a double category C measures the complexity of mixed compositions of globular and horizontal identity squares of C and thus provides a measure of complexity for lifting problems on the horizontal bicategory HC of C. I will explain recent results on the theory of globularly generated double categories and the vertical length invariant. The ultimate goal of the talk is to present conjectures on the vertical length of framed bicategories and possible applications.

Previous Semesters:

##### Fall 2019 - Spring 2020 Categorical Logic Reading Group

##### Fall 2018 - Spring 2019 Topos Theory Reading Group

##### Spring 2017 - Spring 2018 Topos Theory Reading Group

##### Fall 2015 - Fall 2016 Topos Theory Reading Group

##### Fall 2013 - Spring 2015 Homotopy Type Theory Reading Group

##### Spring 2012

##### Fall 2011

##### Spring 2011

##### Fall 2010

##### Spring 2010

##### Fall 2009

##### Spring 2009

Other Category Theory Seminars: