Speaker: Chris Kapulkin, University of Western Ontario
Mini-Course Title: Discrete homotopy theory
Time:Friday, Nov 10th, 9:30am
Monday, Nov 13th, 9:30am
Wednesday, Nov 15th, 9:30am
Abstract: Discrete homotopy theory, introduced around 20 years ago by H.
Barcelo and collaborators building on the work of R. Atkin from the mid seventies,
is a homotopy theory of (simple) graphs. As such, it applies techniques previously
employed in the "continuous" context to study discrete objects. It has foundapplications both within and outside mathematics, including: matroid theory,
hyperplane arrangements, topological data analysis, and time series analysis.
The first part of this mini-course is an introduction to discrete homotopy theory.
We define the main concepts, including homotopy of graph maps and homotopy
groups of a pointed graph. We also briefly discuss some of the applications of thefield.
The second part discusses the recently introduced "cubical setting" for discrete
homotopy theory, which brings a whole host of new tools into the field, including
the topological realization theorem (formerly a conjecture of E. Babson, H.
Barcelo, R. Laubenbacher, and M. de Longueville from 2006), the long exact
sequence of a fibration, and the Hurewicz theorem. It is based on joint work withD. Carranza (arXiv:2202.03516).
The third part zooms in on the notion of the fundamental group and thefundamental groupoid of a graph. Several computational tools are discussed,including the Seifert-van Kampen theorem and the theory of covering graphs. It isbased on joint work with U. Mavinkurve (arXiv:2303.06029).
Speaker: Teresa Hoekstra Mendoza, CIMAT
Title: Configuration spaces of graphs
Time: Monday, Nov 13th, 11:00am
Abstract: Given any topological space, in particular a graph G, we can define its(ordered) configuration space on n points as follows. Consider the product
GxGx...xG n times and remove every tuple in which two or more entries are thesame. In this talk I shall give a few examples of configuration spaces of graphs on
two points, and a description of the homotopy type of the 1-skeleton ofconfiguration spaces of trees on n points. I will also mention no k-equal spaces,
which are like configuration spaces but instead of removing all tuples in which two

or more entries are the same, we only remove tuples in which k or more entries
are the same for k<n.
Speaker: Julian Candela, CIMAT
Title: Central Limit Theorems for Large Random Simplicial Complexes
Time: Monday, Nov 13th, 12pm
Abstract: Stochastic topology is the study of the topological properties ofrandomly formed spaces. In this area, we can find several articles dedicated tothe study of homology for different models of random simplicial complexes. Oneof the most important models was introduced by Costa and Faber in their article"Large random simplicial complexes" where they described and studied a modelof random simplicial complexes that are random in every possible dimension.
For this presentation, we use a normal approximation theorem by Stein's method
to prove central limit theorems for the random simplicial complexes model byCosta and Farber.
Speaker: Daniel Carranza, Johns Hopkins University
Title: Hurewicz Theorem in Discrete Homotopy Theory
Time: Tuesday, Nov 14th, 10:00amAbstract: Homotopy groups, while easy to define, are notoriously hard tocompute (both for spaces and graphs). By contrast, homology groups are much
more computable, despite their more complicated definition. One would like a
way of leveraging homological techniques to compute homotopy groups, and inclassical algebraic topology, this is achieved by means of the Hurewicz theorem.
This theorem states that for a connected space, the first non-trivial homotopy andhomology groups agree (up to abelianization). I will present a discrete analogueof this theorem, and show how it can be used to compute previously-unknown
higher homotopy groups of graphs. This talk is based on joint work with Chris
Kapulkin (arXiv:2202.03516).
Speaker: Maru Sarazola, University of Minnesota
Title: A homotopical framework for path homology of directed graphs
Time: Tuesday, Nov 14th, 11:00am
Abstract: Traditionally, the way we compare mathematical structures is by using
the notion of equality, or even of isomorphism. However, there are many settings
where this is no longer the natural notion of "sameness". A notable example

occurs when our objects admit a homology construction: then, we want twoobjects to be "the same" if they have identical homology.
Homotopy theory provides the framework required to work in these settings. In
this talk, we will describe an algebraic invariant of directed graphs called "pathhomology", and introduce a new homotopical framework that encodes "samenessup to path homology". We will also show how this new framework allows us tomake homology-invariant constructions.
Speaker: Laura Scull, Fort Lewis College
Title: X-homotopy for graphs and Z/2-equivariance
Time: Tuesday, Nov 15th, 12:pm
Abstract: X-homotopy for graphs is defined using the categorical product ofgraphs. In this context, our graphs are rigid and edges cannot be collapsed. Thus
there is a sense in which graphs in this theory are built out of edges, instead ofvertices. The 'flip' which reverses direction of an edge introduces a Z/2-action inthis wordview, and X-homotopy phenomena such as parity can be thenunderstood as Z/2-equviariant features.
In this talk, I will focus on the X-fundamental group. I will introduce thisfundamental group in a classical way, as homotopy classes of paths in our graph,and give examples (and pictures). Then I will show how taking an edge-centric
approach allows us to define a fundamental group as a Z/2-equivariant
fundamental group of paths of edges. I will motivate the equivariant theory andthen explain this second definition. These two seemingly different definitions giveequivalent groups, providing a way of seeing the parity showing up in the
traditional definition as coming from a Z/2-action.
Speaker: Gregory Lupton, Cleveland State University
Title: A Second Homotopy Group in Digital Topology
Time: Wednesday, Nov 15th, 11:00am
Abstract: (Joint work with Musin, Scoville, Staecker and Trevino)
Digital topology refers to the use of notions and methods from (algebraic)
topology to study digital images. A digital image in our sense is an idealization ofan actual binary digital image, which consists of pixels in the plane, or higher
dimensional analogues of such. One aim of digital topology is to provide usefultheoretical background for certain steps of image processing, such as feature
extraction or recognition.

An extensive literature on digital topology includes several treatments of thefundamental group. I will report on our recent work in which we develop a notionof a second (higher) homotopy group and calculate its value to be Z for a digital
image that may reasonably be interpreted as a digital 2-sphere. This calculation
involves some interesting combinatorial ingredients. Our development andcalculation may equally well be applied in the (graph-theoretic) settings oftolerance spaces and X-homotopy theory. The definitions readily extend to higher
homotopy groups of any dimension.
I will also speculate briefly about possible applications of our work.