Jason Polak
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 Apr3 comment Examples of real world situations where mathemematical rigour is needed A level of rigour is definitely required for applying statistical results to the real world. You could get an idea of this at the cross validated site.... Apr3 comment Being a good listener in math Informal setting: ask lots of questions. Lecture: ask some question anyway, but if you have a lot of them that would disrupt the lecture, record/writing everything down, study it and email the speaker later. Mar27 answered Why are cohomotopy groups defined only up to dimension $2m-2$ and not $2m-1$? Mar20 comment Multiple categorifications of structures @QiaochuYuan: Yes. That is true, it is too narrow to include the finite sets example. However, I wasn't aiming at a good general definition of categorification, only to provide some examples for some definition of categorification. Mar20 answered Multiple categorifications of structures Mar16 comment Show $X$ is a H-space? Hint: let $H:Y\times I\to Y$ be a homotopy that shows $Y\to Y\times Y\xrightarrow{m} Y$ is homotopic to the identity, where $Y\to Y\times Y$ is $y\mapsto (*,y)$. Can you use this to show the H-space property for your map $n$? Mar16 comment Show $X$ is a H-space? By definition, that $n$ satisfies the conditions for an H-space is exactly that $n$ composed with each $X\to X\times X$ given by $x\mapsto (x,*)$ and $x\mapsto (*,x)$ gives you a map homotopic to the identity, so that is all you need to show. Feb1 comment How to show that homotopy of chain maps respects composition? @Exterior: Thanks, I just mistyped the subscript 2 and it propagated. I fixed it. It is supposed to be $g_1$ the whole way through. Feb1 revised How to show that homotopy of chain maps respects composition? mistyped an index Jan19 comment Why does the author prefer function names after arguments? Even now, after using the usual convention for so long, I still have to do a bit of mental processing when writing functional composition one way in text but the other on a diagram. Jan18 answered Book recommendation for Measure Theory Jan15 comment Why any short exact sequence of vector spaces may be seen as a direct sum? More generally, every short exact sequence splits is equivalent to every module being projective; i.e. the ring is finite direct product of matrix rings over division rings. Jan9 answered Subring which is not an ideal? Jan4 comment Modern algebra and set theory: ZFC vs. NBG As an algebraist, I feel like this is only an issue when writing foundations. At other times, the set theory one uses is almost always out of the way. Jan3 answered Proof of $\mathbb Z/n\mathbb Z\bigotimes_{\mathbb Z}\mathbb Z/m\mathbb Z \cong Hom(\mathbb Z/n\mathbb Z, \mathbb Z/m\mathbb Z)$ Dec24 comment Is Hoffman-Kunze a good book to read next? I agree with this post. I'd like to add that I read much of the book myself as an undergrad and the only disappointing part of this book is that it has fewer examples than I would have liked. For instance, I would have enjoyed more worked examples on actually finding the Jordan normal form. In this sense I believe the book falls short. Dec17 comment Is my understanding for group algebra correct? Can you explain how you are viewing $Aut_k(V)$ as an algebra? Nov28 answered Restriction of scalars of a torus Nov28 comment Prerequisites for studying Homological Algebra @Sayan: I think you probably do have enough. However, Herstein's book doesn't have much module theory, so if you find the going a bit tough you could keep a book on modules at hand. For instance, the treatment of projective and injective modules in Weibel is a bit terse, and some of the constructions could seem a bit unmotivated if you've not seen many examples of modules before. Nov23 comment Homotopy Groups for Categories All the homotopy groups of an essentially small category can be defined and this is well-known. You take the homotopy of the associated nerve. This is one way for instance to define the algebraic K-theory of a ring: one takes the homotopy groups of the nerve of the Q-construction applied to the category of fg projection modules over the ring.