## Garnet Akeyr

### Supervisor:

Eric Katz

### Project title:

Using Hurwitz Trees to Study the Lifting Problem

### Comments:

This past summer working with Professor Katz has provided me with a rich and useful experience in conducting research in mathematics. My project incorporated knowledge from the various fields in math that I have come to enjoy through course work, such as algebraic number theory and algebraic geometry. It was challenging, but also rewarding in how I was able to see what kind of problems are being researched in fields I am interested in. Any lingering doubts I had about whether or not I wished to pursue my education further in mathematics have long since dissipated; this past summer was the zenith of my experience at UWaterloo.

## Allan Sapucaia Barboza

### Home university:

Universidade Estadual De Campinas, Brazil

### Supervisor:

Ricardo Fukasawa

### Project title:

Lower Bound Frameworks for the Traveling Salesman Problem

## Rutger Campbell

### Supervisor:

Jim Geelen

### Project title:

Dense Triangle-Fee Binary Geometries

## Da Qi Chen

### Supervisor:

Bertrand Guenin

### Project title:

Capacitated Woodall's Conjecture

### Comments:

Over the course of four months this summer, I worked on a capacitated version of Woodall's Conjecture. To be exact, the conjecture states given a digraph G with edges of weight 0 or 1, if every dicut contains at least two weight-1 edges then there exists two disjoint dijoins of weight-1 edges. Professor Guenin and I also worked on a variation of this conjecture where the digraph G contains both a supersource and a supersink. This experience was extremely enriching and rewarding. It was a great opportunity to meet different profs and learn about their research areas through the weekly seminars, to befriend fellow undergrads who shares similar passions and level of curiosity as you, and most importantly to work firsthand with great mentors and be introduced to the world of research which could very well be your future career.

## Breno Lima de Freitas

### Home university:

University of Toronto

### Supervisor:

Daniel Haven Younger

### Project title:

Extending Snarks

### Comments:

This term I have worked with Graph Theory, a very lovely mathematics field with rich problems. I specifically worked with Integer flows and how one could extend our current knowledge about snarks (cubic graphs which do not admit a 4-flow) to non-cubic graphs. At the end of the term, we could describe an algorithm to decide whether or not a graph has a 4-flow. I have learnt plenty of writing and proving techniques that gave me a broader view of the subject. I would definitely recommend the URA program to people who like learning and improving their mathematical skills.

## Honghao Fu

### Supervisor:

Debbie Leung

### Project title:

Transmission Rate of Quantum Butterfly Network

## Wenbo Gao

### Supervisor:

David G. Wagner

### Project title:

Electrical Networks on Graphs and the Burton-Pemantle Theorem

### Comments:

This summer, we obtained a proof of the Burton-Pemantle Theorem that uses only matrix algebra, and attempted to use it to derive new inequalities with relevance to statistical physics.

Unfortunately, most of our results were counter-examples, showing that identifying general conditions under which these inequalities hold is a difficult problem. But such is the way of research.

I strongly encourage anyone interested in mathematics to try this program, and gain valuable experience with research.

##
**
E**dward
Lee

### Supervisor:

Konstantinos Georgiou

### Project title:

Lift and Project Systems performing on the Partial Vertex Cover Polytope

### Comments:

Over the summer I worked with Professor Costis Georgiou on applying lift and project methods on the standard polyhedron of the partial vertex covering problem. The partial vertex covering problem is a generalization of vertex cover in which only some parametrized number of edges need be covered. Lift and project methods are used to systemically tighten 0-1 polyhedra towards their underlying integral hull. The work was very much interesting; we used a whole range of mathematics in order to derive our result, which was very neat. It was certainly hard work; there were times where I was very much stuck, but Costis was a good supervisor and was helpful in getting me 'unstuck'. I particularly enjoyed working with the other undergraduate research assistants in a shared office; it was very nice to have fellow undergraduates around in order to receive help, give help, and to just discuss mathematics and other topics. Overall, I enjoyed the summer working with Costis, and I would recommend this program to people who wish to continue further studies in mathematics.

## Jason Lin

### Supervisor:

Jochen Koenemann

### Project title:

Multicut on Special Directed Graphs

## Henry Liu

### Supervisor:

Chris Godsil

### Project title:

Perfect State Transfer in Quantum Walks

### Comments:

It is fascinating how your brain continually deceives you into believing you fully understand something, such as a theorem, until you try to use and apply it in non-textbook cases. My research term allowed me to realize and correct this. Armed with other insights on research from my supervisor, and exposed to various new subjects and ideas from the other URAs, I feel significantly more mathematically mature. As a bonus, I obtained and co-authored a paper on some new results on Laplacian perfect state transfer with Gabriel Coutinho, one of Chris's PhDs. I hope anyone who is interested in Combinatorics and Optimization will consider spending a summer as a URA; it is definitely worthwhile and enjoyable.

## Hao Sun

### Supervisor:

Henry Wolkowicz,

### Project title:

Eigenvalue, Quadratic Programming, and Semidefinite Programming Relaxations for a Cut Minimization Problem

### Comments:

We study the problem of partitioning the node set of a graph into k sets of given sizes in order to minimize the cut obtained using (removing) the k -th set. If the resulting cut has value 0, then we have obtained a vertex separator. This problem is closely related to the graph partitioning problem. Indeed, the model we use is the same as that for the graph partitioning problem except for a different quadratic objective function. We look at known and new bounds obtained from various relaxations for this NP-hard problem. This includes: the standard eigenvalue bound, projected eigenvalue bounds using both the adjacency matrix and the Laplacian, Quadratic Programming (QP) bounds based on recent successful QP bounds for the quadratic assignment problems, and semidefinite programming bounds. We include numerical tests for large and huge problems that illustrate the efficiency of the bounds in terms of strength and time.

## Yi-Ting Tsai

### Supervisor:

Alfred Menezes

### Project title:

Decentralized Applications with Ethereum

## Jesse Wang

### Supervisor:

Debbie Leung

### Project title:

Characteristics of Universal Embezzling Families

## Youngho Yoo

### Supervisor:

David Jackson

### Project title:

Enumeration of rooted maps

### Comments:

I've had the pleasure of working with Professor Jackson and learning about his work on enumeration of rooted maps - embeddings of graphs onto surfaces with no edge crossings. In the process, I learned about the theory of representations, characters, group algebras, symmetric functions, and how they can be used together to find the generating series for maps. These series are related to many problems in mathematics and theoretical physics, and it was a delight to see the deep connections between these subjects. This experience has broadened my knowledge and it has shown me a glimpse of the intricacy of higher-level mathematics.