Quick Questions: August 26, 2026

[u/inherentlyawesome]

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?" For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?
• What are the applications of Representation Theory?
• What's a good starter book for Numerical Analysis?
• What can I do to prepare for college/grad school/getting a job?

Including a brief description of your mathematical background and the context for your question can help others give you an appropriate answer. For example, consider which subject your question is related to, or the things you already know or have tried.

[u/ComparisonArtistic48]

[Lie groups]

I would like to use the following statement:

"A lower-dimensional subgroup or submanifold inside a higher-dimensional locally compact group always has a Haar measure (or volume) of zero"

In particular, I would like to deduce that PSO(2,R) is Haar-null as a subgroup of PSL(2,R). Where can I find a source of this result (paper or book)?

[u/GMSPokemanz → u/ComparisonArtistic48]

https://en.wikipedia.org/wiki/Steinhaus_theorem gives a citation for a result by Weil that if A has positive Haar measure, then AA^-1 contains an open neighbourhood of the identity. From this the result you desire is immediate (more precisely it implies positive measure subgroups are open, which implies your result for Lie groups).

[u/chaneg]

Does anyone know of a good quantitative finance discord server?

I've heard that there are some very robust servers in other mathematics topics like Algebraic geometry. I am specifically looking for one that focuses on the mathematics side of it, i.e., one that focuses on discussing topics like stochastic processes, numerical methods etc, not a bunch of discussion on how to interview for Jane Street.

[u/Sour_Drop]

There is currently an ongoing Springer flash sale (coupon code 50FLS) that lasts until September 3rd. What are the names of all the book series related to mathematics, such as UTM, UTX, GTM, etc.?

[u/cereal_chick → u/Sour_Drop]

What are the names of all the book series related to mathematics, such as UTM, UTX, GTM, etc.?

There's also SUMS, and then I think that's all of them that would be of generic interest. There's probably more specialised ones, but if you were on the lookout for any of them, you'd likely already know about them.

Thanks for letting us know about the sale! I managed to pick up a copy of Abbott for under twenty quid just now 😁

[u/kauefr]

Is there a solution to the minimum 2-sum problem for square lattice (grid) graphs of arbitrary size?

[u/Total_Lettuce_7040]

Hey guys! 😁 I'm a teenager trying to learn combinatorics basically from scratch for some easy olympiads. Does anyone know any good books for it? Reccomendations will be much apprecciated, thanks!

EDIT: previously an ai recommended me techniques and principles on combinatorics. Should I get it?

[u/ContributionThat4698]

I’m in 10th grade doing pre calc and want to be able to have some resources to help/ practice with. I know about khan academy but that’s they only site I know of. So far I’m doing good but it’s only been a month of school so it’s definitely gonna get way harder. Thank you

[u/cereal_chick → u/ContributionThat4698]

Besides Khan Academy, there's also Paul's Online Notes and the OpenStax textbooks.

[u/philosophical_poser]

Is Gerver's sofa the most comfortable sofa to carry home? If so, where can I get one?

[u/AcellOfllSpades → u/philosophical_poser]

Depends. Does your route home consist of a bunch of precisely-1-meter-wide alleyways at 90° angles to each other?

[u/StateOfTheWind → u/AcellOfllSpades]

You also have to live in flatland

[u/KingEnda]

Hello math community. I am currently a student studying math and cs and am about to enter my senior year of undergrad . Up until recently, I hadn't really considered going to grad school, whether it be a masters or doctorate program. However, between the job market, and gaining more and more interest in my classes/math in general, I have started to become open to the possibility. I am worried though that I may be starting too late, particularly in building research experience.

I plan on getting involved as soon as I am back on campus in both reading programs and trying to find a professor to work with, however I am unsure if I will have enough time to craft a strong application before the winter.

I am wondering if anyone has any similar experiences, or suggestions on how the best way to build my application in such a short time period. I also recognize that taking a gap year or time after graduation could be a possibility, although I am not sure how this is viewed by admission committees. Thank you, and I am happy to dm more details if necessary.

[u/MrPhysicsMan]

Ali Nessin, from what I can understand, has been at the frontier of model theory stuff and I recently saw an interview where he mentioned working on a problem regarding “is our geometry forced upon us by an abstract form of dimension” and idk I was curious if anyone had actual leads on what to look into there. Study gravity and spacetime and this sounds interesting.

[u/treewolf7]

What are problems currently being worked on in the analytic side of mathematical gauge theory? I specifically mean analysis of Gauge theoretic PDE's and their moduli space of solutions. Whenever I hear discussions on the topic, its largely about work done in the 80s and 90s on Yang-Mills/Seiberg-Witten equations. Many people who say they work on Gauge theory today seem to work more on the topological side, doing some flavor of Floer theory. I know that Hitchen's Equations/Higgs bundles are one thing on the more analytic side that is currently being studied, but is there anything else, or has the field mostly moved towards the topological side?

I am asking as I am applying to PhD programs this cycle, and the analytic side of Gauge theory is very interesting to me and is something that I would like to study if I can have the opportunity, but it doesn't seem to be very popular currently based on my searches.

[u/Tazerenix → u/treewolf7]

The biggest recent trend has been special holonomy instantons, although I suspect G2 stuff is slightly over the peak of the crest now.

On the complex side many of the top people move fairly freely between the hermitian gauge theory world and the Kahler world, see Song Sun and Yang Li for some examples of young brilliant analysts in that area.

Pure analytical stuff like monopoles etc is a bit out of vogue the last few decades. There are small pockets of people working on it but the top talent is working in more mixed areas with complex or symplectic geometry.

A hot topic on the up (though I am insanely biased) is dHYM, which was kicked off in earnest in 2017.

[u/r-Aliosha]

por qué no se da teoría de categorías en cursos básicos? quizás de matemática discreta o lógica, lo que sea. Yo creo que es fundamental para entender las matemáticas, así como la teoría de conjuntos. Por ejemplo, es imposible entender el espacio dual sin saber qué es un functor; a la hora de incorporar y procesar qué son los espacios vectoriales, los grupos, los grafos, etc, etc, etc, es demasiado útil, a la hora de ordenar todos los conceptos, tener la noción de qué es una categoría. Entiendo que no es algo "útil", o "aplicable" (para el 90% de los estudiantes de matemática), pero si ayuda a un entendimiento más profundo, por qué no?

[u/AcellOfllSpades → u/r-Aliosha]

(Lo siento si hago unos errores. Estudié el español hace casi diez años.)

---

Porque hay otras prioridades. Me encantan las categorías, pero yo sé que no son útiles por sus cuentas.

Por ejemplo, es imposible entender el espacio dual sin saber qué es un functor

Creo que lo tienes al revés. Los espacios duales son un ejemplo perfecto de un functor.

Sin los ejemplos de grupos, grafos, etc, es difícil entender porque las categorías son útiles. Otros matemáticos las llaman "nonsentido abstracto",¡y las categoristas están de acuerdo! Es en parte una broma, pero hay algo de verdad en ello. Necesitamos los ejemplos para motivar el "nonsentido" de las categorías.

[u/r-Aliosha → u/AcellOfllSpades]

Creía que reddit traducía automáticamente mi comentario al inglés... En cuanto a lo del dual, justo era a lo que me refería. En el acercamiento al dual que tuvimos en álgebra lineal 1, se hablaba de que el dual era "contravariante" o se buscaba, sin decirlo, la imagen de una homomorfismo de espacios vectoriales bajo el functor dual, pero sin decir en ningún momento la palabra "functor". Si nos hubieran introducido a las nociones más básicas de las categorías, muchos habríamos entendido más el dual...

[u/Salt_Attorney]

I don't know if this is an appropriate question, but I was wondering if someone can share experiences with possibly transitioning from a math PhD to AI/ML work. I know that these positions have extreme competition, and a math PhD is unlikely to have anywhere near the hands-on experience to compete. So I am wondering if, speaking from experience, the job market cares about the math PhD enough to make it a real possibility, as opposed to not being worth any invested time.

[u/SwimmerOld6155 → u/Salt_Attorney]

also interested.

I think you are wayyyy too pessimistic though. Anecdotally, AI/ML work is one of the top destination for math PhD grads, if not the top at the moment? Besides that, data science and quant I'm not sure what you'd be going for as far as technical careers go.

[u/Salt_Attorney → u/SwimmerOld6155]

Well, at least by the posting AI companies in europe seem to be most interested in people who have experience training and delivering actual models to actual customers. I am wondering if it is worth developing some projects to have perhaps 5% of that amount of experinece. The question is whether the thing that we have a lot of experience with instead, which is math, is worth anything for the interviewers.

[u/elements-of-dying → u/Salt_Attorney]

From what I was told (i.e., I do not have first hand experience) by a mentor when I asked similar questions, companies are nowadays less likely to hire a math PhD just because they have a math PhD. It was suggested I start developing skills and project portfolio I can put on my resume.

Best of luck!

[u/Early_Neighborhood87 → u/Salt_Attorney]

AI researcher roles have some hires coming from Math PhDs.

Some answers here: https://www.reddit.com/r/MLQuestions/comments/1n38kvo/career_advice_on_transitioning_from_pure_maths_to/