#### What were your key objectives for completing the URSS and have they been met?

My key objectives included my two researcher goals and experience about research through the whole project. My two researcher goals were understanding the connections between random walks and electric network and deriving explicit formula for cover times of random conductance model on a polygon. I accomplished these two goals and learned a lot about how research was done in various aspects.

#### What skills have you gained by completing the URSS project that you will utilise in your studies or other aspects of your working life?

I gained a lot of skills by completing the URSS project.

1. Using Matlab to do symbolic calculations. This improved my ability to use Matlab and there might be similar calculations in future study.

2. Analysing Markov chains in details and applying Doob-h-transform to find certain probabilities. Markov chain is an important topic in my course and Doob-h-transform is also a helpful tool for Markov chains.

3. Understanding important connections between maths and physics. Maths and physics may help each other to solve the problems. This project is a live example that renews my understanding about the interaction between physics and maths.

4. Using Latex and Latex poster template. This is a great tool for typesetting mathematical articles and posters, which will be applied on my fourth year dissertation.

5. The ability to discuss problems with my supervisor clearly. Since my supervisor often asked me to explain the problems and my thoughts before our discussion, it gave me a chance to practise my communication skill. And this is a very fundamental skill.

6. Reading papers to understand what the problem is and what the author tries to do. This gives me a great source to learn from other people, since most of academic work are captured by the papers from the authors.

#### What will you do differently as a result of your URSS experience?

Through my project, I found that research work indeed interests me though there may be unforeseen difficulties along the way. Thus I prefer to do research related work in the future and I feel more confident to do research work.

#### Have there been any unexpected outcomes of taking part in the URSS?

Not many. I did not expect to write such a long report since the template report contained only several pages. But I thought this was indeed a very good practice. My report contained all essential details to understand the content, including propositions, lemmas and theorems. This was what my supervisor required, i.e. self-contain.

#### What would you consider as your highlights of the URSS process?

My supervisor denied one of my findings in the middle of the project. My findings came from my calculations, which were not based on proof. But my finding suggested that two irrelevant terms should be equal. Although I knew that in theory they represented two different concepts, I guessed that they might be equal due to some unknown reasons. So I tried to prove it. However, my supervisor thought that those two terms could not be equal and suggested another method to solve the problem. Instead of learning and applying my supervisor's method directly, I struggled to prove it for a few days. Finally, I found my calculations were correct only in the simple symmetric case and those two terms were only equal in such case. For the general model, which was my target, it could not be generalised. Hence I still needed to use my supervisor's method.

I had another finding which was also based on calculations, similar to the case above. However, this time the two terms were highly related to each other. Even I could see the relationship, but I could not describe them precisely. When I told my supervisor this finding, he responded rapidly and gave a logical deduction.

#### Did you encounter any challenges, issues or difficulties whilst partaking in the URSS? How did you overcome them?

One of the major issuses was described in the above section on highlights.

Some other difficulties appeared at the beginning of the project. We applied Green's function to calculate certain terms. The formula were very complicated and I used Matlab to do the calculation. After calculations, the answer was wrong but the procedures looked correct. We could not find what the problem was. Then I tried another method which seemed to work.

Here are some technical issues

1. Matlab functions

I used some Matlab functions on symbols. It saved me some time. Then under the pressure of a rise in the amount of calculations, I explored the functions further and found more useful functions which saved me more time. However, there was some limitation in the input of the functions, which I did not notice at the first time. Later it gave out some error since I input some too complicated formula.

Therefore, to test functions before using and get to know the limitation of the functions are important.

2. Bugs.

When I did the simulation, there were annoying errors reported. The problem was that the error reported was not the actual error. It was caused by another error. This mislead me and cost much time to understand the problem.

3. Replacing files wrongly.

I worked both on computers on campus and on my own laptop. I replaced some files wrongly, which cost me one night to recover what I had done. So remember to be careful when replacing files.

#### How do you feel about the URSS now that you have completed your project?

The URSS is indeed a great chance for student to know more about research work and experience a different way of learning. I learned a lot through the project and my supervisor helped and encouraged me a lot. The workshops let us know other URSS researchers and their projects. My mentor's comments on my blogs gave me beneficial feedbacks.

I think the promotion of URSS was not very intense, at least in my department. There were only a few notices on the noticeboard, in the workrooms and on the department website. And the difficulty of the application part was to find a supervisor. I felt nervous when I asked an professor to help me to be my supervisor, especially for a professor who never taught me before. I might think whether they would care such a undergraduate project when I looked through their homepage, which were full of their publications. I do feel lucky to find such a nice professor to be my supervisor, who provided me a suitable topic just before the deadline.

Thank you, Dr David Croydon and other people who help me through the URSS.