302 reputation
18
bio website math.rice.edu/~andyp
location Houston, TX
age 35
visits member for 2 years, 4 months
seen 18 hours ago

associate professor at Rice University


Dec
16
awarded  Caucus
Aug
16
comment Making new sense of the three-body problem in the light of Maryam Mirzakhani math contributions
@semiclassical : please don't send questions like this to MO, where it will be quickly closed.
Jun
21
comment examples of polyclic groups
@noah s: So you don't consider zero-dimensional manifolds to be manifolds?
Jun
2
answered Transitive action of $\text{SL}(n,\mathbb Z)$
May
31
awarded  Yearling
May
26
awarded  Editor
May
26
revised Homology subgroups generated by non-intersecting cycles
added 183 characters in body
May
26
comment lifting a product of commutators of standard generators on 2-manifolds
@FilipParker : Fix a piecewise-linear structure on your surface and homotope the curve such that it is piecewise-linear (this is trivial; just fix a bunch of points on it and "pull them tight"). Then jiggle the vertices a little so that all of the line segments in your curve intersect transversely.
May
26
answered Homology subgroups generated by non-intersecting cycles
May
12
awarded  Commentator
May
12
comment lifting a product of commutators of standard generators on 2-manifolds
@FilipParker : My last comment constitutes a complete proof. If you don't understand it, I recommend meditating on the part of any book on algebraic topology that discusses changing the base point in the fundamental group. I remember Massey's book having a particularly nice treatment of this.
May
11
comment lifting a product of commutators of standard generators on 2-manifolds
@FilipParker : The homotopy lifting property tells you that that a loop $f$ lifts to a closed curve if and only if any curve which is homotopic to $f$ while fixing the base point lifts to a closed curve. Freely homotoping the curve lets the base point move, and thus corresponds to conjugating $f$.
Apr
26
comment lifting a product of commutators of standard generators on 2-manifolds
(continued) The second is that the surface is of this form. This can be derived from the classification of surfaces. I recommend perusing Section 1.3 of Farb-Margalit's "Primer on mapping class groups" for the details. 3. You are correct that these are typos; thanks for pointing them out!
Apr
26
comment lifting a product of commutators of standard generators on 2-manifolds
@FilipParker : 1. The surjection depends on $x$; for example, if you are dealing with $\alpha_1^7 \beta_1$ then you choose the homomorphism that takes $[\alpha_i]$ to $0$ for all $i$, that takes $[\beta_i]$ to $0$ for all $i \neq 1$, and that takes $[\beta_1]$ to $1$. 2. There are 2 points here. The first is that this is a presentation for $\pi_1$ of a punctured surface, which is discussed in any book that deals with surfaces and the fundamental group (e.g. Massey's book on the fundamental group). (continued)
Apr
25
answered lifting a product of commutators of standard generators on 2-manifolds
Apr
24
comment why say complex multiplication of elliptic curves is beautiful
You are wrong about the definition of an abelian extension. The class group is (by its very definition) always an abelian group. The word "abelian" in "abelian extension" refers to the Galois group of the extension.
Mar
19
comment Books (and supporting material) that are useful in deconstructing one's intuition?
I could immediately see that there must be a solution. Here's how. It's clear that you can connect A to A and C to C by paths that don't touch. Now imagine "cutting" the square open along those paths. Since those cuts stop in the middle of the paper (i.e. at the positions of the top A and top C), they don't actually end up cutting the paper into two pieces (to do so, you'd have to finish cutting to the edge of the paper!). So there must exist a path in the "cut open paper" from B to B. Tape the cuts back together and you've got your solution.
Dec
19
comment Higher dimensional Euclidean geometry problem
MO is intended for questions at the mathematics PhD student level and above. I've voted to close.
Sep
4
comment Best Books to learn Proof-Based Linear Algebra and Matrices
MO is intended for topics at the graduate school level and above.
Sep
1
awarded  Critic