Mathematics

Department Research Overview: 

The Department of Mathematics focuses its research efforts into eight areas: 

• Algebra and Combinatorics- 

I don't know why these are grouped together, they're not very similar. The algebras are sections of mathematics concerned with the operations and structures, like how Linear Algebra is concerned with vector/matrix operations and vector spaces. Combinatorics is concerned with the methods for determining how many arrangements a group of objects can be in, like the total possible scramblings of a Rubik's cube. 

• Control, Optimization, and Modeling- 

These areas are all concerned with communicating information about systems. Control theory is concerned with how systems behave over time. Optimization is finding the optimal solution to a system, given certain constraints. Modeling is very broad, but it boils down to choosing the best way to describe a system, determining on its characteristics (deterministic vs. probabilistic and discrete vs. continuous). 

• Mathematical Biology- 

This area uses mathematical methods to describe biological systems and processes. 

• Numerical Analysis and Scientific Computing-  

These areas are concerned with using substantial computing power and usually a tremendous amount of data to find solutions that are not exact, but are close enough to still be useful. 

• Ordinary Differential Equations, Partial Differential Equations, and Analysis- 

These areas are concerned with continuously varying systems, the difference between ODEs and PDEs lies in how many variables are changing. Analysis is concerned with proving the rules for working with continuous systems using calculus. 

• Probability, Stochastic Processes, and Financial Mathematics- 

This is where the huge overlap between math and stats is most apparent. Stochastic processes stand out, but it's just when a group of variables (or just one variable) seems to act randomly over time, like how the stock market can go up or down any amount at anytime. 

• Symbolic Computation- 

This area is very cool. This is where the theory behind things like wolfram alpha and the CAS in modern calculators is developed. There are also methods for developing entire proofs with computers using tools like Lean, Agda, and Coq. I'd also personally include the entire concept of lambda calculus in this category, but that may be controversial.

• Topology, Geometry, and Mathematical Physics- 

Topology is concerned with properties of shapes that don't change when the shape is bent or stretched. A common joke is that a doughnut (torus) and a coffee mug are the same shape to a topologist, because no cutting is required to go from one to the other. Geometry deals with shapes, their properties, and spaces. Mathematical physics applies these concepts to physical phenomena. 

Research Samples from Area 8: 

I had a chance to speak to Dr. Manion, who has previously published work in the Journal of Knot Theory and its Ramifications. He gave me a few resources on the basics, history, and present work being done in Knot Theory, which I will summarize. 

A Brief Summary of the History of Knot Theory: 

Knot Theory started as a "Theory of Everything", as its originator, Peter Tait, posited that the elements were simply different knots of ether. He made this misguided discovery while playing with a machine he made that made smoke rings out of various questionably safe chemicals (Silver, 2006, p. 160). This was demonstrated to be false, as the existence of ether was debunked by Michelson and Morley. However, in the time between, Peter Tait tabulated quite a few knots in his mission to build his own periodic table. He successfully found all the prime knots for the crossing numbers up to 7. This inspired a great deal of work tabulating more knots, which, in a way, culminated in JW Alexander, with inspiration from the works of Poincare, defining the Alexander polynomial. The Alexander polynomial is a polynomial that relates the number of crossing a knot has to how the knot crosses over itself. Finally, substantially more recently (~1985), Vaughan Jones (who was Dr. Corey Jones' Ph.D advisor) discovered the Jones Polynomial, while studying algebraic operators for statistical mechanics in physics. Jones realized that his work resembled a new knot invariant. From this, Witten published "Quantum field theory and the Jones polynomial" (1989), which paved the way for TQFTs (Topological Quantum Field Theories), like quantum Chern-Simons theory, which is all the rage in high-energy physics to this day. 

Here are some fun Knot Theory images: 

Current Research Related to Knot Theory at NCSU: 

Dr. Andrew Manion has 5 projects in the works concerning Heegard Floer Homology, which deals with defining invariant for structures (like knots and links) using flows on manifolds. See: Preprints. 

Dr. Bojko Bakalov has a project in the works concerning braided vertex algebras, which can also be used to define invariants for structures (like knots and links). See: Preprint.

Work Cited: 

Daniel Silver. "Knot Theory's Odd Origins." American Scientist 94 (2006). 158-165.

PRESENTATION

I looked at interdisciplinary approaches to Knot Theory. I started with a broad overview of the subject (see above), I then looked at research directly involved with or related to Knot Theory done by people at NCSU. This second step made me realize that it would be much more fun to look at different takes on Knot Theory, so I chose two ways of doing that: "broadening the scope" and "applications". A bit of mental gymnastics can be done to say that this is interdisciplinary comparison, so it will be treated as such going forward. 

Broadening the Scope:

Knot theory is inseparable from topology, by definition the knots cannot be cut. There are a collection of "moves" that can be done to any knot (see gif above) called reidemeister moves. I also wanted to find a way to talk about homologies, because there's a neat bread crumb trail that I discovered. Therefore, the "tools" that I'm going to focus on are Algebraic Geometry and Homologies. 

Algebraic Geometry:

Homologies: 

A homology is an operation done to a chain complex, which is a sequence of abelian groups and isomorphisms. The result is a different collection of abelian groups. The homology of interest to us is called the Khovanov homology, because it is considered to be the categorification of the Jones Polynomial. This ties in to the breadcrumb trail, I previously discussed a neat finding with the Jones Polynomial, and then two of the articles I looked at also involved the Khovanov homology, which makes sense now considering the connection between it and a very important invariant. This "tool" is used to related different structures to each other, that's the point of finding homologies. The preprint by Dr. Manion is about the khovanov homology of a special kind of knot. 

Applications: 

One of the articles I looked at was entirely focused on applying the concepts of homologies and cohomologies to geometric mesh modeling using Gmsh. The researchers found that this resulted in faster and more efficient results when it came to vector potential formulations during the modeling or simulation process. There's a really good example for this with more numbers on page 13 of the first article in the annotated bibliography. Note, this is not a use of Knot Theory itself, but rather a use of one of the concepts we talked about while broadening the scope. 

A Problem with Pure Math Research: 

There is a really large problem with modern pure math research. The field is hyper-overspecialized. There's so much preliminary knowledge for so many of these topics. Something can only be diluted so much. There are better entry level resources on topics like the ones that I've covered, they're not scholarly though. I had much higher hopes going into this topic, because I was under the impression that it would be easily adapted for a general audience, because this is technically pop-math. I don't think this is a problem that the COS or NC State can solve all by itself, but more "abridged" overviews of complex topics like this would be very helpful for general audiences.

Works Cited can be located in the Annotated Bibliography linked on the homepage.