31,985 reputation
165147
bio website math.berkeley.edu/~ianagol
location Berkeley
age 44
visits member for 5 years, 6 months
seen 2 hours ago

I'm a professor at UC Berkeley working predominantly in the field of low-dimensional topology.


4h
comment Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
Nice description. One thing I still don't quite get is why $\Gamma_0$ is generated by the 4 given elements? I don't see how you can deduce this without a bit more information (a free group on 4 elements has many 4 generator subgroups).
15h
comment Can every $\mathbb{Z}^2$ disk be pinball-reached?
It seems that you're searching for something like "Sinai billiards".
16h
comment Avoiding mean-curvature flow dumbbell neck-pinch by inflating a surface
@JosephO'Rourke: Did you mean to discuss Ricci flow? I was confused by the title of your question, since you actually seem to be looking for something related to mean curvature flow.
1d
comment Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
@QiaochuYuan: Ok, I was using the term "Bass-Serre tree" incorrectly. I really meant the tree associated to $SL_2$ of a local field, which is a special case of a Bruhat-Tits building. Serre's book Chapter II section 1 is the usual reference. Shalen's notes (linked in my other comment) Chapter 3 also has a nice discussion.
1d
revised Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
fixed terminology
2d
revised Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
fixed a mistake computing the quaternion algebra
2d
comment Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
You're right, I misapplied the theorem since the group is elementary. I'll fix the answer later
May
2
comment Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
@QiaochuYuan: Some references are found on Wikipedia: en.wikipedia.org/wiki/Bass%E2%80%93Serre_theory Shalen also covers some of the theory here: dl.dropboxusercontent.com/u/8592391/3-manifold%20seminar/…
May
1
awarded  Nice Answer
May
1
revised Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
added 489 characters in body
May
1
answered Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
May
1
comment Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?
Thinking of these quaternions as elements in $SU(2)$, they have trace $-\frac12$. This indicates that there ought to be a non-trivial action on a Bass-Serre tree. It might be possible to show that the action on this tree is free, in which case the group is free.
Apr
30
awarded  mg.metric-geometry
Apr
27
comment Negatively curved metrics minimizing the length of a homotopy class of simple closed curves
@SelimG: Yes, you're essentially correct. In fact, for a fixed $M$, there are boundedly many curves of bounded length. So a subsequence will preserve the homotopy class of $\gamma$. I should fix the answer to reflect this.
Apr
25
comment Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds?
It is canonical, if you're after a classification of discrete subgroups of $PSL_2(\mathbb{C})$. But Kleinian group theorists usually study such groups up to conjugacy, in which case this does not make a canonical choice for a conjugacy class. One may single out finitely many orbits of points (if it's cofinite volume) by taking points of maximal injectivity radius, for example.
Apr
24
comment Reduction of self-intersections without reducing the geometric intersection
Yes, the embedded case is just an illustration.
Apr
24
revised Reduction of self-intersections without reducing the geometric intersection
added 508 characters in body
Apr
24
comment Reduction of self-intersections without reducing the geometric intersection
@Cusp : I think it's a bit easier to think about the graph when $b$ is embedded. I've added an explanation in this case to clarify, and one may forget about lifting closed arcs unless it helps.
Apr
23
answered Reduction of self-intersections without reducing the geometric intersection
Apr
23
revised Negatively curved metrics minimizing the length of a homotopy class of simple closed curves
Added more exposition.