11,068 reputation
32881
bio website math.stanford.edu/~jmadnick
location Stanford, CA
age 24
visits member for 4 years, 4 months
seen 15 hours ago

Third-year graduate student at Stanford University. My interests originate from differential geometry. My advisors are Robert Bryant and Rick Schoen.

My favorite aspects of math are those which appear to generalize calculus. I like the algebraic and analytic machinery of geometry and topology.

I learn best from concrete examples, explicit calculations, and relating modern ideas to their classical counterparts.

Currently learning about:

  • Riemannian manifolds with special holonomy
  • Gauge Theory
  • Calibrated geometries
  • Exterior differential systems
  • Geometric analysis & PDE

15h
comment Differential Geometry and classical mechanics basics.
No, the flow of a vector field is not the tangent vector at each point: the vector field itself is the vector at each point. I will not discuss flows any further, except to say this: all of your questions would be answered by reading books (or taking courses) on: (1) linear algebra, and (2) manifold theory. For (2), probably the best book is John Lee's book, "Introduction to Smooth Manifolds."
15h
comment Differential Geometry and classical mechanics basics.
@JanetthePhysicist: The tangent space $T_pM$ is an example of a vector space. There is no difference between saying "the tangent vector $v$ at a point $p$ in $TM$" and "the tangent vector $v_p \in T_pM$."
1d
comment Vector spaces and bundles in classical mechanics
"The tangent bundle is not a manifold in general, it is a fibre bundle." This is rather misleading. Yes, the tangent bundle is a fiber bundle, but its total space -- which one also calls the tangent bundle -- is a manifold.
1d
answered Differential Geometry and classical mechanics basics.
2d
comment Differential Geometry and classical mechanics basics.
Regarding point 1: the tangent bundle $TM$ is a smooth manifold whose dimension is twice that of $M$. I think that was the intended meaning.
2d
comment $\int_Ef>\int f−\epsilon$
@ErginSuer: You're right that $f\chi_{A_n} \nearrow f$, so that the convergence is monotone. Do you know a theorem you can apply in this situation?
Dec
11
comment Which mathematical topics is knot theory related to?
@MarianoSuárez-Alvarez: Really? When I was an undergraduate who knew nothing other than calculus, linear algebra, and (the basics of) analysis and abstract algebra, the idea of an entire field devoted to studying knots seemed completely out of left field. Only after someone told me that studying knots is the same as studying embeddings of the circle $S^1$ did I begin to see any sort of connection.
Dec
9
comment Geodesic of a curved surface
@AmagicalFishy: Fixed. Thank you.
Dec
9
revised Geodesic of a curved surface
edited body
Dec
8
awarded  Caucus
Nov
24
awarded  Enlightened
Nov
24
awarded  Nice Answer
Nov
18
comment Geometrical interpretation of Ricci curvature
Bonnet-Myers? Cheeger-Gromoll Splitting?
Nov
12
comment Open mathematical questions for which we really, really have no idea what the answer is
@MikeMiller: (1) Any almost complex structure on $\mathbb{S}^6$ would have to be incompatible with both the standard metric (LeBrun) and the standard symplectic structure (Bryant-Chern). (3) The Chern conjecture is true for complete affine manifolds. (2) I remember hearing the same guesses for the Hopf conjecture at one point (by a couple experts), but I honestly don't recall them and won't venture one myself.
Nov
11
answered Open mathematical questions for which we really, really have no idea what the answer is
Nov
11
comment Open mathematical questions for which we really, really have no idea what the answer is
Scorpan mentions that there are no compact simply-connected smooth $4$-manifolds known to admit only finitely many smooth structures.
Nov
1
awarded  Good Answer
Oct
10
comment $K = 0$ and $K = \text{const}$ surfaces produce $k_g = \text{constant}$ intersections?
I don't mean to be difficult, but as the question is phrased, I wouldn't have the slightest idea of how to begin. I imagine that many other mathematicians might have a similar reaction, but I hope I'm wrong; hopefully others are better able to interpret and answer your question. Best of luck!
Oct
10
comment $K = 0$ and $K = \text{const}$ surfaces produce $k_g = \text{constant}$ intersections?
Are "ring" and "thin parallel circle of ring" and "oval" all referring to the same thing, namely a simple closed curve with constant $k_g$? Is it obvious that on every surface of revolution with $K$ constant that there actually exists such a curve at all? By "thin" do you mean "$1$-dimensional," because to me a "circle" is something which has only one dimension, and hence exactly zero thickness. And by "axial plane" do you mean "tangent plane" or "normal plane" or "rectifying plane," or something else entirely? And to me, every "plane" is completely flat, without curving or bending.
Oct
9
comment $K = 0$ and $K = \text{const}$ surfaces produce $k_g = \text{constant}$ intersections?
I think there are some interesting things here, but you have to be more precise with your question. For example, I'm not sure what you mean by the phrase "how can a plane be bent and relatively positioned." Maybe you could phrase your question in the form: "Given [certain things], does there exist [a specific thing you want] satisfying [exact conditions you want]?"