Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?
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1$\begingroup$ Do you want representation theory problems or enumerative geometry problems? $\endgroup$– Ben McKayCommented Jan 29, 2020 at 12:50
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$\begingroup$ Dear Ben, one of them or both will be ok. Since I am interesting in both fields and they have some connections to each other. Thank you so much for your advice! $\endgroup$– MihawkCommented Jan 29, 2020 at 12:55
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4$\begingroup$ I guess that for representation theory you want to be more specific since different people mean different things when they say "Representation Theory". For some open problems in the Representation theory of finite dimensional algebras and quivers, see e.g. math.uni-bonn.de/people/schroer/fd-problems.html. $\endgroup$– Julian KuelshammerCommented Jan 29, 2020 at 13:10
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1$\begingroup$ The work of J.M. Landsberg on matrix multiplication draws from representation theory and complex algebraic geometry, but maybe not much enumerative geometry, and is driven by conjectures from computer science about the speed with which computers can multiply matrices. $\endgroup$– Ben McKayCommented Jan 29, 2020 at 14:37
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5$\begingroup$ I'm kind of amazed this question is still open, given how strict math overflow usually is. Regardless of what happens, just a bit of advice Khanh: this question is way too broad. Even if your question was faithful to the title, it would be far too broad. The only real answer is yes, there are many conjectures and open problems in representation theory. The more thought you put into your question, the better answers you will get. $\endgroup$– Andy SandersCommented Jan 29, 2020 at 18:29
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2 Answers
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The Clemens conjecture in enumerative geometry: a general quintic threefold has only finitely many rational curves in each positive degree.
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There are many open, and seemingly deep, conjectures in modular representation theory (or block theory) in connection with enumerating representation-theoretic invariants: a start of a list might be : Brauer's $k(B)$-problem, the Alperin-McKay Conjecture, the Alperin Weight Conjecture, Dade's conjectures, Isaacs-Navarro conjecture. Gabriel Navarro has several recent survey papers discussing these and other conjectures.
In a different part of (modular) representation theory, with perhaps a more geometric flavor, there are problems such as the Lusztig Conjecture (now known to be false in its original formulation), and work of Geordie Williamson.
As noted by Julian Kuelshammer, Representation Theory is a vast subject, and it might be helpful to point out which specific areas you are most interested in (I only mention two facets of the subject which are most familiar to me).
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4$\begingroup$ For some specific conjectures from the Lusztig-Williamson school, see: arxiv.org/abs/1703.05898 $\endgroup$ Commented Jan 29, 2020 at 15:07