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# Questions and Answers in Geometry and Analysis

## Questions and Answers in Geometry and Analysis - LiveJournal.com

Last Build Date: Tue, 29 Aug 2006 07:52:13 GMT

Tue, 29 Aug 2006 07:52:13 GMT

Is there a geometrical meaning in the following number associated to a finite-dimensional semisimple Lie algebra g: an integral over g of the density rho(x)=Det(Sinh(ad_x)/ad_x) for x\in g, calculated on matrices of adjoint representation of g?

Tue, 04 Apr 2006 18:02:43 GMT

Hello, I'm doing some work, and am having some difficulty with terminology, so I want to get it right. I have two questions for now.

1. I have a surface in three-space, and am looking for a term to describe if it's possible to determine which side a viewer is on. I was thinking about the 33336 tiling, how viewed from one side it has a left twist, and the other side has a right twist, so I thought chiral vs. achiral would work, but if I have a surface that is all black on one side, and all white on the other, compared to one that is white on both sides, that terminology is wrong. So is there a term to describe if a surface looks the same from both sides, or has a different appearance.

2. In a tiling, if A and B both belong to the same Transitivity class, is there an equivalence term to say that. When I've been writing I've been saying that A and B are Transitive, but I'm not sure it that's valid, or correct, so I want to make sure before I use it any more.

Thanks
Nekura

Relationship between L-polynomial and the Stiefel-Whitney class

Fri, 13 Jan 2006 20:35:07 GMT

Let  M  be a closed manifold of odd dimension  n=2k+1. Let  L(p)  denote the Hirzebruch L-polynomial in the Pontrjagin classes of  M. Denote by  L2k(p) the component of  L(p)  in  H2k(M,Z) and by   w2k Î H2k(M,Z2)  the  2k-th Stiefel-Whitney class of  M.  I have strong reasons to believe that the reduction of   L2k(p) modulo 2 is equal to to w2k. . Is it really so? And, if it is, how to prove it?

Remark: if   k  is odd, so that  2k  is not divisible by 4, then, of course,  L2k(p) = 0   by dimensional reason. In this case, Massey [Amer. Jornal of Math. 82, 92-102] showed that  w2k=0. Thus, if  k  is odd, the answer to my question is positive.

Mon, 10 Oct 2005 17:04:35 GMT

Dear ALL!

Do you probably know how to:

1. Find furthest-neighbor or furthest-site diagram out of Voronoi Diagram? (Fast and efficient way for each site to know which one is farthest site from it)?

2. Let P1...Pk be a collection of pairwise disjoint simple polygons with a total of n edges, all enclosed in a given square. Find in O(n*logn) time a largest disk that can be inscribed in this square so that it is disjoint from all the interiors of the polygons Pi ------------ how do I solve this? Provided I know how to find largest empty circle in Voronoi diagram, how can I take care that it does not intersect the EDGES of polygons (staying disjoint with their interior)?

Thanks a lot!!!

Question on Haar Measure/Integration

Sat, 13 Aug 2005 15:40:12 GMT

Ok well you guys said this is for analysis, so don't yell at me since this post is not even remotely geometric :p

If G is a locally compact group and K subse G is compact, and
lim n --> inf int_K f_n(x) dx = 0, then does lim n --> inf int_{K^-1} f_n(x^-1) dx = 0? (f_n subset L^p(G) for some p >= 1) I actually ultimately need this question asked for integration being Bochner integration, but is this true even for Lebesgue integrals? I don't think it is, since the modular function will screw things up even if it's a 'mild' function.

Non-Metrical Geometry

Tue, 29 Mar 2005 06:54:27 GMT

What is Non-Metrical Geometry?

It is mentioned in A. N. Whitehead's "Process and Reality" (pages 490s), in discussion of infinitesimals (Weierstrass), presentational immediacy, strain locus, projection (opposite of injection, I think), God and mentality. Intuitively, it brings me back to the 470s (pages); "Its is to be remembered that two points determine a complete straight line, that three non-colinear points determine a complete plane, and that four non-coplanar points determine a complete three dimensional flat locus." ("Process and Reality", page 472)

Mon, 14 Mar 2005 21:20:02 GMT

I'm really sorry, but I have questions that really aren't directly pertinent to the group.

How are you using the special math text that you used in your first post to this community? Can you give me a link to how to use it?

Would graph theory be included in your interests?

Thank you very much!

naive defitinion of normalisation of complex manifolds

Thu, 10 Mar 2005 17:46:33 GMT

I need a reference to these simple statements about normalisation of complex spaces;
are they correct at all ? I am willing to assume everything lies a very nice ambient space, say
holomorphically convex manifold.

\def\hatZ{{\hat Z}}
Let $\nnn:\hatZ \ra Z$ be the normalisation of a variety $Z$, and
let $\Zz k(\C) = \{z\in Z(\C): \#\nnn\inv(z)\geq k\}$ be a closed
subvariety of points which have $k$ preimages or more.

\begin{fact}\label{fact:zknrm} $\Zz k$ is a closed subvariety of $Z$, for all $k$;
$\Zz {\deg \nnn + 1}$ is empty, $\Zz 1 = Z$. \end{fact} \bp
Reference!!!! \ep

\begin{fact}\label{fact:214} Around each point $z\in Z(\C)$ there exists a
sufficiently small neighbourhood $z\in V$ open in complex topology
such that
V\cap \Zz k= \bigcup\limits_{i_1<..

Flat connection vs. representation of the fundamental group

Sat, 05 Mar 2005 16:01:37 GMT

Let M be a closed manifold. A complex representation α: π1(M) → Cn of its fundamental group  gives rise to a flat vector bundle Eα = M×αC whose monodromy is isomorphic to α. Given a continuous family of connections α(t) the bundles Eα(t) are isomorphic (as vector bundles without a connection). Thus it is not difficult to see that there exists one bundle E → M and a continuous family of connections (image) (t) such that the monodromy of  (image) (t) is isomorphic to α(t). Moreover if α(t) is differentiable then (image) (t) can be chosen to be differentiable. I have 2 questions:

1. Is there any book or paper where these simple facts are proven?

2. How far this relationship between families of representations and families of connections can be pushed? For example, if α(t) is analytic can one find an analytic family (image) (t)?

Reference for variational formula for the determinant of an elliptic operator

Fri, 04 Mar 2005 06:06:50 GMT

Let A be a (note necessarily self-adjoint) invertible elliptic operator. The following variational formula for its determinant is well known

δ Det(A) :=  δ (-∂s Tr ( A-s ) |s=0)  = s  ( s Tr[(δ A ) A-s-1])|s=0

cf., for example, Burghelea, Friedlander, Kappeler, Meyer-Vietoris type formula for determinants of elliptic differential operators. J. Funct. Anal. 107 (1992), 34--65, or Kontsevich, Vishik , Geometry of determinants of elliptic operators. (of course, to define A-s one uses a spectral cut, which I did not write explicitly to simplify the notation).

Is it proven anywhere in the literature? (I do know how to prove it, but I need a reference)