Univ at Albany: Math: William F. Hammond Personal Research Interests and Activity William F. Hammond
General
The range of my strictly mathematical research interests falls within the territory spanned by number theory and algebraic geometry, and I have since 1987 acquired an interest in the metamathematical topic of the design of mathematics in the international universe of networked computing Most of my visible past strictly mathematical investigations have involved modular forms, theta functions, abelian varieties, the geometry of Hilbert modular surfaces, and reduction of SchwartzBruhat functions
Mathematics on the Network
In the arena of mathematics and the network great hope has been present since, say, 1993 when CERN's Tim BernersLee's free World Wide Web (WWW) emerged into the consciousness of the mathematical community at large From the University of Minnesota Paul Lindner offered gopher, an alternative approach with browsing free to everyone and serving free to universities, and with by 1993 the handling of arbitrary contenttypes This alternative had equal potential utility to serious needs of academic mathematics, science, and engineering In a sense not much has happened since that time, while href="http:www.albany.eduhammondgellmuwebmath.html"a great deal more is explicitly desired by the mathematical community It is my opinion that much more will happen if the design of mathematics on the web is regarded as a question of the construction of an abstract metamathematical entity It may not be too late It does need the attention of mathematicians who understand the universe of networked computing Beginning in late 1987 I became interested in optimizing the use of desktop computer processing for the authoring and reading of mathematical documents Many of my ideas about href="mte.html"Mathematical Typewriter Emulation (MTE) were formed at that time Those ideas are evolving inasmuch as they are still somewhat relevant to current issues concerning math on the web The question is how to design the whole structure, as an abstract entity, of mathematical information in the universe of networked computers so as to enable that universe: to contain all information that in 1970 one expected to be able to find available in printed paper form in library buildings to admit sufficiently complex searching that a mathematician may freely locate sources in that universe of current information on a narrowly defined mathematical question beginning from little information and without reliance on human networks This has led to what I call href="igl.html"GELLMU, a name which, as I use it, refers to: a crossplatform href="http:www.gnu.org"free computer software project assembled with interchangeable components an extensible LaTeXlike markup language with provision for mathematical authors serving as href="http:www.tug.orgTUGboatArticlestb223tb72hammond.pdf" A Bridge from to XML that is suitable for single source authoring toward multiple presentation formats (SSATMF)
Reduction of Schwartz Bruhat Functions
Despite all of my work with computing and mathematics on the web, some of which includes very substantial voluntary service to my Department in supervising its IX network since 1992 and serving as both editor and manager of a substantial portion of its web service, I do have a current mathematical focus, which since 1977, has been the topic of product formulas, a phrase that in my context is implied jargon referring to identities that involve one of a certain fuzzy family of infinite products indexed by primes, often called Euler products A simple, but important, special case of such products is an essential invariant of the geometric object that is our contemporary understanding of the solution of a finite number of polynomial equations in finitely many variables And, of course, there are important modular forms and important objects in representation theory that are such products Reduction of SchwartzBruhat functions is a topic handled explicitly in Donald Roby's 1980 Albany thesis, under my direction, that arises naturally as an extension of J.I.Igusa's extension of the principal result of Andre Weil (19061998) in his Acta paper (1964) These results apply to the context of adelic analysis, a (the?) context from which product formulas arise and which should be understood today as part of href="anabasic.html"basic analysis, though that perception has yet to receive a proper dissemination in graduatelevel mathematical education inside many institutions in the United States Weil, as an easy sideline corollary, reached the only truly conceptual proof (at least that I know) of (Hilbert's product formula version of) Gauss's famous law of quadratic reciprocity In the early years after the particular Weil results (his first Acta paper published in 1964) became known, there was a substantial buzz, especially around Harvard, about the possible relation of adelic Haar measure to the study of the Lfunction of an elliptic curve This view led eventually to the Langlands Program.It even led eventually by the time of the Antwerp Meeting on Modular Forms in 1972 to an inadvertently inappropriate use of the term Weil curve (not, I believe, originating with Weil) for an elliptic curve that arises from a modular form The name Shimura curve would have been better except that at the time it had another meaning; today one says modular curve for such a curve, which is understood now as an elliptic curve that is, up to isogeny, in the image of a map the Shimura map first constructed by Goro Shimura that might have been given more highlighting in his 1971 book It was reported in 1999 that Breuil, Conrad, Diamond, and Taylor had proved the modular curve conjecture using an argument along the lines outlined by Wiles in 1993 for the case of a semistable elliptic curve. The approach to the modular curve theorem through adelicanalysis, in contrast with the recent successful approach through equivariant arithmetic geometry, so far has not born fruit I still see the absence of such a proof of this basic fact about elliptic curves as a hole in our knowledge of basic analysis without knowing what to do about it The precise Archimdean dimensions of this hole were measured by Weil and published in Math. Ann. in 1967 The subject of elliptic curves is rich enough to admit more than one approach It is, even today after the work of Wiles, not completely clear that there is no use to be found in the circle of ideas surrounding reduction of SchwartzBruhat functions in the subject of elliptic curves This question may be regarded as a question about the transport of the reduction idea to nonabelian contexts with arithmetic dimension 2 In fact, a useful connection, if any, of this idea with Artin reciprocity (see below), a context of arithmetic dimension 1, has yet to surface, so far as I know It may take time A proper understanding of basic analysis here will certainly take time I suspect that in both cases (reciprocity and elliptic) one wants to look far beyond the Heisenberg group and its Schrodinger normalizer One wants to be alert for interesting operators of finite order, especially order 6 in the elliptic case (Fourier transform is an operator of order 4 that is almost certainly relevant; of course, it is relevant for theta nullwerte.) In the general theory of product formulas, one wants to understand the entire class of identities that ultimately may be regarded as consequences (perhaps, for some cases just in a mirrorlike fashion) of two things: the fact that Haar measure on the adeles is restricted product measure inthecase evaluation of integrals, i.e., adelic and everywhere local evaluation of integrals (whether or not L1, i.e.2, absolutely convergent) that are derived from objects existing over the rational field (or, as appropriate for the context, a global field of characteristic zero) This is a fundamental task in basic analysis I expect that AndreWeil, who appears to me to have been careful about venturing conjectures, as opposed to raising questions, especially perhaps after having perceived a loss with intermediate Jacobians in the 1960's, imagined this as a further task when he posed as a challenge in his Acta paper the explicit task of bringing Artin reciprocity under this banner inasmuch as this is the most obvious more general framework to consider when pondering his treatment of the Hilbert product formula We do not have proper followup on Roby's work
Everyone Seminar, October 1993
The href="oct93.html"official notes on my October22, 1993 Albany Everyone Seminar presentation offer a beginning introduction to the idea that Fermat's Last Theorem is a consequence of knowing as much about plane curves defined by cubic equations with rational coefficients as first year undergraduates have traditionally been expected to know about plane curves defined by quadratic equations with real coefficients
Special Theta Relations
The notes of my 1988 New York City Number Theory Seminar talk have have finally appeared: Special Theta Relations, Number Theory: New York Seminar 19911995, D.V.Chudnovsky et al., eds., Springer Verlag, 1996, pp. 195199 As far as I know, Patrick McNally's 1995 doctoral dissertation on the subject has not appeared For me its chief thrust is that in the context of abelian surfaces with real quadratic endomorphisms the classicallydescribed theta relations obtained as a corollary of Mumford's Tata approach to Riemann's relations (my New York talk) have the same zero loci as the complete set of relations constructed somewhat less classically (in the newer more general Mumford setting) by Zarhin Of course, the whole subject of theta functions has deep ties to the subject of Basic Analysis, properly understood
Plans Regarding Reduction
Unless Roby, unbeknownst to me, has published his thesis, I plan an expansion of some of the material treated there and never published beyond archiving at University Microfilms I have been stalling this writing project since about 1990 while seeking a markup language worthy of the effort, leading to my development of href="http:www.albany.eduhammondgellmu" GELLMU, which, as of 2004, is adequate to this task
Weil's Real Metaplectic Group
Some day there will be available here an href="rmpgp.tex"item from 1976, never published, dealing with the straightforward but tedious calculation of the 2cocycle giving the ShaleMackeyWeil group as an extension of the group of centrally trivial automorphisms of the real Heisenberg group by the one dimensional compact real torus that provided my motivation for one of the questions treated in Roby's thesis The padic case was treated in Roby's thesis This also has been waiting for GELLMU
A Certain Class of Primes
During 199192 I went searching for the href="cpenth"second member of a class of primes that ought to be infinite This is the class of odd primes p for which the smallest positive primitive root c mod p fails also to be primitive mod p2 Such failure is equivalent to the condition cp1 1 p2 In the common case when a number is primitive mod p2 p an odd prime, it is necessarily primitive modulo every power of p and, therefore, it is a topological generator of the multiplicative group of padic units The smallest member of the class is the prime 40487 In April 2001, Professor Stephen Glasby of the University of Central Washington wrote me that he had found another example, which is the prime 6692367337 and that these two primes are the only examples smaller than 1010
Planned Course Notes on Theta Functions
Also I plan to make available here my notes from the course about theta functions that I presented in the Spring of 1995 This will require converting sketchy outlines to markup; it has very low priority right now Like any busywork task on my list it is always at risk of yielding to something more worthwhile
Syracuse3n1 Doodles
In 1992 I wrote some gp code (as in PARIgp from HenriCohen et al., see, e.g., the href="http:math.albany.edu:8010gMathMathComppariview.html" locally archived information regarding PARI) for toying with the problem known variously as Syracuse, 3 N 1, Since I used it for only a month or two without drawing any firm conclusions and it has lain dormant since that time, I am making my href="n32p"package of gp code, which contains about 55 functions in 1200 lines of code, believed to be debugged, available with a href="n32pdoc"doc giving English function definitions but otherwise without explanation In this I think that I may have seen a machine that makes infinite eventually periodic sequences of natural numbers, and another such machine, of course, is a quadratic Hilbert modular cusp, something subordinate to a real quadratic number field It's just a curiosity An online reference for the Syracuse, 3 N 1, problem is an article by Jeff Lagarias that originally appeared in the MAA Monthly: http:www.cecm.sfu.caorganicspaperslagariaspaperhtmlpaper.html
Command Line Utilities
Along the way I became interested in writing certain kinds of href="sftw.html"software, all of which had at some point been relevant to mathematical authoring For example, fwid is relevant both to MTE and to clean textplain or textansi rendering of GELLMU documents, conv is relevant both to efficient printer setting of MTE as well as to articulation between authoring platform base character sets, i.e., ASCII vs. EBCDIC, and the experience of having once needed to furnish myself with xcho led to my philosophy that every nonword character in a platform base character set needs a symbolic name in any sane singlesource authoring language on that platform
About this document
This is an example of a href="igl.html"GELLMU document Various forms of it are available: the href="researchrsch.glm"GELLMU source, the href="researchrsch.sgml"syntactic translation to SGML, the href="researchrsch.xml"subsequent translation to XML from which all end formattings are derived, the href="researchrsch.html"HTML formatting, the href="researchrsch.ltx" formatting, a href="researchrsch.dvi"DVI file for letter paper, a href="researchrsch.pdf"PDF file, and an href="http:math.albany.edu:8010mathpershammondresearchrsch.xhtml"XHTMLMathML formatting