158,535 reputation
15347635
bio website math.berkeley.edu/~qchu
location Berkeley, CA
age 24
visits member for 4 years, 5 months
seen 3 hours ago

I am a third-year graduate student at UC Berkeley. I'm interested in the interplay between homotopy theory, quantum field theory, manifold topology, and higher category theory.


3h
comment Functorial Properties Preserved by Natural Isomorphism
This is the kind of precisification I was worried about: there are many basic and important properties of groups which are clearly preserved by isomorphisms but which cannot be stated in the first-order language of groups. For example, simplicity is not such a property, and neither is the property of being finite (e.g. because neither of these properties are preserved under ultraproducts).
3h
awarded  Revival
4h
comment Functorial Properties Preserved by Natural Isomorphism
This is clearly true but I've never felt the need to have a precise formulation of it. There are similar metatheorems in other parts of mathematics, e.g. "isomorphic groups have the same group-theoretic properties" and so forth. Roughly speaking any property that can be stated in the "language of categories" only involves statements that are preserved by natural isomorphisms, and maybe one can make this precise but it seems that any particular precisification runs the risk of excluding important examples.
16h
comment C*-algebra of polynomials?
@Sergei: no, of course you can adjoin whatever you want. When I write $A[t]$ and so forth I have in mind primarily the commutative case. If $A$ is noncommutative I don't know what the standard notation is for adjoining a new element with no commutativity requirements.
18h
revised Calculation of rate of interest on monthly installments.
edited tags
23h
comment Motivation for Definition of Derived Category
That depends on where you want to go.
23h
comment Motivation for Definition of Derived Category
I think localizing at the quasi-isomorphisms is much more like localizing at the weak equivalences, not the homotopy equivalences. If you want to invert homotopy equivalences you can just pass to the homotopy category. Passing to the derived category is different and harder.
1d
comment C*-algebra of polynomials?
Yes, but that's a choice you've made that you're hiding with the use of the term "polynomial." I'm not saying your definition isn't meaningful, just that using the term "polynomial" for it is misleading.
1d
comment Motivation for Definition of Derived Category
The basic motivation is that we want to identify objects with their resolutions, since to compute derived functors we end up replacing objects with resolutions anyway. To learn more you should probably read a textbook on homological algebra.
1d
comment C*-algebra of polynomials?
I'm saying it's misleading because polynomials of a given degree form a vector space, not an algebra; you've chosen a multiplication so that the variables you're adjoining are nilpotent, but you could choose others.
1d
comment C*-algebra of polynomials?
Dunno. I would call it $A[t]/t^2$, etc. A slight modification of this argument shows that $A[t]$ itself also cannot be given a C*-norm: $t$ has spectrum $\{ 0 \}$, and in a commutative C*-algebra an element with spectrum $\{ 0 \}$ must itself be zero, again by the commutative Gelfand-Naimark theorem.
1d
comment Interesting recurrence equation models?
Study the behavior of the Fibonacci sequence modulo a prime. Lots of good stuff there.
1d
answered C*-algebra of polynomials?
Dec
19
comment Functors that has a natural transformation from identity
The collection of all natural transformations $\text{id}_A \to F$ is one possible definition of the trace $\text{tr}(F)$ of an endofunctor $F$.
Dec
19
awarded  Constituent
Dec
19
comment What is $\Gamma(1+\Gamma(1+\Gamma(1+\dots)))$?
You want the limit, if it exists, of the sequence $x_n = \Gamma(1 + x_{n-1})$ with, say, initial condition $x_0 = 1$. This is perfectly well-defined. It's clear that if the limit $L$ exists then $L = \Gamma(1 + L)$, and if there are multiple $L$ with this property then in general the initial condition will determine which $L$ actually ends up being the limit, again if it exists.
Dec
19
comment What is the principal bundle structure of $O(n)$?
Yes, your guess is correct.
Dec
19
comment Continuity of Modified Hom Functor
The key statement is that limits in functor categories are computed pointwise.
Dec
19
comment What's wrong with this trivial proof that every element of a compact Lie group is contained in a maximal torus?
I agree. If connectedness followed for formal reasons then the exponential map would always be surjective, which isn't true.
Dec
19
comment Why is this generator an element of the Cartan subalgebra of SU(3)?
I mean conjugation by an element of the Lie group $\text{SU}(3)$ (usually when a physicist uses this notation it means the Lie algebra $\mathfrak{su}(3)$). See en.wikipedia.org/wiki/Adjoint_representation.