Paulo Ney de
Souza
desouza@math.berkeley.edu
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Contents of My Pages
Here you will find information
on the courses I am currently teaching, my publications and the projects I am
working, or playing with, on right now ... down in the bottom you can have a live feed from my office! I hope to make this even more
complete in the near future and provide here my bookmarks set and many other goodies, check back ...
Books
If you want to read-me this is the place to go, the first one is out and more to
come soon:
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Courses I am teaching this semester
Publications
You will need Adobe Acrobat Reader to see the papers in here. It is available
for free in here and really worth the download.
- All smooth orientable 4-manifolds embed in R7, is just corollary of one of Donaldson's results that he apparently missed ...
- Van der Blij type equalities for unimodular quadratic forms, is an exploration on how far the equality that produces the embedding of 4-manifolds in R7 is really valid.
- Integral Unimodular Symmetric Bilinear Forms, is a sneak-preview of an Appendix of the forthcoming book on 4-manifolds.
- Computer Algebra Systems, has been published in print three times, but you are much
better of with the on-line version in here. If you are still interested in the
print version you can check them at:
- Notices of the Amer. Math. Soc., 40 (6) 1993, 617-623.
- The Handbook of Software for Engineers and Scientists, P. W. Ross, CRC Press - IEEE Press, 1996, 1005-1027.
- Computer Algebra Systems - A Practical Guide, Michael J. Wester, John Wiley & Sons, 1999, 333-355.
- Problems in Low Dimensional Topology, is a book by Rob Kirby that I helped produce and am extremely interested in producing a live electronic version of the text. Keep coming back to this link, the PDF version is now due soon! The paper version can be gotten here
- Ends - The Geometric Compactification. this is my master thesis at IMPA in 1981, that I am now translating to English and expanding it a bit.
My Current Projects
Some of them are not so current but something that I have lot f hopes to finish sometime soon.
- Geometric Topology of 4-manifolds, there are many beautiful 4-manifolds books out there now but all based on niches and almost none have a full and comprehensive view of the subject. This project is still in its infancy but you can check the table of contents now ...
- A short history of the Closing Lemma. It was brought up for the first time by Poincaré in ----, which observed how difficult the problem was, a few decades down the line Rene Thom assumed it as a triviality in one of his works that was never published, with the mistake pointed out by Peixoto, it became a widely researched problem ...
- Geometric
Topology Problems for the new Millennium. It is a collection of some of
the most interesting open problems in geometric topology.
- The Poincaré Conjecture.
- Chern's Problem: Does S2 x S2 admit a metric of positive sectional curvature ?
- The Closing Lemma.
- The 16th Hilbert Problem.
- 4-manifolds in R6 The celebrated hard-embedding theorem of Whitney states that every closed n-manifold embedds in R2n; guaranteeing R8 as the embedding space for all closed 4-manifolds. With the additional assumption of orientability the embedding space can be reduced to R2n - 1, the case of n = 4 is a consequence of the hard-Donaldson Theorem (see [P. de Souza]. For every dimension higher than 4 the manifold of ---- is an example that Whitney's theorem is sharp. In 1982 Tim Cochran gave the first example of an orientable manifold that embedds in R6 but not in R5 [Invent. Math. 77,
173-184 (1984)]. Up to this day it is not known if the problem of embeddability is a failure of the extension of the embedding to the last point or not, that is, if there is a 4-manifold that does not embedd in R5 but embedds punctured in R5 .
- Entropy of Knot
Polynomials I conjectured in 92 that the entropy of a polynomial grows
with the crossing number of a knot, it would be nice to see a relation between
thenm.
- Neumann's Conjecture for Homology 3-Spheres He formulated the conjecture back in the beginning of the 80's and slowly as computer power has climbed I verified it for an enormous amount of examples of the type Sigma(p,q,r). I would like to push the envelope on this verification even further by implementing new algorithms for quadratic forms classification.
- Computational Geometric TopologyThis article is a walk on some of the most significant moments of Topology from the point of view of computational methods, from the bridges of Konigsberg to the computation of Toda lattices and the classification of low crossing number knots.
- Computational Knot TheoryThis is probably the most developed area in Computational Geometric Topology and the methods and results have their own character, here I make a brief description of the developments in the area from ancient tabulation of knots to modern classification of the 16-crossins and the search for Jones-knots.
- Group Theory with Magma. I would like to find out how much Group Theory Magma really knows and can help students experiment with in a classroom environment... probably following with either Fraleigh or Herstein.
- Classification of Small Integer Unimodular Quadratic Forms with Magma. I am interested in writing a set of routines for classification of unimodular quadratic forms in small dimensions, so it can be used in further exploration of 4-dimensional manifolds.
- Linear Algebra with Maple. This is one of the weakest Maple packages and can be improved quite a bit and at the same time is an excellent point for insertion of technology in the classroom because students are advanced enough to deal with it from an advanced standpoint and examples that require long computations abound. There are quite a few books that touch on the subject, I would like to get my hands on all of them and test it out ...
- Algorithms in Graph Theory. Skiena's book and package are a nice playground for experimentation in Graph Theory, but some of the choices of algorithms can be sharpened, specially implementing Busacker-Saaty for computation of the number of connected components and better algorithms for bicolorability. I have started on it and will be done with it soon as part of my Graffiti project below.
- Can we improve the computation of n! ?, or better “How dumb are the Computer Algebra Systems right now ?” The rough answer is that some of them are incredibly dumb and I am using this example to test many of them, including CAS substrates.
- Computer Algebra Substrates - How do they stack up ? This wheel has been re-invented many times and some times with not so pleasant results. My proposal is to come up with a decathlon-like series that these systems can be made to race each other to sort out what is really good.
- Soletrador em Português, that is a Portuguese speller. English has many spellers and most of the free software, but Portuguese, specially Brazilians never had access to a good free speller in their native language; even the ones developed with public funds are now blatantly sold or profit, and now the effort seems to be going in many different directions at the same time. Uniformizing it will probablky be a good idea for all so we can have access to a good and free version of "ispell" for portuguese. Some useful links for a starting point are:
- Mathematical Illustration LaTeX You can also have a preview of this book in here. My plan is to cover as much ground as possible on all sorts of freely available tools like MetaFont, PS, CASes, PovRay, and how they can change the quality of graphics for mathematical publication.
- KnotPlot One of these days I'll get KnotPlot installed on my machine so I can produce the dazzling knots I need for some of the projects above.
- Graffiti To install Graffiti too
and use it to test conjectures in Knot Theory, just as it has been done for Graph Theory.
- Measuring Thermal Properies of Concrete This is a joint project with Jorge Zubelli of IMPA and Paulo Monteiro of UCB Enginnering.
- Topological Museum of Computing for the preservation of code and ideas that have shaped the way we experiment and do topology in the second half of this century. If have any ideas for holdings, send me e-mail.
- Santos Dumont The life and myth around this man and the state of the brazilian psych on the issue is most intriguing. He flew his for (his) first time in Bagatelle, on the outskirts of Paris on Oct 23, 1906, to the celebration of the french and the world. Santos maneuverability approach proved later to be doomed, specially when compared to the stability approach used by the Wright brothers. After flying side by side on ... the french forgot Santos Dumont and the brazilians are the only ones to venerate this man with certainly a lof of help and hypocrisy from the military that has choosen his as one of his patrons.
- Planning Design and Construction at UC Berkeley Even tough we exibit one of the most talented faculty body around the world, our campus is certainly one of the architecturely-challenged around the world. Not even the vicinity of great architects like Julia Morgan and astoning buildings around campus have been enough to influence developements in our campus.
Starting with the Architecture Department building itself, a huge block of cement without windows, passing by Evans Hall which looks more like a war bunker with its cannons ready to pop-out at any minute, and going all the way to the multi-million dollar big bathroom that is Soda Hall, we continue to develop the land without a coherent plan, and without having the invidual as the objective of our planning. If you know of anything that is non-functional at your building or know of one the rarer moments of excellence in our developments send me e-mail, I would like to visit it and include in this essay.
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A new battle-front has oppened now with the contruction of suburbia-style overpriced
housing for students in what used to be one of the best environments for living in
the world. See all about the contruction project and University's arrogance in the
Express article.
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Those in My Esteem
Click here to get to the page of some of my friends with a presence on the web, in order of appearance:
- Marise Velmovitsky, is the longest running of all my friends, now at Furnas.
- Silvio Levy, is the longest running one with a web-page, now quiet at MSRI.
- Alexandre Santarosa Freire is the runner-up!
- Othon Monteiro, incredibly enough we have been in the same place in the world almost continuously for the last 25 years ...
- Renato Duarte Carneiro Monteiro We had a great time at Berkeley and now he
is so far away in Georgia-Tech.
- Elon Lages Lima is simply the guy that brought me to Math, and because of that some of my friends wants to know him (reasons not give!) so here is the pointer!
- Jorge Passamani Zubelli after some unexploited forays in beautiful Santa Cruz, is now at IMPA, and believe it or not, PREGNANT!.
- Jonas de Miranda Gomes is the guys that really know the cool graphics stuff.
- José Felipe Voloch I am mad
at him lately for inflating the market of you know what... but that I'll go soon!
- Robion C. Kirby the coolest advisor in town and from whom I learned this pure-beauty math that is topology.
- Jorge Nuno Silva is another one that has moved away and is now collecting books in Lisbon and trying to convince me to get going with another book.
- André de Carvalho has been a friend since caming to Berkeley a few years back, he is now busy at SUNY, and so close to NYC.
- Henrique Bursztyn I'll let him push me out of pedra da Gávea one day!
- Dan Reznik is my guru on all things multimidia.
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Phone: 510-643-8638 | FAX: 510-642-6726 | e-mail: desouza@math.berkeley.edu
Department of Mathematics | University of California @ Berkeley | Berkeley CA 94720-3840
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