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Yes, the place of publication can absolutely hurt your reputation. Specifically, I can tell you from having served on many hiring committees (and from conversations with professors at other universities about their hiring committees and tenure processes), that publications in predatory journals can hurt you. I'm talking specifically about journals whose ...

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(1) It depends a lot on the field. In fields that rely on specialized techniques discovered relatively recently or known only to a few, or fields where the questions involve recently-introduced objects, it's much easier to keep abreast of current research. On the other hand, in fields with elementary questions that could have been studied a hundred years ...

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Doesn't a symmetry argument make this question rather simple? If $n$ is odd, the second player (blue) can always mirror the first player's (red) moves (reflected through, or rotated by $\pi$ radians around the origin of the grid, i.e. its center vertex), so the first player loses. If $n$ is even, the first player can draw one of the diagonals of the ...

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$\mathbb{Z}$ is not first-order definable in $\mathbb{R}$ (as an ordered field) or $\mathbb{R}^n$ (as a vector space). That is, they cannot even "talk about" the integers. As proven by Tarski, the theory of real closed fields is decidable. But even the weak theory of arithmetic Q is essentially undecidable (any consistent extension of it is undecidable).

10

As was echoed in the comments by YCor and Mikhail Borovoi, the question of originality (especially for results that could have been stated a long time ago) is one that is relevant to all mathematicians. An interesting recent example that comes to mind is this story on Terence Tao's blog. So how can you guard yourself from this? I think that there are two ...

9

“Conway’s knot is not slice”, by Lisa Piccirillo, Annals of Mathematics 191-2 (2020) 581-591, settled a problem (we can call it a conjecture) which was at least four decades old. Read in informal account of the result here: https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/

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The Axiomatic Quantum Field Theory from the 1950's has been rebranded as Algebraic QFT (keeping the same abbreviation), and the emphasis has shifted somewhat from quantum fields to local observables. The Wightman axioms for fields are then replaced by the Haag–Kastler axioms for the algebra of observables. For two relatively recent overviews see Current ...

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If you want to build a reputation as a mathematician, post your preprint on the arXiv (this is a preprint server, not counted as publication, the posts are not refereed, and there are essentially no "barriers"). Then send your paper to a mainstream mathematical journal. Avoid those journals which are not reviewed in Mathscinet. (Recently, many journals ...

7

Two of my favorite recreational papers are: J-F Mestre, R Schoof, L Washington and D Zagier, Quotients homophones des groupes libres. Homophonic quotients of free groups Experiment. Math. 2 (1993), no. 3, 153–155. Norman Wildberger, Real fish, real numbers, real jobs, The Mathematical Intelligencer 21(2), (June 1999), 4-7. Here is a youtube recording of ...

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I just realized that the OP links to a YouTube video and to some slides, but the two don't match—they're two different talks by Buzzard. For completeness, let me therefore mention some results by James Arthur, which are mentioned in the linked slides but not the linked YouTube video. On page 13 of Abelian Surfaces over totally real fields are ...

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This MO question The Riemann zeros and the heat equation describes the Newman conjecture. Very briefly, a deformation parameter is introduced into an integral representation of the Riemann zeta function, creating a function of two variables which satisfies the backward heat equation. Newman made the conjecture that any infinitesimal deformation with this ...

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Richard Friedberg, then an undergraduate pre-medical student, independently solved Post's problem (of whether there are intermediate Turing degrees) by the priority-injury method. This was a significant open problem at the time, so the result made news: 1956 news article "Senior solves logic problem, astounds mathematicians" In Gödel's now famous letter ...

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If you just started your graduate study, you probably have an adviser. If you don't have one yet, try to find one as soon as possible. Anyway, in most universities that I now an adviser is required to defend a PhD. Then show your result to your adviser. Even if it is from a different area from his/her scientific interests. Adviser will probably make a ...

3

It's not that hard nowadays to do a fairly exhaustive literature search to make sure something hasn't been done before. This is a skill which can take a bit of time to develop but will help and saves a lot of time in the long run, especially if you do not have someone you can show the result to. This will also help you to be independent, something which is ...

3

Consider a root system of type D. The Hasse diagram is built up by writing, as bottom row, a node for each simple root, and then on each row above, connect up two roots if their sum is a root. The Hasse diagram has an obvious symmetry in the last two roots (7 and 8, in the picture), from the symmetry of the Dynkin diagram. We can picture that symmetry as ...

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Mohammed Ghomi has a beautiful list of open problems on curves and surfaces, but perhaps most of them are out of reach.

2

May I offer the four-vector notation as an example from physics? Quoting Feynman: The notation for four-vectors is different than it is for three- vectors. [...] We write $p_\mu$ for the four-vector, and $\mu$ stands for the four possible directions $t$, $x$, $y$, or $z$. We could, of course, use any notation we want; do not laugh at notations; ...

2

The direct connection between Poisson bracket and non-commutativity is pretty clear, at least if you agree that the (later introduced) Lie bracket $[X,Y]$ of vector fields measures the non-commutativity of their flows. The original setting is symplectic manifolds, where each function (“hamiltonian”) $a$ defines a vector field $X_a$ (its “symplectic gradient”)...

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In 2017, a counterexample to Conway's $1,000 conjecture “Climb to a Prime” was discovered by James Davis, “not a mathematician by any stretch”. (details) Also in 2017, Dmitry Zakharov (then a 9th grader!) published a new bound for the 1962 problem of Danzer and Grünbaum, particularly improving a result of Erdős. His proof is elementary and takes just half a ... 2 For me the most basic compact spaces are$\{x_n: n \in \Bbb N\} \cup \{x\}$when$x_n \to x$, (the countable cofinite space is a special case), of course all finite spaces, and all ordered topological spaces with a suprema for all subsets ("very order complete", usual order completeness being: "every nonempty subset that is bounded above has a supremum"). (... 2 I like to use figures to represent quantities. For example, in this recent preprint, my coauthor and I use simple diagrams to represent certain weighted sums (polynomials). Writing out the sums explicitly would be extremely cumbersome to parse, and any sane reader would just convert it back to a generic figure anyway, in order to understand the sum. 1 G. Eric Moorhouse's list of open problems, mainly focusing on finite geometry. Douglas B. West's list of open problems in Graph theory. The open problem garden. 1 Elaborating on Michael Greinecker's comment: if$X$is a compact Hausdorff space then the map$i \colon X \to \Pi_{C(X,I)} I$given by$i(x)_f = f(x)$, where$I = [0,1]$, is a homeomorphism onto its image. So every compact Hausdorff space can be expressed as the closed subspace of the product of closed unit intervals, which is a partial negative answer to ... 1 Richard Stanley’s symbol for number of ways to make choices with replacement. Looks like a binomial coefficient but with double parentheses. (More here.) It's a calculation that comes up frequently -- I became more aware of just how frequently it comes up when I started giving it it's own symbol -- and it helps to give it its own notation, even though it ... 1 I understand the question as a request for pointers in the literature to research in the discretization of spacetime. There are two reasons why this is an active research topic, a fundamental and a practical reason: Fundamentally, spacetime might be discrete at the smallest levels (Planck scale); practically, to simulate relativistic quantum field theories (... 1 If you are interested in the largest prime factor of$ab(a+b)$, there is xyz conjecture. Smooth solutions to the abc equation: the xyz Conjecture This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor ... 1 A "hot spot" on a sufficiently regular domain is an interior extremum of the first nonconstant Neumann eigenfunction of the Laplace operator. The Hot Spots conjecture states that hot spots do not exist on convex planar domains. Chris Judge and Sugata Mondal have settled the Hot Spots conjecture in the affirmative for all Euclidean triangles: Euclidean ... 1 The Homology of Iterated Loop Spaces (Thomas Joseph Lada, J. Peter May, Frederick Ronald Cohen), The Geometry of Iterated Loop Spaces (J. Peter May),$E_\infty$ring spaces and$E_\infty$ring spectra (J. Peter May),$H_\infty\$ ring spectra and their applications (R. R. Bruner, J. Peter May, James McClure), Equivariant stable homotopy theory (L. ...

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For beginners 'A Course on Geometry Group Theory' by Brian Bowditch is nice although quite terse in places, but good for 'setting the scene'.

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The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009.

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