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bio website cornellmath.wordpress.com
location Ithaca, NY
age 30
visits member for 3 years, 11 months
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I was a graduate student at Cornell University studying harmonic analysis with Camil Muscalu. Now I'm a postdoc at Washington University in St. Louis.

Dec
20
comment When is a collection of exponentials dense in $L^2(K), |K|<\infty$
Alexei Poltoratski gave a very nice series of lectures with a number of comments in this direction this summer at Clemson. I don't think he really discussed the R^d case, but notes similar to the lectures he gave are available here: internetanalysisseminar.gatech.edu/lectures_uncp Some of the references may point you in the right direction.
Sep
14
awarded  Informed
Jun
25
awarded  Citizen Patrol
May
25
awarded  Necromancer
Apr
30
awarded  Yearling
May
8
comment Square roots of the Laplace operator
Now that I think about it, the analytic continuation might need some additional work... there might be some poles at certain integers $\alpha$, so you would need to add an additional term to balance things out. This computation has to be written down someplace. Sorry that I don't know of a good place.
May
8
comment Square roots of the Laplace operator
Yes, much of this stuff is fairly disjointed, as I recall, which is unfortunate. Many of the basic computations are not terribly difficult (just tricky), so I think they are not written down in many places. The issue you are having in the computation in your comment comes from having to use some analytic continuation technique to extend the range of possible $\alpha$s in the inverse transform of $|\xi|^\alpha$. $|x|^{-n-1}$ is not locally integrable, as Tom pointed out, so its a bit more delicate.
May
8
answered Square roots of the Laplace operator
Apr
30
awarded  Yearling
Nov
23
comment Carleson's Theorem (on the Adeles and other exotic groups)
I don't think you can accept more than one answer, but that's ok. I just hope what I wrote helped in some way!
Oct
19
comment Can the supremum of continuous functions be discontinuous on a set of positive measure?
Hm, I suppose you could also have just made it 1 on $K_n$ and 0 on $E$ and avoided that $\sup(-f_n)$ business...
Oct
19
answered Can the supremum of continuous functions be discontinuous on a set of positive measure?
Sep
13
comment Approximating high-dimensional integrals by low-dimensional ones
If X is very nice, perhaps you can utilize Gaussian quadrature? For example, in a fixed interval of $\mathbb{R}$, by sampling n special points one can compute exactly the integrals of all polynomials of degree <2n. It seems like this might be generalizable to, say, rectangular regions in higher dimensions; it is computationally infeasible to do this in high dimensions, though, as it would require an absurd number of samplings. That is ultimately why Monte Carlo methods are used to do integrations in $\mathbb{R}^100$
Sep
12
comment Function space between uniform continuity and Hölder continuity
What do you mean by containment in this case? Contained as sets of functions? Must there be some kind of isometric vector space embedding-type relationship? If so, what are the topologies on each of the spaces?
Sep
1
comment Applications of PDE in mathematical subjects other than geometry & topology
Solving PDE is rather tricky business; even some of the simplest ones are so rich! The Cauchy-Riemann equations are innocent looking yet produce an incredible theory. Since there is much interest in solving these problems from an applied standpoint, people work hard and come up with ingenious ways to solve them, and in doing so produce a host of great abstract problems which don't relate that closely to the original problem anymore.
Sep
1
comment Applications of PDE in mathematical subjects other than geometry & topology
A professor here at Cornell gave a talk several years ago where he began by briefly estimating that the majority of mathematics (as measured by number of articles) of the last 50-60 years relates to differential equations (he includes most of applied math, numerical analysis, much of analysis itself, and so on). I think by this measure, differential equations represents the greatest source of interesting problems in all of mathematics. It is probably rather difficult to find an area of mathematics which does not relate in some (potentially very loose) way to or use ideas from PDE.
Aug
31
answered Applications of PDE in mathematical subjects other than geometry & topology
Aug
29
comment What does the σ in σ-algebra stand for?
Couldn't the $\sigma$ have originally come from the word for sum (which, in latin, is summa)? In the end, $\sigma$-algebras are designed to produce sets which play well with summation (capital $\Sigma$). There is the additional connection that the sign for the integral also comes from the latin word summa (and is supposed to be an elongated s), and $\sigma$-algebras are used to generate the modern integral.
Aug
26
comment Analytic functions with algebraic Taylor coefficients at some point.
Ah, of course you're right. A careless error on my part. It thankfully doesn't change the main points, though.
Aug
26
revised Analytic functions with algebraic Taylor coefficients at some point.
added 29 characters in body