Ryan Reich

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Name	Ryan Reich
Member for	2 years
Seen	1 hour ago
Website	ryancreich.info
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.

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1h	comment	objects which can’t be defined without making choices but which end up independent of the choice Isn'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.
May 18	answered	Terminology: complex of sheaves with cohomology sheaves concentrated in degree zero
May 17	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.
May 17	answered	Is there any proof that you feel you do not “understand”?
May 16	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 $K$ :)
May 13	comment	Applications of visual calculus I honestly didn't think I'd ever learn anything about calculus this new to me at this stage in my education.
May 11	revised	Show that this ratio of factorials is always an integer improve title; added 2 characters in body
May 5	comment	Advantages of the sequence definition of limits This 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.
May 3	comment	the position of strings Shouldn't you be asking, rather than demanding?
May 1	comment	Reference request - localisation de g-modules I've got an English translation I TeXed for myself. Not sure if I am "supposed" to distribute it, though.
May 1	revised	Is there a “right” proof of Riemann’s Theta Relation? Apparently you have to use \cr
Apr 23	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.
Apr 23	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.
Apr 23	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.
Apr 16	revised	Does any lower bound on proofs of FLT improve Shepherdson 1965? insert connecting words to decrease humorous ambiguity
Apr 12	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.
Apr 9	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.
Apr 5	revised	Is there a “mathematical” definition of “simplify”? more on polynomials; deleted 2 characters in body
Apr 5	answered	Is there a “mathematical” definition of “simplify”?
Apr 4	comment	Explaining the concept of projective space: notes for students I 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?
Apr 4	revised	Does Euclidean space have a compact factor? fix tex???
Apr 3	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.
Apr 3	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.
Mar 31	revised	Difference between parallel transport and derivative of the exponential map fix TeX
Mar 31	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.
Mar 31	comment	Confusions over the definitions of universal bundle and characteristic class Hom(F,C), right?
Mar 22	comment	Demystifying complex numbers I 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...
Mar 19	comment	Cosheafification You 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.
Mar 17	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.
Mar 17	answered	When are two operators simultaneously diagonalisable?
Mar 17	comment	When are two operators simultaneously diagonalisable? @David: Two diagonal operators definitely commute, and two matrices commuting is basis-independent.
Mar 16	comment	Reference for a nice proof of “undetermined coefficients” Oh, I have that book. I'll go check.
Mar 16	comment	Reference for a nice proof of “undetermined coefficients” Ah. I think that boils down to Terry Tao's proof, then.
Mar 16	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.
Mar 16	revised	Reference for a nice proof of “undetermined coefficients” remove repetition
Mar 16	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?
Mar 16	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? :)
Mar 16	asked	Reference for a nice proof of “undetermined coefficients”
Mar 7	awarded	● Popular Question
Feb 26	awarded	● Nice Answer
Feb 13	revised	Computer Algebra Errors prettify latex
Jan 27	answered	Are there any books that take a ‘theorems as problems’ approach?

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