Theo Belaire
Supervisor:
Eric Katz
Project title:
Computational Verification of a Conjecture on Bipartite restrictions of the Lattice of Flats annotated by Mobius Functions
Comments:
It was a great experience working with Professor Katz and interacting with all the other Undergraduate Research Assistant (URA) students. I learned a lot about Matroids and the python sage library.
Anyone who is considering going into academia should definitely try this.
Rutger Campbell
Supervisor:
Peter Nelson
Project title:
Linkages in a Directed Graph with Parity Restrictions
Wenbo Gao
Supervisor:
David Wagner
Project title:
The Strong Rayleigh Property in a Class of Symmetric Matroids
Comments:
My project began with trying to show that certain symmetric matroids have the "Strong Rayleigh Property" (that is, a special polynomial associated to the matroid is positive semidefinite) by finding explicit sums-of-squares representations for those polynomials. It quickly became clear that this problem is far more subtle than we had anticipated, and touched upon diverse areas such as symmetric functions, algebraic graph theory, and semidefinite optimization. Using an assortment of techniques, we managed to derive a set of conditions on the matroid which determine when its polynomial has a sum-of-squares certificate. My advisor, Professor David Wagner, was extremely helpful whenever I was stuck and always had a fresh insight to offer. I highly enjoyed this experience and would recommend the URA program to anyone interested in mathematical research.
Sanjith Gopalakrishnan
Home university:
Indian Institute of Technology – Madras
Supervisor:
Bertrand Guenin
Project title:
Minimal Blocking Pair Free Signed Graphs
Comments:
My experience as a URA in the Combinatorics and Optimization (C and O) Department at Waterloo was a very enriching one. I had a great working relationship with my advisor and we were able to make reasonable progress on the research problem at hand. Apart from my advisor, I was also able to gain substantially from the weekly seminar and from my interactions with fellow URAs. It is an excellent summer program for those who are interested in working on mathematical research problems or for those who are just curious to explore what mathematical research is all about.
Sean Hunt
Supervisor:
Chris Godsil
Project title:
Searching for perfect state transfer in quantum walks.
Comments:
This was my first summer working with original math research, and it was a very neat experience. I got to study an area in much more depth that is normally covered in an undergraduate curriculum. It was a very neat exposure to the way that math research works, and all the pitfalls and false starts that you experience. I would definitely recommend a URA to any undergraduate students considering graduate studies or even private-sector research.
Hao (Billy) Lee
Supervisor:
David Jao
Project title:
A New Complexity Reduction Algorithm for Gaussian Normal Basis Multipliers
Comments:
This summer, I had the honour of working with Professor David Jao and his post-doc Reza Azarderakhsh. I learned a lot in the field of cryptography and how to deal with the struggles that came with research. I made many new friends and got to know many professors, staffs and other faculty members. I loved the experience and would definitely recommend it to anyone interested in doing research.
Victor Liu
Supervisor:
Henry Wolkowicz
Project title:
Degeneracy in Cone Optimization and the Geometry of Slater Points
Ritvik Ramkumar
Supervisor:
Kevin Purbhoo
Project title:
Wronskians and Tableaux: "Structure Preserving" embeddings of Quadrics in Grassmanians
Comments:
This summer was my first time being a URA, and it was a wonderful experience. The problem posed to me was to generalize to higher dimensions a bijection established by Professor Purbhoo. In particular a bijection of points in the Grassmanian with a certain Wronskian to "symmetrical" tableaux. But as we tried to do this, we realized that the techniques used in the lower dimensional case weren't applicable in higher dimensions. To fix this, Professor Purbhoo came up with many new ideas on how to approach the problem. Seeing how he comes up with new ideas (the intuition behind it) taught me quite a lot, and implementing them was a lot of fun. Throughout the term I had to read a lot of background material, which made me more knowledgeable, and more importantly exposed me to ideas/results I would not have come across in my Undergraduate Career. More specifically, I learnt about Lie Theory, Algebraic Geometry, Differential Geometry, and how they all related to Tableau Combinatorics (which is quite amazing!). Finally, I would recommend the URA program to anybody who wants to go into Academia, as it is the best way to get a real feel for research and learn a lot from your supervisor.
Hao Sun
Supervisors:
Henry Wolkowicz and Stephen Vavasis
Project:
Eigenvalue and Semidefinite Programming Bounds for Vertex Separators
Comments:
This semester I worked on the graph partitioning problem, that is given a graph we wish to partition it in a way that minimizes the number of edges between different partitions. This has strong applications in chip design, in particular multileveled chips. This also has applications in solving sparse linear systems by rearranging the rows and columns to put entries close to the main diagonal. In this project we studied formulations of this problem that used cool theorems from linear algebra like the Hoffman Wielandt theorem and Sylvester's inertia theorem. It is really enlightening to see the theory learned in linear algebra applied in the real world. It really makes one appreciate how linear algebra is alive when it's outside its own sphere. I really enjoyed learning about the theory in this area especially about the related Quadratic Assignment Problem and Max-cut problem. This was a very good experience to explore research in C and O; I strongly recommend this to any aspiring mathematician.
Ricci Tam
Supervisor:
Eric Katz
Jesse Wang
Supervisor:
Debbie Leung
Project title:
Characteristics of Universal Embezzling Families
Miaolan Xie
Supervisor:
Laura Sanita
Project title:
An Improved Undirected Flow Formulation for Steiner Forest Problem
Comments:
I really enjoyed my time as URA this term. Throughout the four months I had a taste of what doing research is really like and also getting to know myself better. Working with Professor Sanita is an amazing experience. She is a very encouraging and experienced researcher! We were mainly looking at one formulation for the Steiner Forest Problem, proved a conjecture related to it and moved on with some other interesting questions. Laura and I met once a week to discuss progress and possible directions to proceed. I gained much more than I have expected! I would definitely recommend it to anyone who is interested in research and considering applying to graduate school.