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The University of New Mexico

Mathematics directed reading program (DRP)

UNM Math DRP

(Mathematics Directed Reading Program)

  • Alexander Fritschi, MS Pure Math Student
  • Tommy Denny Martins, BS Pure Math Student

About

The Math DRP is a UNM student organization since Spring 2021, which pairs interested undergraduates and graduate students for independent study projects. Undergrads and math grad students will work through a math book or paper together over the course of a semester, culminating in presentations from the undergraduate on a fun thing they learned. In select cases, undergrads maybe able to conduct research with the grads.

Our goal is to enable motivated undergraduates to engage with math in more depth and breadth than a classroom setting typically allows, and to foster a tight-knit mathematically community at UNM.

Join us

Fill out this form if you want to participate in DRP as a mentee! For prospective mentors, please email Alexander or Tommy to sign up as a mentor.

If you would like to get to know other members of the UNM math community in a more informal setting, join UNM's unofficial math club Discord group!

Benefits

Why should you join DRP? If you're an undergrad, DRP is an opportunity to

  • Explore math outside of UNM's standard curriculum,
  • Interact with graduate students and meet other mathematically motivated undergrads,
  • Develop mathematical maturity,
  • Practice formally presenting mathematics,
  • Develop a topic for a thesis or potential future research project.

And for grad students, DRP is an opportunity to

  • Give back to the mathematical community,
  • Act as a guide for the next generation of mathematicians,
  • Grow as educators,
  • Raise undergrad awareness of your personal favorite areas of higher mathematics.

 

Expectations

An undergraduate looking to participate in the DRP should have at least four hours per week to think/read about math and do practice problems (in addition to, not in place of, their coursework). We also ask that after two semesters of participation in DRP, undergrads give a 10-30 minute presentation on a fun topic they learned about at the biannual UNM DRP Symposium. Grads should be prepared to feed the cravings of a curious mind, as well as help mentees choose reasonable topics given their background, and offer feedback/assistance as they design their end-of-semester presentation. Each week, mentees and mentors will meet for at least one hour to discuss progress, ideas, problems, etc. Availability of students may allow for longer or more frequent meetings if desired.

These expectations are important! It is possible to make very meaningful progress in mathematics by following them diligently.

Eligibility

If you're curious and motivated to learn about cool topics in math, by all means join us!

There is no catch-all prerequisite for DRP. If you're interested in a particular area of math but aren't sure whether you're prepared to study it, feel free to contact a member of the DRP Committee for advice. That being said, for most projects, it is desirable for the undergrad to have completed the calculus sequence and/or at least one proof-based math class.

Students from underrepresented backgrounds are particularly encouraged to join.

 

Past Projects

  • Undergraduate Andrew Geyko studying real analysis under graduate student James Legg (since Spring 2021)
  • Undergraduates Franklin Pezzuti Dyer and Kioshi Morosin studying complex analysis under graduate student Owen Davis (since Spring 2021)

 

Student Work

Past student work from DRP groups:

 

Project Ideas

Some ideas for future projects:

  • Topic: Knot Theory
    Book: The Knot Book by Colin Adams.
    Recommended background: No background required.
    Description: Perfect for a student interested in Knot theory, a nice project based off introductory Knot Theory could be the pros and cons of different knot invariants. In mathematics in general an overarching theme is the general notion of when are two things the same, in knot theory there have been a number of invariants which produce different notions of how to knots can be the same, as such the uses of these invariants vary drastically!
  • Topic: Topology
    Book: Topology: A Categorical Approach by Tai-Danae Bradley.
    Recommended background: A first course in Point-Set Topology is necessary for this project, however, sufficient knowledge of category theory would allow the student to avoid the difficulties that arise from not having had a first course in Point-Set Topology. That being said, it would be unlikely to encounter someone with sufficient category theory knowledge that had not taken a first course in topology. (If you are reading this thinking "wow, that person is me!" please sign up for this project!)
    Description: If mathematics was a city, Category Theory could be described as its infrastructure. This project would involve revisiting topology with the new tool of category theory and comparing the differences in these approaches. Are there some categorical definitions that are more easily usable than their standard point-set counterparts? Choose this topic and let's find out!
  • Topic: Analytic Number Theory
    Book: Introduction to Analytical Number Theory by Tom Apostol.
    Recommended background: A first course in analysis would help ease in the student to learning about analytical number theory but it is not necessary, should the student be willing to spend sufficient time independently studying.
    Description: Perfect for a student interested in the intersection of analysis and number theory. Analytic number theory is pretty much what it sounds like - investigating the field of number theory by using analysis to prove bounds or other results. Some of the biggest results in number theory come from analytic number theory. For example, the prime counting function is primarily studied using analytical methods.
  • Topic: Algebraic Number Theory
    Book: A Classical Introduction to Modern Number Theory by Kenneth Ireland and Michael Rosen.
    Recommended background: A first course in abstract algebra would be nice but is not essential.
    Description: Modern Number Theory has become very closely related to the study of algebraic geometry. A project using this book would investigate that relationship and provide a solid foundation for further exploration into elliptic curves and modular forms.

 

Resources

Are you interested in math, but unfamiliar with the different areas of higher mathematics? Here's a short list of the major fields of higher math, along with short descriptions of what they entail:

  • Number Theory studies the properties of integers and sequences of integers, and is often concerned with divisibility and properties of prime numbers, as well as finding integer solutions to equations (e.g. Pythagorean Triples). There are two main approaches to advanced number theory: analytic number theory, which makes use of methods from real and complex analysis, and algebraic number theory, which uses abstract algebra techniques to generalize the integers and their properties to other number rings/fields.
  • Abstract Algebra studies algebraic objects in general, and is extremely broad. It often deals with structures analogous to the integers, real numbers, or complex numbers in various ways, with different definitions of "addition" and "multiplication" that may behave differently from the traditional arithmetic operations. For instance, rings are equipped with addition and subtraction but not necessarily division, whereas fields are rings in which division is defined.
  • Geometry formalizes and studies concepts related to space, such as distance, shape, and size (e.g. length, area, volume... hypervolume?). Until the 19th century, geometry was mostly studied in a Euclidean context, but more recently, geometry has experienced a boom with the advent of non-Euclidean geometries, such as spherical and hyperbolic geometry. Now it's entrenched in almost every other field of math.
  • Linear algebra studies systems of linear equations, the algebra of matrices and vectors, and the properties of higher-dimensional (vector) spaces. It's strongly related to geometry and as such has a wide range of uses in applied mathematics and engineering, but is ubiquitous among fields of higher mathematics in general. No matter where you go, linear algebra needs to be in your toolkit.
  • Real analysis studies the properties of real numbers and real-valued functions. It is often concerned with sequences and series of real numbers (and whether they converge/what they converge to), and formalizing limits, derivatives, and integrals and studying their properties rigorously. Real analysis is the proof-based older brother of calculus.
  • Complex analysis studies the properties of complex numbers (an extended version of the real numbers in which the square roots of negative numbers exist) and generalizing calculus to functions of a complex variable. It gives rise to many amazing results in geometry, number theory, physics, and even more disparate fields, making it a foundational subject in mathematics.
  • Dynamical systems studies how systems (often inspired by physical/biologial processes, such as the swinging of a pendulum or the fluctuation in animal populations) evolve over time. Knowledge of calculus and basic differential equations is preferable. Fractal geometry and chaos theory are two well-known areas that fall under the purview of dynamical systems.
  • Topology studies the geometric properties of objects that persist when they are deformed in a continuous way. A famous example: to a topologist, a coffee cup is the same as a donut (because one could be deformed into the other, with the handle of the cup corresponding to the hole in the donut). Much of topology is motivated by the desire to generalize analysis to "weirder" spaces than the real and complex numbers, in which the notion of distance is called into question (this leads to the study of manifolds).
    • Knot theory is a subdiscipline of low-dimensional topology with surprisingly wide-ranging applications in biology, physics, and chemistry (and, more recently, the exciting field of topological data analysis). Common techniques include representing knots as special polynomials or matrices.

And here's a map of mathematics, designed by UNM's own Dr. Buium!

 

If you're frustrated about being stuck on a problem or feel like you're "bad at math" (you're probably not, we promise!) see the following pages:

On the other hand, if you're a hotshot and think you're the world's next great mathematical genius, check out the following websites:

 

FAQs

Mentees

  • Q: Will this count for course credit?
    A: Nope.
  • Q: Will I always be paired one-to-one with a grad student?
    A: As much as we appreciate injective mappings, we will occasionally allow small groups of undergrads to be paired with one grad student when multiple undergrads are interested in the same topic.
  • Q: I'm reading about the different areas of math offered and I don't know what they are. Can you give me some introductory information about each of them?
    A: Yes! Please read the short excerpts about each of the very broad sections of math. If this only causes you to have more questions (I know this was the case when I started studying mathematics!) please feel free to reach out to a member of the DRP Committee, they will be happy to help you find your math home.
  • Q: I don't know what to do as a project OR I'm not a math major OR I don't have a background in the subject I want to study - can I still participate?
    A: Absolutely! The biggest determining factor in how much you will get out of UNM DRP is your interest and enthusiasm towards mathematics. As long as that curiosity is there, you'll have a fun and math filled semester. Your mentor will help guide you towards a reasonable project which is of interest to both of you, and accessible given your background knowledge.
  • Q: How will I be paired with a mentor?
    A: If you request a particular grad student by name, it is very likely that you will be paired with them. If many students request the same grad student, priority will be given to those who have worked with that grad student the longest. Otherwise, your mentor will be selected from a pool of graduate students who have offered to teach your topic of interest.
  • Q: Is the symposium presentation mandatory?
    A: Strictly speaking, no - but it is an integral part of the DRP experience, and you cannot run for undergraduate positions in the committee or vote on DRP matters unless you have either presented once or committed to present at the next Symposium.

Mentors

  • Q: What are the requirements for being a DRP mentor?
    A: The only real requirements are that you're a UNM mathematics graduate student (or in extremely special cases, a sufficiently advanced undergrad) and that you are willing to put in the effort to making a productive and supportive learning environment for your mentee(s).
  • Q: Is there any financial reimbursement for being a DRP mentor?
    A: No - however, although direct financial compensation is unlikely, we may be able to provide mentors and mentees with necessary books and other resources, depending on funding.
  • Q: Can I be a DRP mentor as a first-semester grad student?
    A: Yes, but be careful to manage your time responsibly. Neither undergrads nor grads should ever put the DRP above their coursework or other official responsibilities.
  • Q: If I've already been in touch with a particular undergrad and we know that we want to work together, can we be paired together under the DRP program?
    A: Yes! You just need to do two things - firstly, have your mentor indicate that they want to work with you specifically when they fill out their application form, and secondly, inform the Chief of Operations about this situation so that they can ensure you are paired according to your wishes.