Seminar in Probability
Random walks and uniform spanning trees
Course 18.604, Fall 2025
Professor: Jacopo Borga (jborga@mit.edu)
Office hours: I am available to help you with your presentations (more details below) – email me about it! For questions about exercises in the homework please ask our course assistant Alexis Zhou (leqizhou@mit.edu) during her office hours or send an email (more info below).
Communication specialists: Susan Ruff (ruff.susan@gmail.com)
Course assistant: Alexis Zhou (leqizhou@mit.edu)
Office hours: The office hours for each pset have been scheduled, 1 or 2 days before the pset deadlines:
10/09 Thu 6:30-7:30pm 2-136
10/29 Wed 6:30-7:30pm 2-136
11/20 Thu 4-5pm 2-136
You can also find these times in the course calendar. If you cannot come to office hours, feel free to email me (leqizhou@mit.edu) with any pset questions! Please don’t email me at the last minute though.
Class Schedule: MW 11.00 -12.30 (2-146) & Calendar. Attendance and active participation in all classes are mandatory and will be enforced. If you are unable to attend a class, please email me in advance with a clear explanation of the reason for your absence.
There is no final exam!
Prerequisites: I expect that you are very familiar with all the material from 18.600 (Probability and Random Variables). In particular, you should review the following three important topics/concepts:
1. Conditional expectation [slides]
1. (Discrete time and space) Markov chains [slides]
3. (Discrete time and space) Martingales [slides]
I will also devote the first few lectures to review some preliminary material from 18.600, but it is very important that you first review the above material independently.
Course Goals and Description: In a seminar, mathematicians help each other to learn a body of advanced material. The main object of this undergraduate seminar is for you to help each other learn the content presented in this lecture notes on Random walks and uniform spanning trees (which are based on that of Lyons and Peres’s excellent book Probability on Trees and Networks), while simultaneously helping each other to better present, discuss, write, and read mathematics. To this end, you will take turns collaboratively presenting sections of the material to each other.
Your final (20-minute) presentation will be on a topic of your choice, about which you will also write a 5-7 page paper.
You will present to your classmates 4 times during the course of the semester, following this calendar:
• Three 30-minute (15min each) collaborative presentations given collaboratively with another classmate, each on an assigned section from the notes. The first presentation will be a black board presentation, the second one a slides presentation and the third one a Zoom presentation.
• One 20-minute final presentation given by yourself on the topic of your written paper. You are free to chose what type of presentation you want to use (black board/slides/Zoom).
During the semester you will also have a reading assignment and 3 homework assignments.
Collaborative presentations: For the collaborative presentations, you and your classmate must present approximately equally. It is fine for one of you to give the first half of the talk and the other to give the second half, but you should work closely together to ensure that all parts of the talk are presented clearly and work well together.
For the first presentation, I strongly encourage you and your collaborators to arrange a practice with me. You are expected to practice all subsequent presentations with each other, but you can ask me for (mathematical) advice. Susan Ruff (ruff.susan@gmail.com) is also available for brief Zoom meetings to review parts of your presentation material and provide constructive feedback.
Each 30-minute presentation will be based on a preassigned section of the lecture notes (in the “Files section” you will find a version of the lecture notes where it is clearly indicated what section you should cover during your talk). You should assume that classmates have read and thought about the text: design the talk to help your classmates with the most challenging aspects of the text.
During each presentation, you will be handed a comment form on which you will give feedback to your classmates presenting (more details are given below).
Before Tuesday, September 9th, each student must send me an email (jborga@mit.edu) containing the following information:
- Your availability on your preassigned day for the rehearsal (as indicated in the calendar). Please indicate all the time slots during which you are available, and try to be as flexible as possible.
- Confirmation of your ability to present on the three assigned dates (again, as indicated in the calendar). If one of the dates does not work for you, clearly explain the reason.
Written paper and Final presentation: You will write a paper on a topic of your choice, to be completed in stages. You will propose a topic or choose one from the list that is provided at the end of this version of the notes. You will have various deadlines (see the dates in the calendar) for the topic, an abstract and plan, and full drafts for feedback from me and other students.
The paper need not contain original results, but the writing must be your own and all sources must be properly acknowledged (ask if you have questions). The paper must be successfully written in the style of a research or expository journal article and must be about 5/7 pages long (using the templated that I will provide in the Files section).
You will give a 20-minute final presentation given by yourself on the topic of your written paper (following the schedule on the calendar).
Before TBA, each student must send me an email (jborga@mit.edu) containing the following information:
- A ranking with your favorite topics for the paper and final presentation. You are allowed to propose topics that are not part of the list at the lecture notes.
Example: My ranking is 1st-topic 3, 2nd-topic 5, 3rd-topic 7, 4th-topic 1, 5th-topic 2, 6th-topic 13, 7th-topic 15, 8th-topic 6, ….
If possible my preference would be to present of this alternative topic: “Kirchhoff’s theorem: how do we count the number of spanning trees of a graph?”
Your ranking should include at least 8 topics.
Reading assignment: We will provide a well-written paper on a topic related to our seminar. You will be asked to read it and answer a few questions designed to help you understand what constitutes “effective writing.” This is a self-directed assignment, meaning we will not provide feedback on your work. However, completing the assignment is one of the criteria for your final grade (see below for more details).
Homework: There will be 3+1 homework assignments during the class. About the first three assignments: Each one will cover a different section of the lecture notes. These assignments are designed primarily to help you assess your understanding of the material. Please note:
-Your homework will be graded by Alexis Zhou.
-Solutions to the exercises will be provided after the submission deadline.
The last homework assignment is for students to check their understanding of the strategies taught in the revision workshop (November 17th). Susan will comment on your work, but not grade the responses to that homework.
AI Usage Policy: Students are permitted to use AI tools as a resource to aid their learning, brainstorming, and problem-solving. However, AI-generated content should not be copied and submitted as original work. Instead, students should critically engage with AI outputs, refine them, and ensure they fully understand the material. Misuse of AI, including direct copying without proper attribution or comprehension, may be considered academic dishonesty.
Feedback: During each talk, you will be asked to provide feedback to the presenter using the Comment Form which can be found in the “Files” section. Please ensure your feedback is always respectful. We are all here to learn, support one another, and improve our mathematical understanding and presentation skills.
Additionally, you will be asked to review two of your colleagues’ final papers. Your feedback should focus on “effective communication for the target audience” and will be shared during the “Peer Review Session”.
Important dates: All the important dates can be found in this calendar. You will have two 24-hour grace periods in case of last-minute problems or difficulties for any of the deadlines. If this is the case, you should send me an email (jborga@mit.edu) saying that you want to use one of your 24-hour grace periods. Any further delay will not be accepted and will affect your final grade.
Grading: The final grade is based on 40% for the group talks, including the practices, attendance, and participation after the presentations; 5% for the reading assignment and feedback; 25% for the homework and 30% for the paper and final presentation, including drafts and peer critique.
Resources: In additional to the assistance you will receive from your peers and from me, help with presenting and writing is available from our communication specialist, Susan Ruff. For help with presentations and writing, please email either me (jborga@mit.edu) or Susan (ruff@math.mit.edu).
General help with writing and presenting (not specific to mathematics) is available from MIT’s Writing Center: http://cmsw.mit.edu/writing-and-communication-center.
Student Support Services (S3): If you are dealing with a personal or medical issue that is impacting your ability to attend class or complete work, you should contact a dean in Student Support Services (S3). S3 is there to help you. The deans will verify your situation, provide you with support, and help you work with me to determine next steps. In most circumstances, you will not be excused from coursework without verification from a dean. Please visit the S3 website for contact information and more ways that they can
provide support. Website: https://studentlife.mit.edu/s3.