17,457 reputation
23476
bio website utm.edu/staff/jdevito
location Martin, TN
age 30
visits member for 4 years, 1 month
seen 2 hours ago

I'm an assistant professor at the University of Tennessee at Martin, in my fourth semester here.

My research interests include compact Riemannian manifolds with positive/non-negative sectional curvature, and, in particular, those with a "large" isometry group. I love working with compact Lie groups, especially homogeneous spaces and biquotients.

At some point, I intend to learn some set theory.


2h
comment Can one prove that $\Bbb{CP}^\infty$ is a $K(\Bbb Z, 2)$ without invoking the long exact sequence of a fibration?
I don't know how to make this work, but if one could directly show the loop space $\Omega \mathbb{C}P^\infty$ was homotopy equivalent to $S^1$, that would prove the claim. I don't have the first idea how to write down such a homotopy equivalence, however.
1d
answered homotopy groups of wedge sum
2d
answered Second Stiefel-Whitney Class of a Five Manifold
Aug
26
awarded  Necromancer
Aug
26
comment When a group acts on a metric space isometrically, transitively, and faithfully, is it a closed subspace of the isometry group?
studiosus: Do you know whether it's true that, under the OPs hypothesis, $G$ necessarily contains a subgroup closed in $Iso(M)$ which still acts transitively? All the counterexamples I know to the OPs original question have the form you gave: two different compact Lie groups, one a subset of the other, both act transitively and faithfully.
Aug
26
comment $T^2\times S^n$ is parallelizable
See also math.stackexchange.com/questions/665779/…, for products of more spheres.
Aug
25
comment Can we measure how close a vector bundle is to being trivial?
It may be worth mentioning that characteristic classes allow you to bound $\eta$, but generally don't allow you to compute it directly. The relationship is the following. For a rank $k$ complex bundle over $M^n$, if there are $l$ independent sections, then $c_{i} = 0$ for any $i>\max\{n, k-l\}$. A similar fact holds for real bundles using Pontryagin or Stiefel-Whitney classes. Very often, the bounds are not sharp. For example, any odd sphere other than $S^1$ or $S^3$ has vanishing Pontrygain and Stiefel-Whitney classes, but is never parallelizable.
Aug
15
answered Implied relationships between Lie groups and Lie algebras.
Aug
7
comment quotient by a group that acts almost freely
This post is rather longer and more complicated than I anticipated. I'd really like to know if I'm implicitly using finite isotropy groups somewhere. Note that I do use compactness all over the place, so perhaps the correct problem is "non-compact Lie group (probably with another condition or two) and finite isotropy implies Hausdorff quotient"
Aug
7
answered quotient by a group that acts almost freely
Aug
7
comment quotient by a group that acts almost freely
Do you know of an example of a compact Lie group on a manifold $M$ for which the quotient $M/G$ is NOT Hausdorff? I also don't understand the relation of your comment about discrete Lie groups to the question. Could you clarify?
Aug
6
comment quotient by a group that acts almost freely
Is finite isotropy necessary? By compactness of $G$, the action is proper, and then I belive math.stackexchange.com/questions/50044/… answers the question.
Jul
30
comment True or False: Topological Group and $S^1 \vee S^1$
I just wanted to note that there are topological groups which cover things with non-abelian $\pi_1$. Probably the simplest example is that $S^3$, thought of as the unit quaternions, covers the Poincare dodecahedral space.
Jul
28
awarded  Yearling
Jul
28
answered Is there a retraction of a non-orientable manifold to its boundary?
Jul
18
awarded  Sportsmanship
Jul
18
answered Find a CW complex with prescribed homology groups
Jul
16
comment R-linear functionals on manifolds
I should point out that this is a standard style of proving that tensorial things only depend on points, and not on neighborhoods.
Jul
16
answered R-linear functionals on manifolds
Jul
13
comment Question on unitary representation of non-compact simple Lie groups
Why is the map $\mathfrak{g}\rightarrow \mathfrak{su}(n)$ surjective?