11,590 reputation
32982
bio website math.stanford.edu/~jmadnick
location Stanford, CA
age 24
visits member for 4 years, 5 months
seen 2 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 parts of math arise from the geometric aspects of 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

Jan
23
answered Expression for Hamiltonian vector field!
Jan
23
comment When is a linear map of 1-forms a pullback?
Ah, OK, I initially thought that $M$ and $N$ were just smooth manifolds; I see now that they are oriented Riemannian 2-manifolds.
Jan
23
comment When is a linear map of 1-forms a pullback?
What is a "coexact form" in this context?
Jan
18
comment What's your favorite proof accessible to a general audience?
I like the dominoes picture. The picture I first learned was similar: "Suppose you're climbing a ladder. If you know you can reach the first rung, and you know that if you can reach any rung then you can reach the next one up, then in fact you can climb the whole ladder."
Jan
18
revised How to define orthogonal complement in an arbitrary vector space
added 404 characters in body
Jan
18
comment How to define orthogonal complement in an arbitrary vector space
@NateEldredge: Thank you for catching that; you're absolutely right. In points 1 and 2, I meant for the vector space in question to be finite-dimensional. I'll edit.
Jan
17
comment How to define orthogonal complement in an arbitrary vector space
@student: You should be asking the answerer on that thread why the image of $T$ is closed, not here. But to save you the trouble, the Open Mapping Theorem implies that the operator $T'$ is a homeomorphism, hence maps closed sets to closed sets. The subspace $X \oplus 0 \subset X \oplus C$ is closed, so $T'(X \oplus 0) \subset Y$ is closed, and $\text{Im}(T) = T'(X \oplus 0)$ by the definition of $T'$.
Jan
16
comment How to define orthogonal complement in an arbitrary vector space
@Aahz: Sure. But I must be missing something: how does your comment relate to my answer?
Jan
16
answered How to define orthogonal complement in an arbitrary vector space
Jan
16
answered A fan, a horn, and a snowflake - unusual math terms
Jan
15
comment Why does $p(a)=0$ imply $(x-a) \mid p$?
Why does $p(x)$ have linear factors?
Jan
13
revised constant-curvature Riemannian metric for Bring's surface
added formatting, added two tags
Jan
7
revised Spivak Calculus on Manifolds, Theorem 5-2
deleted 30 characters in body
Jan
7
answered Spivak Calculus on Manifolds, Theorem 5-2
Jan
7
comment Is there a domain “larger” than (i.e., a supserset of) the complex number domain?
pbs is talking about the Cayley-Dickson construction: this is (one of) the construction(s) that builds the complex numbers from the real numbers. It can be repeated to build the quaternions (from the complex numbers), and then the octonions (from the quaternions). And as pbs said, after each step, one loses an algebraic property.
Jan
5
awarded  Good Answer
Jan
4
awarded  Nice Answer
Jan
4
comment Is there a vector field that is equal to its own curl?
I see now that the linked paper by GPerez and David K essentially carries out this proof in Section 3. Interestingly, Section 2 shows that any differentiable vector field with $\text{curl}\,F = F$ is automatically real-analytic -- essentially because of $F = \text{curl}(\text{curl}\,F) = -\Delta F$ and elliptic regularity -- and so the real-analyticity restrictions above are not as strong as they seem :-)
Jan
4
revised Is there a vector field that is equal to its own curl?
added 556 characters in body
Jan
4
revised Is there a vector field that is equal to its own curl?
added 35 characters in body