\documentclass[10pt]{beamer} \include{macros} \mode { \usetheme{Warsaw} %\usecolortheme{rose} \usecolortheme{seahorse} \setbeamercovered{transparent} % or whatever (possibly just delete it) } \usepackage[english]{babel} % or whatever \usepackage[latin1]{inputenc} % or whatever \usepackage{times} \usepackage[T1]{fontenc} % Or whatever. Note that the encoding and the font should match. If T1 % does not look nice, try deleting the line with the fontenc. \title{SAGE: Software for Algebra and Geometry Experimentation} \author{William Stein} % - Use the \inst command only if there are several affiliations. % - Keep it simple, no one is interested in your street address. \date[Jan 5] % (optional) {\vspace{-6ex} January 15, 2007, UW CSE Colloquium\\ \includegraphics[width=22em]{sage-car.png}} \subject{Talks} % Delete this, if you do not want the table of contents to pop up at % the beginning of each subsection: \AtBeginSubsection[] { \begin{frame} \frametitle{Outline} \tableofcontents[currentsection,currentsubsection] \end{frame} } % If you wish to uncover everything in a step-wise fashion, uncomment % the following command: %\beamerdefaultoverlayspecification{<+->} \begin{document} \begin{frame} \titlepage \end{frame} \begin{frame} \frametitle{The Genesis of SAGE} \large \begin{itemize} \item {\dblue 1997--1999:} (Berkeley grad student) HECKE -- my free C++ program for my thesis research. \item {\dblue 2000--2005:} (Harvard) I wrote $>25,000$ lines of Magma code. I had really wanted a sufficiently powerful open source math software system with a modern language... but none existed. \item {\dblue Feb 2005:} Started SAGE (job offers with {\dgreen tenure}). \item {\dblue Feb 2006:} {\dred UCSD SAGE Days 1} -- SAGE 1.0. \item {\dblue June 2006:} {\dred UW High school} workshop -- Notebook. \item {\dblue August 2006:} {\dred MSRI Grad student} workshop. \item {\dblue October 2006:} {\dred UW SAGE Days 2} workshop. \item {\dblue Now:} {\dred SAGE-1.7}; and {\dred SAGE 2.0} by end of month! \end{itemize} \end{frame} \begin{frame}[fragile] \frametitle{Welcome to SAGE!} {\small \begin{verbatim} rank4:~/talks/2007-01-15-sage-cse was$ sage -------------------------------------------------------------- | SAGE Version 1.7, Release Date: 2007-01-16 | | Type notebook() for the GUI, and license() for information.| -------------------------------------------------------------- sage: 2 + 2 4 sage: notebook() ************************************************** * Open your web browser to http://localhost:8000 * ************************************************** .... \end{verbatim}} \end{frame} \begin{frame}[fragile] \frametitle{Is Open Source Math Software Important for Mathematical Research?} ``You can read Sylow's Theorem and its proof in Huppert's book in the library [...] then you can use Sylow's Theorem for the rest of your life free of charge, but for many computer algebra systems license fees have to be paid regularly [...]. You press buttons and you get answers in the same way as you get the bright pictures from your television set but you cannot control how they were made in either case. \vspace{1ex} With this situation {\dred two of the most basic rules of conduct in mathematics are violated}: In mathematics {\dblue information is passed on free of charge} and {\dblue everything is laid open for checking}. Not applying these rules to computer algebra systems that are made for mathematical research [...] means {\dred moving in a most undesirable direction}. Most important: Can we expect somebody to believe a result of a program that he is not allowed to see? Moreover: Do we really want to charge colleagues in Moldava several years of their salary for a computer algebra system?'' \vspace{1ex} -- J. Neub\"user (1993) (he started GAP in 1986). \vspace{-.5ex} \end{frame} \begin{frame} \frametitle{Who is Writing SAGE?} {\dblue Contributors Include:} {\small Martin Albrecht, Nick Alexander, Tom Boothby, Robert Bradshaw, Iftikhar Burhanuddin, Craig Citro, Alex Clemesha, John Cremona, Didier Deshommes, David Harvey, Naqi Jaffery, David Joyner, Josh Kantor, Kiran Kedlaya, David Kirkby, Emily Kirkman, David Kohel, Jon Hanke, Bill Hart, Robert Miller, Bobby Moretti, Gregg Musiker, Andrey Novoseltsev, Bill Page, Fernando Perez, Yi Qiang, David Roe, Michael Rubinstein, Nathan Ryan, Kyle Schalm, Steven Sivek, Jaap Spies, Gonzalo Tornaria, Justin Walker, Mark Watkins, Joe Weening, Joe Wetherell, ...} \vfill \begin{itemize} \item {\dred Undergraduates}: have many {\dblue interesting} ideas; superb at researching available free software; amazing programmers. \item {\dred Graduate students}: excellent at implementing optimized code and finding fast algorithms. \item {\dred Faculty and computer professionals}: general direction, great writing, and quality control. \end{itemize} {\dgreen Funding:} Miniscule compared to that of Magma, Maple, etc. UW Math department startup money, NSF VIGRE money, my NSF grant, the MSRI (workshops), IPAM (workshop) \end{frame} \begin{frame} \frametitle{Active UW Student SAGE Developers} \begin{enumerate} \item {\dblue Tom Boothby} (undergrad) -- The SAGE Notebook, powering algorithms, elliptic curves \vfill \item {\dblue Robert Bradshaw} (grad) -- Linear algebra, number theory \vfill \item {\dblue Josh Kantor} (grad) -- Numerical computation; 3d graphics \vfill \item {\dblue Emily Kirkman} (undergrad) -- some linear algebra; lots of work on the SAGE graph theory package. \vfill \item {\dblue Robert Miller} (grad) -- Graph theory \vfill \item {\dblue Bobby Moretti} (undergrad) -- Symbolic calculus, some group theory, packaging, chroot jails. \vfill \item {\dblue Andrey Novoseltsev} (grad) -- combinatorial geometry, polytopes \vfill \item {\dblue Yi Qiang} (undergrad) -- Distributed SAGE \vfill \end{enumerate} \end{frame} \begin{frame}\frametitle{SAGE Days 2 at UW: Coding Sprints...} \includegraphics[width=\textwidth]{sagedays2moment.png} \vfill {\small Bobby Moretti (UW undergrad), Robert Miller (UW grad), David Harvey (Harvard grad), Joel Mohler (grad), David Joyner (USNA), Bill page (Axiom).} \end{frame} \begin{frame} \frametitle{Upcoming SAGE-related Workshops I'm Organizing} \begin{itemize} \item {\dred Parallel Computation Workshop} at MSRI, Jan 29--Feb 2, 2007. I'm the organizing committee chair. \item {\dred SAGE Days 3} at IPAM (in LA) Feb 17--21, 2007. \begin{center} \includegraphics[width=0.4\textwidth]{sd3-motivator-mozaic.jpg} \end{center} \item {\dred AIM} workshop, databases in SAGE, July 30-Aug 3, 2007. \end{itemize} \end{frame} \begin{frame} \frametitle{The SAGE Website} \begin{center} {\dblue\LARGE The SAGE Website} \end{center} \begin{enumerate} \item {\dred Website:} \url{http://modular.math.washington.edu/sage} \item {\dred Free online SAGE notebook:} \url{http://sage.math.washington.edu:8100} \item {\dred Documentation:} Tutorial, Install Guide, Programming Guide, Reference Manual, Constructions. \item {\dred Targeted Platforms:} OS X, Linux, and MS Windows (Cygwin). \item {\dred Mailing Lists:} {\dblue sage-devel}, sage-announce, sage-forum, sage-support, and sage-uw. \item {\dred Wiki:} \url{http://sage.math.washington.edu:9001/} \item {\dred Bug Tracker:} \url{http://sage.math.washington.edu:9002/sage_trac} \item {\dred IRC Chatroom:} {\tt \#sage-dev} on irc.freenode.net \end{enumerate} \end{frame} \section{What is SAGE?} \begin{frame} \frametitle{What is SAGE?} {\Large SAGE has {\dred three} complementary aspects:} \vfill {\Large \begin{enumerate} \item {\dred A Free Distribution} of open source math software. 80MB source tarball that builds {\dblue easily} on OS X, Linux, and MS Windows. \item {\dred New Functionality} that fill in gaps in what is available elsewhere. \item {\dred A Unified Interface} to most math software: to Magma, Macaulay2, Singular, Maple, MATLAB, Mathematica, Axiom, etc. \end{enumerate} } \end{frame} \begin{frame} \vspace{-1ex} \frametitle{What is SAGE?} \begin{center} {\dblue\LARGE 1. A Free Distribution} \vspace{-2ex} \end{center} \begin{center} \begin{tabular}{|l|l|}\hline Arithmetic & {\dred GMP, MPFR} \\\hline Commutative algebra & {\dred Singular} (libcf, libfactory) \\\hline Cryptography & {\dred OpenSSL, PyOpenSSL, PyCrypto}\\\hline Group theory and combinatorics & {\dred GAP} \\\hline Graph theory & {\dred NetworkX} \\\hline Number Theory & {\dred PARI, NTL}\\\hline Numerical computation & {\dred GSL, Numpy} \\\hline Calculus, Symbolic comp & {\dred Maxima} \\\hline Specialized math & many C/C++ programs...\\\hline Command Line & {\dred IPython} \\\hline Graphical Interface & {\dblue Notebook}, {\dred jsmath, Moin wiki} \\\hline Plotting & {\dred Matplotlib, Tachyon, libgd} \\\hline Networking & {\dred Twisted} \\\hline Database & {\dred ZODB}, Python Pickles \\\hline Programming language & {\dred Python}, {\dblue SageX} (compiled python) \\\hline \end{tabular} \end{center} \vfill The components of SAGE are all active open source projects. \end{frame} \begin{frame}[fragile] \frametitle{What is SAGE?} \begin{center} {\dblue\LARGE 2. New Functionality} \end{center} Python and SageX code --- {\dred readable}: \small \begin{verbatim} algebras edu lfunctions monoids sets categories ext libs plot structure coding functions matrix quadratic_forms tests combinat geometry misc rings crypto groups modular schemes databases interfaces modules server Over 100,000 new lines (including docstrings): $ cat */*.py */*/*.py */*/*/*.py */*.pyx \ */*/*.pyx */*/*.pyx |sort |uniq | wc -l 102932 May Documentation Examples: $ cat */*.py */*/*.py */*/*/*.py */*.pyx \ */*/*.pyx */*/*.pyx |sort|uniq|grep "sage:" | wc -l 11899 \end{verbatim} \end{frame} \begin{frame} \begin{definition}[Bernoulli Numbers] The \defn{Bernoulli numbers} $B_n$ for $n\geq 0$ are {\em defined} by: \begin{equation}\label{eqn:def_bernoulli} \frac{x}{e^x - 1} = \sum_{n=0}^{\infty} B_n \frac{x^n}{n!}. \end{equation} \end{definition} Expanding the series further, we obtain the following table: \vspace{0.5ex} \noindent{}$\ds B_{0}=1,\quad B_{1}=-\frac{1}{2},\quad B_{2}=\frac{1}{6},\quad B_{4}=-\frac{1}{30},\quad B_{6}=\frac{1}{42}, \quad$\vspace{1.5ex} \noindent{}$\ds B_{8}=-\frac{1}{30},\quad B_{10}=\frac{5}{66},\quad B_{12}=-\frac{691}{2730},\quad B_{14}=\frac{7}{6},\quad $\vspace{1.5ex} \noindent{} $\ds{}\!\!B_{16}=-\frac{3617}{510},\quad B_{18}=\frac{43867}{798},\quad B_{20}=-\frac{174611}{330},\quad $ \noindent{} $ B_{22}=\frac{854513}{138},\quad B_{24}=-\frac{236364091}{2730},\quad B_{26}=\frac{8553103}{6}. $\vspace{1ex}\label{tbl:bern} Relevant to computing integrals, infinite sums, and very important in number theory. \end{frame} \begin{frame}[fragile]\frametitle{SAGE Demo: New Code (interactive help)} {\small \begin{verbatim} sage: bernoulli? # one ? for help Return the n-th Bernoulli number, as a rational number. INPUT: n -- an integer algorithm: 'pari' -- (default) use the PARI C library; by *far* the fastest. 'gap' -- use GAP 'gp' -- use PARI/GP interpreter 'magma' -- use MAGMA 'python'-- use pure Python implementation EXAMPLES: sage: bernoulli(12) -691/2730 sage: bernoulli(50) 495057205241079648212477525/66 ... AUTHORS: David Joyner and William Stein \end{verbatim} } \vfill \end{frame} \begin{frame}[fragile]\frametitle{SAGE Demo: New Code (interactive help)} {\small \begin{verbatim} sage: bernoulli?? # two question marks for source code File: ... python2.5/site-packages/sage/rings/arith.py ... if algorithm == 'pari': x = pari(n).bernfrac() # Use the PARI C library return Rational(x) elif algorithm == 'gap': x = sage.interfaces.gap.gap('Bernoulli(%s)'%n) return Rational(x) elif algorithm == 'magma': x = sage.interfaces.magma.magma('Bernoulli(%s)'%n) return Rational(x) elif algorithm == 'gp': x = sage.interfaces.gp.gp('bernfrac(%s)'%n) return Rational(x) elif algorithm == 'python': return sage.rings.bernoulli.bernoulli_python(n) else: raise ValueError, "invalid choice of algorithm" \end{verbatim} } \vfill \end{frame} \begin{frame}[fragile]\frametitle{SAGE Demo: Unique New Code} \begin{verbatim} sage: bernoulli_mod_p? Computes bernoulli numbers B_0, B_2, ... B_{p-3} modulo p. PERFORMANCE: Should be complexity O(p log p). INPUT: p -- integer, a prime OUTPUT: list -- the bernoulli numbers modulo p. EXAMPLES: sage: bernoulli_mod_p(37) [1, 31, 16, 15, 16, 4, 17, 32, 22, 31, 15, 15, 17, 12, 29, 2, 0, 2] AUTHOR: David Harvey (2006-08-06) \end{verbatim} This implements a famous algorithm of Buhler et al. \vfill And there is much much more that is unique in SAGE. \end{frame} \begin{frame} \frametitle{What is SAGE?} \begin{center} {\dblue\LARGE 3. Unified Interfaces} \end{center} \vfill\large \begin{itemize} \item SAGE {\dred interfaces to}: Axiom, GAP, GP/PARI, Kash, Macaulay2, Magma, Maple, Mathematica, MATLAB, Maxima, Octave, Singular, etc. \vfill \item This gives SAGE a wide range of {\dred functionality}. \vfill \item Unified {\dred command completion and help}. \vfill \end{itemize} \end{frame} \begin{frame}[fragile] \frametitle{SAGE Demo: Interfaces} {\dred HOW THEY WORK:} Use buffered psuedo-tty, files, and Python objects that wrap native objects. This makes it possible to wrap {\dred all} math software that has a command line interface using similar code. \vfill {\blue \begin{verbatim} sage: x = gp('9+6') # the GP/PARI math software \end{verbatim}} This fires up one copy of GP/PARI (if it wasn't already started) and sends the line {\tt 'sage[1] = 9+6'} to it. It also creates a Python class x with a field set to {\tt "sage[1]"}. {\small\blue \begin{verbatim} sage: !ps ax |grep gp 16389 p5 Ss+ 0:00.02 /Users/was/s/local/bin/gp --fast ... sage: type(x) sage: x 15 sage: x.name() 'sage[1]' sage: x.factor() [3, 1; 5, 1] \end{verbatim} } \end{frame} \begin{frame} \frametitle{Overall Structure of SAGE} {\dblue\Large The Overall Structure of SAGE} \vfill \begin{itemize} \item {\dred Custom package management system } -- 53 standard packages and 24 optional ones. Automated upgrades. \vfill \item {\dred Interactive command-line} interface -- IPython. \vfill \item {\dred Graphical user interface} -- via your web browser (AJAX app). \vfill \item {\dred Fast underlying arithmetic} -- built on mature C/C++ libraries (GMP, NTL, PARI, GSL). New code in C, SageX and Python. \vfill \item {\dred Interfaces with other software} use buffered {\dred psuedo-tty}'s. \vfill \item {\dred Special purpose components} -- {\dred Lcalc}, {\dred GMP-ECM} and {\dred FlintQS} (integer factorization), {\dred PALP} (integer polytopes), and many more. \vfill \item {\dred Mercurial revision control system} -- included standard; encourages users to be developers. \end{itemize} \end{frame} \begin{frame} \frametitle{The SAGE Notebook: GUI For Mathematics Software} \begin{enumerate} \item The SAGE Notebook -- an {\dred ``AJAX application''} like Google maps or Gmail.\vfill \item {\dred Written from scratch} by me, Alex C. and Tom B.\vfill \item Uses Python's built-in {\dred BaseHTTPServer} web server (we will switch to Twisted for robustness and security). \vfill \item Works well with Firefox and Safari.\vfill \item Client/server model which works {\dred over network} or locally. \vfill \item Current version is {\dred stable and in use by many people}. \vfill \item Try it: {\tt http://sage.math.washington.edu:8101}\vfill \end{enumerate} \end{frame} \begin{frame} \frametitle{What Next? Goals for SAGE-3.0} {\Large\dred SAGE in 2007: ``Speed and Polish''} \vfill SAGE has a huge range of features, but is overall {\dred not fast enough} to solidly compete with Magma and {\dred not polished enough} to compete with Maple/Mathematica/MATLAB. \vfill \large \begin{enumerate} \item {\dblue Optimize everything} in SAGE. (Example: SAGE devs Hart and Harvey have nearly finished the first version of a new C library for vastly faster arithmetic in $\Z[x]$ than in any other math software in the world.) \vfill \item {\dblue Design and implement parallel algorithms} at many different levels to take advantage of multicore and SMP computers. \end{enumerate} \end{frame} \begin{frame} \frametitle{Summary} \begin{enumerate} \item SAGE is {\dblue free software} for research in {\dred algebra}, {\dred geometry}, {\dred number theory}, {\dred cryptography}, and {\dred numerics}. \vfill \item SAGE is an {\dred environment for rigorous mathematical computation} built using Python, GAP, Maxima, Singular, PARI, etc., and provides a {\dred unified interface} to Mathematica, Maple, Magma, MATLAB, etc. \vfill \item There have been {\dred several successful SAGE workshops}, and there are many active SAGE developers. \vfill \item The {\dred primary goal} of SAGE is to make powerful mathematical algorithms available in an integrated package with a modern interpreted language and graphical interface. %\item {\dred SAGE is growing}. The sage-devel mailing %list has {\dblue 85 members} and about {\dblue 500 posts} %per month. \end{enumerate} \end{frame} \end{document}