Favorite Mathematics Field?

Forum for the GRE subject test in mathematics.
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Favorite Mathematics Field?

Postby ReQuiem » Sun Jan 28, 2018 7:52 am

I am an undergraduate mathematics student and I am exploring fields of mathematics. All I am asking you is to tell which field is your favorite and why you have a passion for it.

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Re: Favorite Mathematics Field?

Postby petrokov » Sun Jan 28, 2018 8:52 am

I'm applying for a PhD to study math with an emphasis on graph theory.

I like graph theory, as well as combinatorics, for a few reasons. First, combinatorics was the first taste that I had of real math. I took a combinatorics class that was way too hard for me while studying abroad in Korea, and I didn't realize I was in over my head until it was too late to drop. With no other options, I asked the TA for help with some binomial identity proofs that I was not at all equipped to do. However, she didn't speak English, so everything she told me was in Korean. I still managed to learn a lot from her and make a huge amount of progress on problems that were previously impossible to me. I eventually did pass the class, even though my grade was very good. The whole of feeling of struggling through math that was too hard for me and resorting a language that I barely knew to get help and still making tangible progress really stuck with me. I felt strongly that no matter how hard things seem, I can learn a lot if I just push through it, and that became my motivation to study more math. I've liked combinatorics ever since.

Combinatorics is closely related to graph theory, which is why I took a graph theory class. I feel that while many branches of math, such as algebra and analysis, really emphasize the rigor of mathematics, graph theory really emphasizes how creative math can be. Any idea can be invented, such as graph colorings, list colorings, perfect graphs, various embeddings of graphs, and random graphs, and there will always be amazing results waiting to be found. I can define a new class of graphs or a new process related to graphs, and I will almost definitely find something new and interesting. Graph theory still has all of the rigor of other types of math, but it's so easy to invent new things, and I feel like the possibilities are endless. I'm sure that other types of math are like this too, but I feel that graph theory has a very inviting and simplistic feel while still being incredibly complex. Graph theory pulls me in with seemingly simple concepts, and then it forces me to attain a deep understanding in order to reach results.

Also, I feel like the ideas of graph coloring and graph structure have a very deep connection that we don't fully understand, and I feel that with enough effort, we'll be able to find the relationship between the way graphs are put together and properties like chromatic number. I want to be a part of this effort.

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Re: Favorite Mathematics Field?

Postby ReQuiem » Sun Jan 28, 2018 1:43 pm

Thank you very much for answering ,your answer is exactly what I was looking for, you seem very passionate about the field you've chosen and I think you will do well.

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Re: Favorite Mathematics Field?

Postby smiledaniel » Sun Jan 28, 2018 6:11 pm

the field of complex numbers of course, as it is the easiest algebraically closed field to do algebraic geometry hahahah

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Re: Favorite Mathematics Field?

Postby EGA » Sun Jan 28, 2018 6:53 pm

Homotopy groups arose as a tool to detect holes in topological spaces. If you live in the xy-plane, and someone removes the origin, how do you know that there's a hole there? Well a pretty reasonable method is to take a rope, start at some point and leave one end of your rope at this point, and then walk in a circle around the origin. Once you get back to where you started, grab both ends of the rope and try to pull the rope back to you - it will get stuck on the hole and you won't be able to pull it back in!

It turns out that you can endow the collection of all 'loops' (up to homotopy - which essentially corresponds to pulling the rope back in) with an algebraic structure, and it detects holes (which turn out to be not only a topological invariant, but a homotopy invariant). So this is useful for distinguishing spaces.

Ok, so suppose you remove the origin from xyz space. Now if you try to wrap a rope around it, the rope won't get caught on the hole - it will just slip off. So this idea fails to detect holes in 3 dimensions. So instead of a rope, we need a net! If you cast a net out around the hole, and try to pull the net back to yourself, it gets caught on the hole. And as we increase the dimension, we just need higher dimensional nets to find any higher dimensional holes that might be lurking, and these all can be endowed with an algebraic structure just like the 1-d case. You might expect that if we cast out a net of some dimension into a space of lower dimension (in a continuous manner), that we can't really catch anything because the dimensions will sort of cause the net to collapse in on itself.

You would be wrong.

For example, if we cast a 4-dimensional net onto a 2-dimensional sphere in the right way, it fucking detects a hole. But this isn't any hole like you've seen before, it has torsion. This means that I can wrap my net around this hole and successfully catch the hole - but if I wrap my net around twice, suddenly the hole can escape the net!

Of course, my description here isn't perfectly accurate - but it gives you a sense of the weirdness of this subject. Determining the algebraic structure that casting high dimensional nets into spaces is an impossibly difficult unsolved problem - we don't even know the answer for spheres, and many expect we never will.

I find algebraic geometry to be equally as interesting, but I don't have a nice analogy to describe the basic ideas.

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Re: Favorite Mathematics Field?

Postby djysyed » Sun Jan 28, 2018 7:21 pm

I really enjoyed learning about Abelian Categories, Cochain Complexes, Triangulated and Derived Categories. Now that I'm studying about the fundamental group and basic homotopy theory, I'm quite enjoying everything i'm learning.

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Re: Favorite Mathematics Field?

Postby petrokov » Mon Jan 29, 2018 2:05 am

smiledaniel wrote:the field of complex numbers of course, as it is the easiest algebraically closed field to do algebraic geometry hahahah

I think I prefer the surreals; after all, OP asked for a Field. :D

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Re: Favorite Mathematics Field?

Postby mathisawesome » Mon Jan 29, 2018 2:38 am

Algebraic topology. It's a demanding subject and one needs to spend some time understanding every single detail. It can be an extremely fascinating subject or it could be a frustrating maze of complicated maps that make no sense at all. I have grown to like symplectic geometry, which is the study of symplectic manifolds. It uses differential geometry and manifold theory along with some algebraic topology. :D

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Re: Favorite Mathematics Field?

Postby regularitylemma » Mon Jan 29, 2018 3:34 pm

Last edited by regularitylemma on Sat Feb 09, 2019 1:04 am, edited 1 time in total.

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Re: Favorite Mathematics Field?

Postby Bayesian1701 » Tue Jan 30, 2018 12:45 am

While statistics isn't exactly a field in mathematics it's close. I am a senior mathematics student applying to statistics PhD programs. I don't rergret majoring in math, but a graduate degree in math isn't for me. I see the value in pure mathematics but I dislike it personally. Statistics definitely has some quirks compared to math. Statistics is always wrong from the standpoint of you can never exactly estimate something without sampling the entire population. Statistics is also uncertain meaning you cannot be 100% sure of anything. But it has its benefits as well. You still use a lot of math, but you get to focus on applying math instead of proving it (which may be a pro or a con). You can study anything (political science, economics, biology) which I think is kinda cool. You take a problem, collect data, and build a model. If you want to do something math related but you learn that you do not enjoy pure mathematics statistics is a great option.

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