# Ryan Reich

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## Registered User

 Name Ryan Reich Member for 2 years Seen 1 hour ago Website Location Los Angeles, CA Age 30
I am a postdoc at U Michigan (Ann Arbor). I was a graduate student at Harvard, where my advisor was Dennis Gaitsgory, and I graduated in the spring of 2011. In short, I do geometric representation theory, which in the case of my thesis means perverse sheaves on the affine grassmannian with some gerbes thrown in, but could also mean something much more elementary. I'm also active on tex.stackexchange.com.
 1h comment objects which can’t be defined without making choices but which end up independent of the choiceIsn't it true that the theorem "every field has an algebraic closure" is somewhere between plain ZF and ZFC? That is, you seem to need to make some arbitrary choices (of representatives of each isomorphism class of algebraic extensions) to form a general algebraic closure, even though it's unique. 1h comment objects which can’t be defined without making choices but which end up independent of the choice@zeb Does it count as making choices if you do so by constructing $\operatorname{Ext}^n$ the usual way and showing it's isomorphic to Steven's definition? After all, it's not the definition that contains the choices, but simply a proof that the definition has a certain property. 1h comment objects which can’t be defined without making choices but which end up independent of the choice@paulgarrett It's not that simple. Running through the construction of a composition series in my head, the best precise statement I can come up with is "if simple, then the whole group; else, a composition factor of a normal subgroup or a quotient". The problem is that normality is not transitive. This also obstructs the otherwise appealing definition "a normal simple subgroup or a composition factor of the quotient by the normal subgroup (the socle) generated by all normal simple subgroups". 2h comment What does a singular simplex with real coefficient mean I edited your question to make it more clear; tell me if I've changed the meaning. 2h revised What does a singular simplex with real coefficient mean Extensive rewrite for clarity 16h answered objects which can’t be defined without making choices but which end up independent of the choice 19h asked What is the meaning of the cospecialization map? 20h asked Fiberwise acyclic, locally acyclic morphisms 2d comment Is there any proof that you feel you do not “understand”?Toby, FWIW, I think the book we used in high school was by a set of authors named Larson, Hostetler, and Edwards. It still seems to be successful, but of course Stewart is number one, and I don't remember how he does it. Now, as to why one would choose this order: quite possibly, because the first theorem as I stated it is the one used all the time in computations, while the second one is sort of a curiosity at the level of high school calculus. Nowadays, Stewart seems to have put a lot of thought into his logic, and my #1 does (modulo hypotheses) follow from #2, so switching makes sense. May18 answered Terminology: complex of sheaves with cohomology sheaves concentrated in degree zero May17 comment Is there any proof that you feel you do not “understand”?That's actually a great point, since Euler's method is the physically intuitive interpretation of the fundamental theorem, with $f'$ = velocity and $f$ = position. May17 answered Is there any proof that you feel you do not “understand”? May16 comment How to memorise (understand) Nakayama’s lemma and its corollaries?@Toink: Because by flatness we know that the sequence you wrote is actually of the form $0 \to K \otimes k \to R^n \otimes k \to M \otimes k \to 0$. Therefore $K \otimes k = 0$, so by Nakayama $K = 0$. Without flatness we'd only have $K \otimes k \to R^n \otimes k \to M \otimes k \to 0$, and thus $K \otimes k \twoheadrightarrow \operatorname{ker}(R^n \otimes k \to M \otimes k) = 0$, which is exactly zero information about $K$ :) May13 comment Applications of visual calculusI honestly didn't think I'd ever learn anything about calculus this new to me at this stage in my education. May11 revised Show that this ratio of factorials is always an integerimprove title; added 2 characters in body May5 comment Advantages of the sequence definition of limitsThis is a good response to Christian Blatter's answer. It may be harder to prove continuity, but by the same token, it is much easier to disprove it. May3 comment the position of stringsShouldn't you be asking, rather than demanding? May1 comment Reference request - localisation de g-modulesI've got an English translation I TeXed for myself. Not sure if I am "supposed" to distribute it, though. May1 revised Is there a “right” proof of Riemann’s Theta Relation?Apparently you have to use \cr Apr23 comment Intuitionistic logic as quantization of classical logic?Okay, one more comment. The activity you say classical mathematicians refuse to acknowledge is what I mean by identifying what we find important: suppose we tried to prove the existence of bases constructively. We'd find we need something to work with, and that would lead us to finitely-generated vector spaces, where the theorem is constructed. We might then ask "so what are infinite-dimensional vector spaces?" and in learning the difference, decide whether the concept is fundamentally cool or merely abstract nonsense. I posit that no one works in a field they honestly think is the latter. Apr23 comment Intuitionistic logic as quantization of classical logic?@Paul: It's just a philosophical observation. Non-commutative algebra differs from commutative by rejecting certain constraints on the structure of the theory. Some reject non-commutative rings since comm. rings have a geometric side, as per their personal mathematics. Likewise some reject non-associative "groups" because of the rep.-theoretic connection to regular groups. Some (you?) specialize in constructivizing classical results. Non-EM math has a constraint: directness; and a connection: plausible intuition. Thus, it is like commutative algebra. Apr23 comment Intuitionistic logic as quantization of classical logic?Of course, the proof style " Thm: P; Proof: Suppose $\lnot P$; proves P ; Contradiction" is inane and stems from laziness. And the prevalence of EM obscures the constructive truth of those propositions that may be demonstrated as in Bishop-Bridges. Because of it we may miss that some non-constructive theorems (existence of bases) have constructive weakenings (when finitely generated). Analyzing the difference helps identify what we find important in our personal math; perhaps non-constructive math is the real analogy to non-commutative algebra, not the reverse. Apr16 revised Does any lower bound on proofs of FLT improve Shepherdson 1965?insert connecting words to decrease humorous ambiguity Apr12 comment What is the purpose of section 3 of BBD?Note that the filtered derived category is of practical use elsewhere: for example, Beilinson's "On the derived category of perverse sheaves", where he invokes a functor from the eponymous category to the usual derived category, constructed using the filtered derived category. It is this functor that is shown to be an equivalence. Granted the definition is pretty technical, but even just for the sake of this result it seems like it is a valid piece of the theory and not just architecture. Apr9 comment Elliptic curve over a scheme is a group scheme?A summary as I understand it: "The same construction that makes an elliptic curve over a field, namely $x \mapsto [x] - [e]$, into a group scheme, works for a relative elliptic curve as well." Then there are the details, of course. Apr5 revised Is there a “mathematical” definition of “simplify”?more on polynomials; deleted 2 characters in body Apr5 answered Is there a “mathematical” definition of “simplify”? Apr4 comment Explaining the concept of projective space: notes for studentsI would have been very surprised if many of your students had heard of projective space. There is no standard course that teaches geometry in the Euclidean sense outside of 9th grade, or in the case of non-Euclidean geometry, at all. Occasionally some geometer on the faculty will opt to give such a course, but who knows? Apr4 revised Does Euclidean space have a compact factor?fix tex??? Apr3 comment Where to buy premium white chalk in the U.S., like they have at RIMS?This answer amazingly manages not to be spam despite being an advertisement. Apr3 comment Where to buy premium white chalk in the U.S., like they have at RIMS?I think you should edit this into your last answer; it's not really different. Mar31 revised Difference between parallel transport and derivative of the exponential mapfix TeX Mar31 comment Domain of the wedge product in Little Spivak@KConrad: Too Long; Didn't Read. I'm never sure if writing it is condescending or accommodating. Mar31 comment Confusions over the definitions of universal bundle and characteristic classHom(F,C), right? Mar22 comment Demystifying complex numbersI just used basically this in a first course in differential equations to prove Euler's formula for them. One of us thought it was really cool, anyway... Mar19 comment CosheafificationYou mean $F \colon \mathrm{Open}(X) \to \mathrm{Vect}^\mathrm{op}$, I think. You could alternatively reverse the category $\mathrm{Open}(X)$, but that destroys the open covers and presumably creates more problems. Mar17 comment When are two operators simultaneously diagonalisable?@David: This is where my understanding of topological linear algebra (e.g. analysis) fails, but the answer to your first question is yes for purely formal reasons. For the second question I refer you to Robert Israel's answer concerning whether positivity is at all useful. The "invariant" thing is just this: suppose $P$ is a property of linear operators implying diagonalizability and preserved under restriction to invariant subspaces. Then $P$ works in place of "compact and self-adjoint" in the above argument. Mar17 answered When are two operators simultaneously diagonalisable? Mar17 comment When are two operators simultaneously diagonalisable?@David: Two diagonal operators definitely commute, and two matrices commuting is basis-independent. Mar16 comment Reference for a nice proof of “undetermined coefficients”Oh, I have that book. I'll go check. Mar16 comment Reference for a nice proof of “undetermined coefficients”Ah. I think that boils down to Terry Tao's proof, then. Mar16 comment Reference for a nice proof of “undetermined coefficients”It is, but I don't know that term. I'm not really asking for a suggestion "what's the best proof of undetermined coefficients?"; I gave two others previously in the course. I was testing out this one, which is (in my opinion) a great advertisement for almost every technique and idea introduced to attack linear equations. Mar16 revised Reference for a nice proof of “undetermined coefficients”remove repetition Mar16 comment Reference for a nice proof of “undetermined coefficients”@Gerald: I'm curious about your favorite proof, so perhaps you could give the link in a comment? Mar16 comment Reference for a nice proof of “undetermined coefficients”This is a very nice, not at all messy "proof by integration", but I was more asking for a reference to the argument I gave. Or, can I consider this post an earliest reference to the proof you give? :) Mar16 asked Reference for a nice proof of “undetermined coefficients” Mar7 awarded ● Popular Question Feb26 awarded ● Nice Answer Feb13 revised Computer Algebra Errorsprettify latex Jan27 answered Are there any books that take a ‘theorems as problems’ approach?