| February 27, 2019 | Theme: Mathematics |
Distinguished Professor Vladimir Tonchev
Coding Theory, Combinatorial Designs, and Finite Geometry
TechTalks presented by the Department of Mathematical Sciences:
- Missy Keranen, Mathematical Sciences
- William Keith, Mathematical Sciences
- CK Shene, Computer Sciences
This special iteration of the Research Forum series will be a combination for a Distinguished
Lecture (20 minutes) and several TechTalks presentations (2 Minute. 2 Slides)
Error-correcting codes are used to protect data from random errors in satellite and wireless communication systems, audio and video recording, and data storage. A large class of codes with many practical applications are based on finite geometry. The most notable example of such codes are the famous Reed-Muller codes that are being used in deep space and mobile communications. The subject of this talk is a class of codes based on combinatorial designs. These combinatorial codes posses remarkable error-correction properties, admit efficient decoding, and may provide a viable alternative to some of the Reed-Muller codes.
Six Questions with Professor Tonchev
Coding theory was a hot and relatively new topic at the time when I was a student. As a sophomore, I started attending a weakly Coding Theory Seminar, that lead to the writing of my first research paper as a junior student. I learned about combinatorics and finite geometry as a student by reading the first edition of the book "Combinatorial Theory" by Marshall Hall, Jr., who was a professor at California Institute of Technology at the time. Later, I published some of my early papers on combinatorial designs in the Journal of combinatorial Theory, Series A, whose Editor-in-Chief was Marshal Hall, Jr., the author of my favorite combinatorics book. I was able to construct some combinatorial designs whose existence was unknown according to Marshall Hall's book, and I was elated when one of these designs found its place in the second edition of the book.
In the graduate courses, as well as in some lower level courses that I teach, I often mention related open research problems. There have been some lucky occasions when a couple of weeks after I mentioned an open problem in class, a student would come up with a solution of the problem, and that would result in writing a publishable research paper by the student.
The rapid development of quantum computing that motivates the need of quantum error-correcting codes.
Many important longstanding open problems in combinatorics, coding theory and finite geometry are of exponential complexity, and that makes them very hard or impossible to solve even by the fastest contemporary digital computers. Some of these problems provably could be solved on a quantum computer, at the time when general purpose programmable quantum computers become a reality.
Michigan Tech has been very supportive for my professional activities related to participation and organizing scientific and educational workshops and conferences. Given the geographic location of Houghton, we are lucky that in comparison with many other even larger universities that are located relatively far from the closest airport, we enjoy two daily flights to Chicago.
Develop fast algorithms for quantum error-correction.
Tech Talks -February 27, 2019
Melissa Keranen, Department of Mathematical Sciences, "Combinatorial Designs and Graphs”
Combinatorial designs have their roots in the design of statistical experiments and in recreational math. In this talk, I will give a brief introduction to the ﬁeld of Combinatorial Designs and explain how you may have constructed a design without even knowing it! I will discuss how my research relates designs to graphs, and has applications to communication and coding theory.
William Keith, Department of Mathematical Sciences,"Partitions Across Mathematics"
In how many ways can one divide half a dozen objects: four and two, three two and
one, and so forth? What if the divisions are constrained by desired properties?
These simple questions are the basis of partition theory, a subject that requires
a mix of cutting-edge mathematical techniques and creative intuition. The benefit
of studying such a fundamental object is that the results can inform many areas in
mathematics and other sciences: partitions index energy states in particle systems,
symmetric functions, irreducible representations, and a wide variety of other objects
where a counting is useful to have.
CK Shene, Department of Computer Science, "Visualizing Everything in Computer Science - What We have Done
An old saying: "A picture is worth a thousand words" indicates that seeing is perhaps a better way of understanding. The goal of Visualizing Everything in Computer Science is to investigate ways of showing the unseen nature and inner working of important topics/algorithms in computer science so that the hidden nature of an event could be understood easily. To visualize, something has to be rendered by a computer. This something is of course geometric. Consequently, how to make something into a geometric object so that it can be rendered by a computer is a key question. The study of geometric objects is an important topic in geometry and topology, standard topics in mathematics. Therefore, visualization requires knowledge in computer science and in mathematics. We have received several National Science Foundation grants for the development of many tools and algorithms about visualization. This talk will discuss briefly what we have done in the past 20+ years.