bio | website | gilkalai.wordpress.com |
---|---|---|
location | Jerusalem | |
age | 59 | |
visits | member for | 5 years |
seen | 11 hours ago | |
stats | profile views | 17,172 |
Professor of Mathematics at the Hebrew University of Jerusalem and at Yale University
Nov 21 |
comment |
The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.
Well, even in the original high dimensional generalization of Gallai Sylvester to have the simplest intersection with k-flat you need the ambient space be 2k-dimensional. Also for the conjectured generalization we need the abmient space be large. Now, k-faces of n-polytopes cannot be too complicated when n is large. (Or at least, so we believe).For example, n-polytopes always have 2-faces which are triangles of quadrangles. I do not see how an example like the one suggested in the answer works when the ambient space is high dimensional. |
Nov 21 |
asked | What is an integrable system |
Nov 21 |
revised |
Fundamental Examples
added 167 characters in body |
Nov 21 |
accepted | The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”. |
Nov 21 |
comment |
The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.
In any case Greg's answer pushed the problem enough to be accepted. More answers are nevertheless welcome. |
Nov 21 |
comment |
The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.
I do not see yet how the example [or such an example] allows 1-dimensional intersection of a 3-planes [ot 2-plane] with V with arbitrary complicated topology, when the abbient space is of high dimension. It looks that indeed it shows that for 1-dimensional intersection with a 3-plane for a variety in 4 space, the topology cannot be as tight as in the original G-S thm. (Unbounded number of connected components seems anavidable.) I dont see if in this example the topology can be arbitrary complicated. I'll be happy to hear more on the last paragraph. (several constructions and open-ended probs) |
Nov 21 |
comment |
The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.
This is an excellent answer and let me give some more detailes about it. First, it raises interesting special cases and generalizations of the problem. 1. Certainly a case which deserves to be singled out is the case where V comes from an arrangelemt of subspaces. I.e., when V is the union of affine subspaces of an n-dimensional space. (The original case deals with 0-subspaces.) 2. It suggests to study the problem (in the real case) for semi-algebraic varieties. Those include sets like skeleta of polytopes and skeleta with added diagonals. 3. To consider smooth varieties. |
Nov 21 |
comment |
Combinatorics of the Stasheff polytopes
The Stasheff polytope is simple, right? so the number of edges is d/2 times the number of vertices, where d=n-3 is the dimension. I think there are formulas for all the face numbers. |
Nov 21 |
comment |
Super-linear time complexity lower bounds for any natural problem in NP?
Here is a citation and a link : Uri Zwick, A 4n lower bound on the combinatorial complexity of certain symmetric Boolean functions over the basis of unate dyadic Boolean functions SIAM Journal on Computing 20, 499-505 (1991) An explicit lower bound of 5n-o (n) for boolean circuits, K Iwama, O. Lachish, H Morizumi, and R. Raz springerlink.com/index/XCP1TCRY1C236RDT.pdf The introduction of the second paper and the slow incremental improvements may give some idea on the difficulty of the problem. |
Nov 21 |
revised |
Something like mathoverflow in other sciences
added 15 characters in body; edited tags |
Nov 21 |
answered | Theorems for nothing (and the proofs for free) |
Nov 21 |
comment |
Theorems for nothing (and the proofs for free)
Here is a related discussion gilkalai.wordpress.com/2009/08/03/… |
Nov 21 |
answered | A single paper everyone should read? |
Nov 21 |
comment |
The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.
Excellent answer, Greg, thanks. |
Nov 21 |
revised |
A single paper everyone should read?
deleted 4 characters in body |
Nov 20 |
answered | A single paper everyone should read? |
Nov 20 |
comment |
Definition of “simplicial complex”
right, i meant 7 vertices... |
Nov 20 |
answered | Definition of “simplicial complex” |
Nov 20 |
comment |
Math Vs Social Science
This is very interesting, and Nate Silver looks like an extraordinary guy and certainly a good statistcian. (See also en.wikipedia.org/wiki/Nate_Silver .) Of course, we need to be cautios about this story and possible interpretations. (For example, quite a few academic statisticians are involved in polling planning and analysis and they probably are aware of graduate level statistics.) Also for role models of using mathematics in social sciences, I would still prefer the standard academic avenues of papers in professional journals etc. |
Nov 20 |
comment |
How Does Random Noise Typically Look?
I agree that this is not realistic - neither in the quantum case neither in the classic case. (At least not without much further explanation.) I only asked about the factual matter: what is the situation in the classical case. It was strange for me that I do not know the answer (and not precisely the question) for a question which was easy in the quantum case. It is still not clear to me how to do the computation for a random stochastic map and what the answer is. |