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Professor Jeremy Brazas

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Professor Jeremy Brazas

My research interests include algebraic topology and general topology. More specifically, I am interested in homotopy groups of locally complicated spaces using topological algebra and generalizations of covering space theory.

What is Topology?

Topology is one of the major areas of modern mathematics. Despite its importance, if you ask a random person on the street what topology is, they may first think you said "topography" and then after realizing that you're describing an area of mathematics will admit to never hearing of it before. The reason for this is probably because topology is largely an abstract subject that is usually only studied by graduate students or undergraduate students taking senior level mathematics or physics courses.

Topology is a beautiful subject that is used by nearly all other areas of mathematics. If you are unfamiliar with topology but are curious about it, below is some content that might give you an idea of what it is about.

Topology is the abstract study of space and shape

A "space" is a set of points and some information that tells you how the points are meant to be related - perhaps telling you how close points are to teach other. Topology is a theoretical field of mathematics, which is the study of spaces and how to tell when two spaces are equivalent or not. 

In classical geometry, you use numerical values like distance, angle, area, volume, etc. as characteristics that describe some property of a figure. A large circle is different than a small circle, because the radii are different. In topology, you loosen the rigidity demanded by classical geometry. You become less specific, imagining that perhaps your objects are made out of some flexible material, and you begin to look at general characteristics of the object - a small circle and large circle are each just circles so they topologically equivalent. Similarly, a square (without interior) and a circle are topologically the same because what really matters is that you have a one-dimensional curve with a single hole in it. 

To a topologist, a donut (solid torus) and a coffee cup are the equivalent because each is a 3-dimensional solid with a single hole (the donut hole and the hole in the mug handle). If you see a topologist struggling at breakfast, this may explain the issue. Kindly remind them their breakfast is in Euclidean space.

Why is a circle are a solid disk different (i.e. non-equivalent) spaces? Easy, one has a hole in it and the other does not. This may seem like a trivial observation but to actually have mathematics which rigorously detects the difference between the two is not so simple. It's important to remember that intuition is not the same thing as mathematics and that sometimes our intuition is wrong. Topology is the area of mathematics which helps us rigorously classify spaces and shapes.

What does a topologist actually do? Think of it this way. When biologists study organisms, they keep track of the observed differences between the organisms. The general differences can be used to place organisms into broad classification groups such as the taxonomic rank: species, genus, family, order, class, phylum, kingdom. If the DNA sequence of an organism could be fully decoded, we could use it to uniquely characterize that individual.

In a similar way, topologists want to know how to classify spaces. We want to understand what kinds of spaces and shapes exist (even abstractly) and how to distinguish between them. By doing this, we're unraveling the limits of the human ability to apply logic and understand the universe. Topologists are like shape-biologists, we answer questions about topological spaces that allow us to distinguish between them and characterize them at various levels of classification from very general characteristics to decoding the DNA of the space (called the homeomorphism type of the space).

Topology is the study of continuity

The idea of continuity is pervasive in mathematics. A relationship (function) is continuous if small differences of input lead to small differences of output. Anywhere that continuity is used, even in applied mathematics, topology is involved in some form or another. It may be hiding in the background, but topology is probably being used. For example, the proof of the Extreme Value Theorem (every continuous function on a closed interval has a maximum and minimum) in Calculus is often glossed over in undergraduate classes. The reason the EVT and its higher dimensional analogues are true in the first place hinges on the topological concepts of compactness and linearly ordered spaces

The objects we study in topology are called topological spaces. Each space can be thought of as its own world, with it's own set of points and data about how close the points are to each other. Some of these worlds may seem quite strange to us. The way in which we study how topological spaces are related is by using the notion of a continuous function. Often, we can think of a continuous function from one space X to another Y as a way of mapping the first into the second in a way that does not violate the closeness rules of the spaces - nearby points in X must be sent to nearby points in Y.

With this in mind, two topological spaces are equivalent if one of the spaces can be continuously deformed (by a bijection) into the other and if the deformation can be inverted continuously as well. If you had a donut made out of playdough you could mold it (without tearing or puncturing the dough) into a coffee cup - I've done it. You could also start with the coffee cup and turn it back into the donut. So the two are equivalent. 

Now imaging smashing your playdough donut into a solid ball. That operation is continuous because you didn't have to tear it at all - just smash it into a ball. On the other hand, try to start with a ball and turn it into a donut. To do this, you must puncture it at some point to get the hole in the middle. When you do this, some points that were very close to each other in the ball are now far apart on opposite sides of the hole. This violates continuity. So it is not possible to continuously turn a ball into a donut - hence the two spaces are different.

A topology is a thing

The word "topology" is used to describe the entire field of study, however, the world topology is also used to define a mathematical thing.

Start with a set of points X. This could be a finite set or an infinite set.

We want to think of X as a space - as it's own universe where the only thing that matters are points in X. But we need more data to know how close points are to each other within this universe.

A topology on X is this data of "closeness". It contains the information that tells us how points are related in X. In particular, a topology T on X is a set of subsets of X - these subsets, elements of the topology T, are called open sets. These open subsets in T need to follow some rules: the intersection of two open sets must be open and an arbitrary union of open sets must be open. For logical purposes, we also insist that the empty set and X itself are open. ,

A topological space is a pair (X,T) consisting of X with some topology T on X.

This set-theoretic definition may seem weird and abstract at first, but after studying topology for a while you may find that it is, in fact, well-motivated.

There may be lots of different topologies on a given space X.You could have two topologies on the same set X and the two resulting spaces may be completely different. The question in topology is then: how can we tell when spaces are the same or different?

What is Algebraic Topology?

Coming soon

Mathematics Research Publications

  • J. Brazas, H. Fischer, Test map characterizations of local properties of fundamental groups. Preprint. 2017. arXiv
  • J. Brazas, Generalized covering space theories, Theory and Appl. of Categories 30 (2015) 1132-1162. Open Access
  • J. Brazas, P. Fabel, On fundamental groups with the quotient topology, J. Homotopy and Related Structures 10 (2015) 71-91. arXiv
  • J. Brazas, P. Fabel, Strongly pseudoradial spaces, Topology Proc. 46 (2015) 255-276. arXiv
  • J. Brazas, P. Fabel, Thick Spanier groups and the first shape group, Rocky Mountain J. Math. 44 (2014) 1415-1444. arXiv
  • J. Brazas, Open subgroups of free topological groups, Fundamenta Mathematicae 226 (2014) 17-40. arXiv
  • J. Brazas, Semicoverings, coverings, overlays and open subgroups of the quasitopological fundamental group, Topology Proc. 44 (2014) 285-313.
  • J. Brazas, The fundamental group as a topological group, Topology Appl. 160 (2013) 170-188. arXiv
  • J. Brazas, Semicoverings: a generalization of covering space theory, Homology Homotopy Appl. 14 (2012) 33-63. Open Access
  • J. Brazas, The topological fundamental group and free topological groups, Topology Appl. 158 (2011) 779-802. arXiv

Miscellaneous Notes

Selected Presentations

  • A characterization of the unique path lifting property for the whisker topology, Joint Meeting of the German and Polish Mathematical Societies (DMV-PTM), Invited talk in Wild Algebraic & Geometric Topology session, Poznan, Poland, September 18, 2014. Slides
  • Topological fundamental groups and open subgroup theorems, Lloyd Roeling Lafayette Mathematics Conference, Invited talk, UL Lafayette, November 9, 2013.
  • Quasitopological fundamental groups and the first shape map, 28th Summer Conference on Topology and Its Applications, Invited talk in Continuum Theory Session, Nippissing University, July 26, 2013. Slides
  • Open subgroups of free topological groups, 47th Spring Topology and Dynamics Conference, Central Connecticut State University, March 24, 2013. Slides

Undergraduate Research

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