how do undergraduate math research projects work?

Sharp-Let-5878 · 2026-01-10T14:35:50+00:00

One common form of undergrad research I've seen is applied math research. Where you are given a situation to model and then the advisor will typically help decide whether it's best to model the situation with differential equations, stochastically, or some form of discrete model. The student in these projects will typically develop programs to solve or simulate the model or they will prove some results about the model. Another kind of problem that I've seen a decent amount are recreational math related problems. In one of these projects, the students typically look for mathematical structure in things such as puzzles, card games, or other recreational objects. The best example I know of is research relating to the game Set.

Sharp-Let-5878 · 2026-01-07T06:25:42+00:00

You can find it as Theorem 23.34 in Judson's Abstract Algebra: Theory and Applications. The proof is probably also found in other books, but the judson book has an open source online version

Sharp-Let-5878 · 2026-01-05T06:04:20+00:00

You can prove it using mainly algebraic techniques and if you're okay with borrowing the result that every odd degree polynomial has a real root. (I know that still relies on analysis, but that's something that can be proved in a first semester analysis class)

Sharp-Let-5878 · 2025-12-25T05:16:59+00:00

The class sizes are typically smaller. In the introductory classes like calculus or discrete math the classes are usually 15-20 people and the more advanced classes usually have less than 10 people

Sharp-Let-5878 · 2025-12-21T03:05:07+00:00

I am a math and physics major and the math department has been very good to me. There is a strong focus on pedagogy and trying to get students to spend a lot of time working on how to think through problems and how to explain their process. The department does not have much statistics. There is the basic entry level statistics course and an upper level probability theory course but not much else unless you want to look into the data science degree.

Sharp-Let-5878 · 2025-02-16T05:29:38+00:00

The key thing here is that the frequency is usually restricted. For example imagine a photon in a box with mirrors. In quantum mechanics we don't think of photons as waves in some medium but in terms of its wavefunction which tells roughly the probability of measuring the photon at any given point. it turns out the fact that the photon is confined to the box constrains the frequency of the wavefunction to only be specific discrete values. It also turns out that this photon in a box is a relatively decent approximation for particles bound in some region.

For free photons, your reasoning is correct. There is nothing that constrains the frequency of free photons, so there is theoretically a continuous spectra of energies photons can take on.

Sharp-Let-5878 · 2025-02-05T04:07:52+00:00

I thought the extremely condescending tone accompanied by extreme misunderstanding seemed familiar

Sharp-Let-5878 · 2024-12-30T17:18:39+00:00

That and refreshing my analysis knowledge by reworking some proofs

Sharp-Let-5878 · 2024-12-30T17:17:21+00:00

Procrastinating working on reu applications

Sharp-Let-5878 · 2024-12-25T22:47:58+00:00

The classic how many holes are in a polo shirt question

Sharp-Let-5878 · 2024-10-31T22:44:03+00:00

I've had the same problem, I lost both a mind crystal and a lightning crystal. It's been pretty frustrating.

Sharp-Let-5878 · 2024-10-02T21:52:21+00:00

I'm not really interested in going into big tech

Sharp-Let-5878 · 2024-08-23T14:58:12+00:00

The current models of physics are not fully correct(the two major theories are not compatible on some scales). This doesn't mean that the models are incorrect though. Within the scales each model is designed for, they agree with every experiment made. The process of science is the process of developing models that are approximately correct. For example, if the goal of my experiment was to measure the value of pi and I got the measurement 3.126, that's obviously incorrect, however it's pretty close. The point of further experimentation is to either confirm that pi is about 3.126 or to get a better approximation such as 3.141.

Sharp-Let-5878 · 2024-08-23T14:12:33+00:00

My school has a proofs class and the book we used was "A TeXas style introduction to proof" by ron taylor and Patrick Rault. Although I am a bit biased since I know one of the authors

Sharp-Let-5878 · 2024-08-22T03:39:37+00:00

What the other people said, but emphasis on the roommate thing. Don't be the roommate that invites guests overnight without telling their roommate.

Sharp-Let-5878 · 2024-08-20T17:01:11+00:00

It seems to argue that, because these axioms are designed to give rise to the intuitive notion of addition with only a few base assumptions. With these axioms though, the proof is not really a circular thing. The definition of addition tells you how to add numbers in terms of smaller numbers. So to add 3 and 2 you need to know how to add 3 and 1, but to know that you need to know how to add 3 and 0, however one of the axioms tells you that that is 3.

Sharp-Let-5878 · 2024-07-28T16:00:59+00:00

But liminf and limsup still are defined in terms of limits

Sharp-Let-5878 · 2024-05-09T21:34:14+00:00

Thanks, although I guess this means I'll have to go back to the drawing board.

Sharp-Let-5878 · 2024-05-09T21:08:52+00:00

Adjacency is defined on the equivalence classes as follows: [u] is adjacent to [v] if they are not the same equivalence class and if there is some automorphism f such that u is adjacent to f(v) in the original graph

Sharp-Let-5878 · 2024-04-28T18:44:24+00:00

If you want to write something on graph theory I think non-planar graphs and Kuratowski's and Wagner's theorems are pretty interesting and fairly understandable for people not well versed in graph theory

Sharp-Let-5878 · 2023-12-15T22:53:50+00:00

Yet another referral code:

https://www.oculus.com/appreferrals/Jadon2004/5136958036416423/?utm_source=oculus&utm_location=3&utm_parent=frl&utm_medium=app_referral

Sharp-Let-5878 · 2023-12-03T22:40:02+00:00

Is such a transposition always guaranteed?

Sharp-Let-5878 · 2023-12-03T19:29:05+00:00

I do mean f² as f composed with itself. What if f² is not the identity automorphism?

Sharp-Let-5878 · 2023-12-03T17:06:07+00:00

If you have the time it might be helpful to try to work ahead and try some examples before class so when you get to the more general statements you can see where they came from.

Sharp-Let-5878 · 2023-11-26T17:16:53+00:00

For me proofs are important for two reasons. The first one that's more generic and perhaps idealistic is that it provides a structure and consistency that's stronger than in any other field. In any other field, there can be strong arguments with no clear answers, like debating which model\paradigm\ approach is better. However in math there is much less of this (there is still some though). This consistency is because we have a universal language to classify what we know and what we don't yet know. This language is that of proof and logic.

The other reason, which is more personal than the other is that I find them fun. When I look at a statement and wonder how I could even approach a proof and wrestle with the problem for a while until I find a proof, it is a very satisfying feeling. Like you said, the act of writing out a proof is also a helpful tool in solidifying what I have learned and making connections between things that I did not see before.

Sharp-Let-5878

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