62

As a department chair, I can say you aren't alone. These situations come up all too often. Please do reach out to your department chair, graduate program director or grad student ombudsperson if your institution has one. We want to know when our faculty are behaving badly and often we can help.


40

I strongly disagree with the "find a list of open problems" approach. Usually open problems are quite hard to make progress on, and I'm thoroughly unconvinced that good research is done by tackling some hard but uninteresting problem in a technical area. That being said, of course solving an open problem is really good for academic credentials. But that's ...


28

You should switch advisors. Since you are independently writing papers and have a track record, it should be possible to find a fair-minded theory advisor in a different technical area who is willing to do the administrative aspects of handling a PhD thesis. Your department chair should also help in this matter.


27

There have been recent developments in dependent type theory which relate type systems to homotopy types. This is now a relatively small field, but there is a lot of exciting work being done right now, and potentially a lot of low hanging fruit, most notably in porting results from algebraic topology and homological algebra and formalizing the notion of ...


25

Algebraic geometry is used heavily in algebraic complexity theory and in particular in geometric complexity theory. Representation theory is also crucial for the latter, but it's even more useful when combined with algebraic geometry and homological algebra.


17

Simon Peyton Jones has an excellent web page devoted not only to advice on writing introductions, but whole papers, and there is a cool video as well. On page 18 of his slides he says that the purpose of an introduction is to: Describe the problem. State your contributions. He then goes on for a while to explain what precisely that means in pracrtice. But ...


15

Your knowledge of field theory would be useful in cryptography, while category theory is heavily used in the research on programming languages and typing systems, both of which are closely related to the foundations of mathematics.


13

You may be working with rude elitists, but from my experience, being reluctant to explain technical details depends more on the context than on the person I am talking to. I would usually avoid giving technical details of a proof/algorithm over lunch or in the corridor even if my interlocutor were Alan Turing himself. The reason is that details heavily ...


12

Understanding papers outside your field can be surprisingly difficult --- every field develops its own specialized terminology and shared background knowledge. A much better advice I once received from a very productive researcher was: Try to attend research seminars outside your field. These are typically more accessible than "cold" paper reads, and you ...


11

Field theory and algrebraic geometry would be useful in topics related to error correcting codes, both in the classical setting as well as in studying locally decodable codes and list decoding. I believe this goes back to work on the Reed-Solomon and Reed-Muller codes, which was then generalized to algebraic geometric codes. See for example, this book ...


10

Speaking as a theorist who has occasionally collaborated with systems researchers, mathematical details are usually the very last thing I want to talk about! It's really, really, easy to formalize the wrong thing, and in addition to being a huge waste of time, it's really dispiriting. Mathematics is a needle, not a hammer, and each jab of the needle is very ...


9

Another avenue for interesting exploration is when you're trying to understand the proof of a theorem or lemma. If you dig really deep into the proof to understand exactly how it works (and I don't mean literally, in the $A\Rightarrow B$ sense, but intuitively), you'll often realize that the proof is more ungainly than it needs to be. Asking whether it can ...


9

One way I have often found theory problems to work on is by reading about an area and trying to figure out exactly what the state of the art is on a problem. Invariably, some basic questions end up being left unanswered, and that’s where I will start my research. Such questions are sometimes left unanswered not because they are too difficult, but because ...


9

Scenario 1: You spend several months tinkering around with colorings, not reading any literature. After many failed attempts you finally discover one that works. Before you can write a paper about it, you need to study literature so that you can properly situate your result. You feel bitter and lonely for some reason. Scenario 2: You read the literature, but ...


8

Almost certainly there are lists of open problems in your particular subfield. Find them and read them. Although it's rather unlikely you will be able to solve these problems --- at least right away ---, use them as a starting point. Can you solve some particular cases? Can you solve a less general problem? Can you show a more general problem is ...


8

Sure, here's a basic checklist. It would make for a very dry read to actually follow these to the letter, but maybe you should first try to write extremely formally, then see where it's reasonable to relax the writing without risk of misunderstanding or vagueness. Preface. The high-level goal of formality is to make your proof closer to "machine-checkable". ...


8

David Hilbert is a renowned mathematician. He put forth a list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. I just want to quote part of Yuri Manin interview entitled "Good Proofs are Proofs that Make us Wiser" about Hilbert and his list: This year’s International Congress is the last ICM in this century. Do ...


8

Unfortunately, there is little you can do -- a PhD adviser exercises a great deal of control over the students' careers. I think at this point, you're better off placating her and adding her as a co-author. When you get your degree, avoid whenever possible requesting recommendation letters from her. A major piece of advice I offer all beginning graduate ...


7

There are some problems in computational learning theory, machine learning and computer vision that can be solved using commutative algebra and algebraic geometry. For instance, convergence of the Belief Propagation algorithm, a message passing algorithm for Bayesian inference, can be formulated in terms of characterizing the affine variety of system of ...


7

I am going to try and answer this with my limited experience. Disclaimer I am just a senior phd candidate myself. The question you are asking is by no means a trivial one nor are you the only one wondering about it. Every single phd student, in almost any field, that preceded us and that will succeed us, has/will wonder the same. So, as a first piece of ...


6

I struggled with the same question while pursuing my undergrad in CS. I performed a fair amount of research, but currently work as a software engineer. I can tell you that for the most part, you won't do too much theoretical CS in the industry (assuming a typical SE, not something tailored for R&D or the like). You may want to consider looking into R&...


6

I agree with Sasho Nikolov's comment: there are two issues here, whose outcomes might influence one another, but they really are two separate issues: writing better proofs, and coming up with good research questions. (1) For writing better proofs, practice practice practice. But also get feedback. If you know this is a weakness, don't be afraid to ask for ...


6

Have you thought about looking at computer algebra? Axiom is a computer algebra system where the type system is modelled after Category Theory (or Universal Algebra, depending on your view). There are two further derivatives of Axiom FriCAS and OpenAxiom. If you're interested in Category Theory, then the type system may be one thing to look at. In Axiom, ...


6

You mention TCS+, and as Aryeh's answer suggests talks and seminars may be a much easier and accessible way to learn about other results. From my point of view (inherently subjective, and not too experienced, so take with a grain of salt) reading papers outside your subarea/subfield may be a good idea*, but going to as many talks and seminars as possible is ...


5

I would say there is no "one best/most appropriate route" to becoming a theoretical computer scientist, so it is more a matter of your personal tastes, and even what part of theoretical computer science you are most interested in, which you may not even know yet. I know of great TCS researchers who started out in pure math, and great TCS researchers who ...


5

Finding good problems to work on is a very difficult task, almost every graduate student struggle with it. That is one of the main reasons you have an adviser who can use his experience and help you find problems that worth working on and can lead to some results in reasonable time. It is also a matter of trial and error. You have to fail a lot till you get ...


5

Here are a lot of interesting answer, but nobody mentioned that every language $L \subseteq X^{\ast}$ is naturally associated with a monoid structure via the Nerode-Myhill congruence relation. The following people have used this algebraic view in the case of regular languages: Samuel Eilenberg on Automata Theory, Jean Berstel, Jean-Eric pin, Marcel ...


5

As a grad student make use of the opportunity to take classes outside of your area and outside of CS. You can audit classes or attend a course partially. This is different from reading papers.


5

There are no metrics, but an excellent discussion of many concurrency models, in Tony Garnock-Jones PhD thesis. See the (HTML version of the) chapter "Approaches to coordination". This studies concurrency models with a particular focus, namely how information is exchanged for coordination.


4

It's a generic question so I'm going to give a generic answer: Speak with your academic adviser! You do have one, right? Are you familiar with his/her research projects? Do any of them interest you? If none do, perhaps you should look for a different adviser...


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