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PhD Program

During their first year in the program, students typically engage in coursework and seminars which prepare them for the  Qualifying Examinations .  Currently, these two exams test the student’s breadth of knowledge in algebra and real analysis. Starting in Autumn 2023, students will choose 2 out of 4 qualifying exam topics: (i) algebra, (ii) real analysis, (iii) geometry and topology, (iv) applied mathematics.

Current Course Requirements: To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297.

Within the 27 units, students must satisfactorily complete a course sequence. This can be fulfilled in one of the following ways:

Math 215A, B, & C: Algebraic Topology, Differential Topology, and Differential Geometry

• Math 216A, B, & C: Introduction to Algebraic Geometry
• Math 230A, B, & C: Theory of Probability
• 3 quarter course sequence in a single subject approved in advance by the Director of Graduate Studies.

Course Requirements for students starting in Autumn 2023 and later: 

To qualify for candidacy, the student must have successfully completed 27 units of Math graduate courses numbered between 200 and 297. (The course sequence requirement is discontinued for students starting in Autumn 2023 and later.)

By the end of Spring Quarter of their second year in the program, students must have a dissertation advisor and apply for Candidacy.

During their third year, students will take their Area Examination, which must be completed by the end of Winter Quarter. This exam assesses the student’s breadth of knowledge in their particular area of research. The Area Examination is also used as an opportunity for the student to present their committee with a summary of research conducted to date as well as a detailed plan for the remaining research.

Typically during the latter part of the fourth or early part of the fifth year of study, students are expected to finish their dissertation research. At this time, students defend their dissertation as they sit for their University Oral Examination. Following the dissertation defense, students take a short time to make final revisions to their actual papers and submit the dissertation to their reading committee for final approval.

All students continue through each year of the program serving some form of Assistantship: Course, Teaching or Research, unless they have funding from outside the department.

Our graduate students are very active as both leaders and participants in seminars and colloquia in their chosen areas of interest.

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A PhD Student’s Perspective on landing an industry position

Yuxin Wang (UM Math PhD 2021) wrote to share her perspective and process on finding a rewarding career after graduation. Yuxin’s advisor was Sijue Wu. Yuxin’s words:

My PhD Student Perspective: landing an industry position

The purpose of this narrative is to provide a data point about career choices outside of academia. In particular, it tells the story of how a pure Math PhD student ended up with a position in the quantitative finance industry. 

Why industry 

Many people come to graduate school knowing that they would be a professor someday, and I was never one of them. While being a Mathematician is the goal, I was also aware of the scarcity of jobs in academia, so the idea of an industry job has also always been on my mind. The situation worsened when it came closer to graduation, since the academic job market had been hard hit by the COVID pandemic. With the fierce competition for postdoc positions, an industry position started to sound more like a real possibility. I thus started my industry job search in August just before my fifth year as a PhD student. 

How I prepared 

While the job search happened in my fifth year, the preparation started much earlier. Throughout the years as a graduate student, I attended career workshops every now and then to learn about the options that a Math PhD student typically have. One resource that has been particularly useful is the “Invitation to Industry” series held by the Erdös Institute and the Math department, where people with STEM PhD degrees would talk about what their jobs are, what it takes to succeed in their roles, how they ended up in their current positions, and thoughts on their career development process in general. 

From these talks, I came to understand that some of the most common industry job options for Math PhD students are quants, software engineers, and data scientists, all of which are computation related. As such, I have been taking computational courses either in school or online since my first year in grad school. These include, for instance, Scientific computing (where I learned Bash scripting and C/C++), Numerical linear algebra and differential equations, Stochastic process, Computational finance, Convex optimization, etc. None of these had been essential for landing an industry job, but a breadth of knowledge in these fields would certainly help. 

In hindsight, while taking classes was helpful, one could certainly learn faster through practice. In my case, I attended the bootcamp organized by the Erdös Institute, which was made available to us completely free, and which covered all the necessary programming, data science, and traditional machine learning knowledge. I had the chance to work in a team on a company-sponsored real world data problem, and as a pleasant surprise, the sponsor was impressed with our work and was willing to consider us for full-time opportunities. I did not end up working for the sponsor company eventually, but the exposure to real world problems proved helpful in my job search process. 

The bootcamp was by no means the only way of gaining practical experience. Many companies and research institutes offer internship programs, and these are excellent opportunities for one to practice their coding and analytical skills, to enrich their resumes, and to provide something to talk about in job interviews. 

About the job search 

When I attended the career talks, many speakers would depict their job search process as being smooth and straightforward. It is not true in my case. The first few weeks of my job search was daunting. I started applying to all kinds of industry positions in August, ranging from machine learning engineers to data analysts. I wanted to test the water first, so none of these positions were in my dream company, but it was still disappointing to receive rejection without even being interviewed. 

Stressed out, I reached out to more people on LinkedIn and used more help with reviewing my resume (U of Michigan career center and the Erdös Institute both offer such services). Then finally I started to receive interview invitations. As a hindsight, my previous rejections were probably partially due to the fact that I wasn’t as devoted to those positions; nonetheless, there are two things that I learned from this process:

• Companies often receive hundreds and thousands of resumes for a single position, and most resumes are not even read by a real person. And the way to get my resume read is to either reach out to the hiring managers directly, or to use as many keywords from the job description as possible. I found a website called ResumeWorded (it is free thanks to U of Michigan career center), that rates one’s resume and makes suggestions according to the job description. I was shocked to see that my resume only got a 64% match to the job description, which probably explained all the rejections I got. I started to refine my resume according to every job description, and finally started to get invited to interviews. 
• This is perhaps cliché, but the importance of networking is never to be underestimated. When people think of “networking”, many have the (understandably unpleasant) impression of crowded career fairs, networking events, or awkward phone calls; while in fact, networking can be spontaneous and even fun. I learned this through the “Designing Your Life Series” offered by Rackham – networking is like asking for directions in a foreign country. All we have to do is to be willing to introduce ourselves and talk with others. 

Eventually, my resume started to get noticed, and I interviewed for some quantitative researcher and data scientist positions. I did not get a single rejection from the companies that interviewed me, so I’m glad that I persisted despite all the rejections at the very beginning. Sometimes it’s not that we are incompetent, it’s that the right opportunity hasn’t come yet, and all we need to do is to learn from our own mistakes and keep trying. 

About the interviews

I will focus on my interviewing experience for the quantitative researcher positions only. 

My experience with SIG 

A year prior to my job search, I met Joey Thompson, a recruiter at Susquehanna International Group (SIG), at an “Invitation to Industry” talk (this is the same Joey as in Mark’s post ). Joey and two quantitative researchers at SIG came to give a series of talks about the company and their work as a quantitative researcher. I remember in the first talk with the Quant Finance master students, Joey asked if anyone knew the difference between an investment bank and a hedge fund, and nobody wanted to answer, so I raised my hand. At that moment, I was just trying to let the speaker feel welcome, having no idea that I would talk with him again in my job search. I guess this might have been a form of networking in a broad sense – and all I did was being willing to talk. 

After that, they held a brain teaser battle at Arbor Brewing Company, where Math grad students worked in groups to solve puzzles. The brain teaser battle was fun, and my group won the first place. Joey invited me, as well as some other Math grad students, to apply for their internship position; I ended up not applying right away since I felt under prepared. Looking back, it perhaps would have been a good idea to at least try to apply for an internship in my fourth year – check Mark’s post for how that would look like. 

I contacted Joey a year later to apply for the full-time quantitative researcher position, and was directed to an online assessment, which consisted of questions on Probability, Statistics, Calculus and Linear algebra. I received an invitation to the first phone interview shortly after completing the online assessment. 

Other than the online assessment, my interview experience with SIG is very similar to Mark’s . There are two rounds of phone interviews, a data exercise, and a full day of onsite interviews, which are virtual this year due to the pandemic. The level of difficulty increased gradually throughout the process, and I got the chance to work on some really interesting questions. I did stumble on some problems in the process, but the interviewers were willing to point me to the right direction, so not solving all the problems in the first attempt did not automatically disqualify me. 

General suggestions on interview preparation 

While I am not allowed to discuss any specific interview question, I would say that many of the quantitative researcher interviews contained questions in:

• Probability. Most questions are just on probability calculation (i.e., not measure theoretic probability). Two resources that have been very useful in preparing for these kinds of questions are: A Practical Guide to Quantitative Finance Interviews , and 150 Most Frequently Asked Questions on Quant Interviews . 
• Statistics. In addition to probability, knowing the basics of Bayesian statistics will be useful. Many companies also tested my knowledge on all kinds of regression techniques, so knowing how to derive them from scratch and how to interpret the results in a statistical sense would be helpful. 
• Linear algebra. Math PhD students are in general acquainted with Linear algebra, but we tend to be more familiar with the abstractness than with the computation aspect. While the high level of understanding is crucial, it never hurts to refresh our memories of calculating eigenvalues, eigenvectors, matrix factorizations etc. I find the AIM QR exam in Linear algebra quite useful for this purpose. 
• Coding. Since one common career path for Math PhD students is software engineering, I did a lot of practice on coding through Leetcode. A good understanding of the undergraduate-level algorithm and data structure is as important as the fluency in a coding language. In addition, knowing the basics about how computers work in general would be helpful. I learned some of these through GSI-ing for EECS 376, but there are other more direct ways to prepare. One resource that many people use to prepare for the coding interviews is Cracking the Coding Interview by Gayle McDowell. 

Final remarks  

People make career transitions due to various reasons. For me, the hit of the pandemic is a pretty random direct cause, but my previous efforts exploring the industry possibilities are not irrelevant. In general, transitioning into industry as a Math PhD, though not trivial, would not be difficult either, since the solid background in Math puts one into a great position to learn all the skills that are required to perform an industry job. But just like any endeavor, it does take practice and preparation, and sometimes perseverance. If you would like to talk about any part of my story, feel free to email me at [email protected] . 

The purpose of this narrative is to provide a data point about career choices outside of academia. In particular, it tells the story of how a pure Math PhD student ended up with a position in the quantitative finance industry. 

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Learn more about Harvard’s Math community and our statement on diversity and inclusion.

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Building next generation autonomous robots to serve humanity

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Recent Grant/Competition Partners

Since completion of the Subterranean Challenge, faculty and students have been conducting follow-on research and competitions with multiple corporate and government partners.

• Lockheed Martin
• National Institutes of Standards and Technology - 2023 First Responder UAS 3D Mapping Challenge
• National Science Foundation
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Robotics Partners at CU Boulder

Research further advancing the capabilities of the Subterranean Challenge Robots is being led by numerous CU Boulder laboratories.

• Nisar Ahmed (Aerospace) - Cooperative Human-Robot Intelligence Lab
• Nichole Barger (Ecology and Evolutionary Biology) - Aridlands Ecology Laboratory
• Nicholas Correll (Computer Science) - Correll Lab
• Eric Frew (Aerospace) - Research & Engineering Center for Unmanned Vehicles
• Chris Heckman (Computer Science) - Autonomous Robotics and Perception Group
• Sean Humbert (Mechanical Engineering) - Bio-Inspired Perception and Robotics Laboratory

One thousand feet underground, a four-legged creature scavenges through tunnels in pitch darkness. With vision that cuts through the blackness, it explores a spider web of paths, remembering its every step and navigating with precision. The sound of its movements echo eerily off the walls, but it is not to be feared – this is no wild animal; it is an autonomous rescue robot.

Initially designed to find survivors in collapsed mines, caves, and damaged buildings, that is only part of what it can do.

Created by a team of University of Colorado Boulder researchers and students, the robots placed third as the top US entry and earned $500,000 in prize money at a Defense Advanced Projects Research Agency Subterranean Challenge competition in 2021.

Going Futher

Two years later, they are pushing the technology even further, earning new research grants to expand the technology and create new applications in the rapidly growing world of autonomous systems.

“Ideally you don’t want to put humans in harm’s way in disaster situations like mines or buildings after earthquakes; the walls or ceilings could collapse and maybe some already have,” said Sean Humbert, a professor of mechanical engineering and director of the Robotics Program at CU Boulder. “These robots can be disposable while still providing situational awareness.”

The team developed an advanced system of sensors and algorithms to allow the robots to function on their own – once given an assignment, they make decisions autonomously on how to best complete it.

Advanced Communication

A major goal is to get them from engineers directly into the hands of first responders. Success requires simplifying the way the robots transmit data into something approximating plain English, according to Kyle Harlow, a computer science PhD student.

“The robots communicate in pure math. We do a lot of work on top of that to interpret the data right now, but a firefighter doesn’t have that kind of time,” Harlow said.

To make that happen Humbert is collaborating with Chris Heckman, an associate professor of computer science, to change both how the robots communicate and how they represent the world. The robots’ eyes – a LiDAR sensor – creates highly detailed 3D maps of an environment, 15 cm at a time. That’s a problem when they try to relay information – the sheer amount of data clogs up the network.

“Humans don’t interpret the environment in 15 cm blocks,” Humbert said. “We’re now working on what’s called semantic mapping, which is a way to combine contextual and spatial information. This is closer to how the human brain represents the world and is much less memory intensive.”

High Tech Mapping

The team is also integrating new sensors to make the robots more effective in challenging environments. The robots excel in clear conditions but struggle with visual obstacles like dust, fog, and snow. Harlow is leading an effort to incorporate millimeter wave radar to change that.

“We have all these sensors that work well in the lab and in clean environments, but we need to be able to go out in places such as Colorado where it snows sometimes,” Harlow said.

Where some researchers are forced to suspend work when a grant ends, members of the subterranean robotics team keep finding new partners to push the technology further.

Autonomous Flight

Eric Frew, a professor of aerospace at CU Boulder, is using the technology for a new National Institute of Standards and Technology competition to develop aerial robots – drones – instead of ground robots, to autonomously map disaster areas indoors and outside.

“Our entry is based directly on the Subterranean Challenge experience and the systems developed there,” Frew said.

Some teams in the competition will be relying on drones navigated by human operators, but Frew said CU Boulder’s project is aiming for an autonomous solution that allows humans to focus on more critical tasks.

Although numerous universities and private businesses are advancing autonomous robotic systems, Humbert said other organizations often focus on individual aspects of the technology. The students and faculty at CU Boulder are working on all avenues of the systems and for uses in environments that present extreme challenges.

“We’ve built world-class platforms that incorporate mapping, localization, planning, coordination – all the high level stuff, the autonomy, that’s all us,” Humbert said. “There are only a handful of teams across the world that can do that. It’s a huge advantage that CU Boulder has.”

Additional Images

PhD student Kristen Such reviewing LiDAR maps generated by the robots in a mine.

A subterranean robot navigating through the Edgar Mine in Idaho Springs, CO.

PhD student Kyle Harlow inspecting one of the robots before activation.

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