ÿþ<html xmlns="http://www.w3.org/TR/REC-html40"> <head> <meta name=Title content="Roy Smith's Home Page"> <meta http-equiv=Content-Type content="text/html; charset=unicode"> <title>Roy Smith's Home Page</title> <style> <!-- h2 { font-size:18.0pt; font-family:Times; font-weight:bold; } a:link, span.MsoHyperlink { color:blue; text-decoration:underline; text-underline:single; } a:visited, span.MsoHyperlinkFollowed { color:blue; text-decoration:underline; text-underline:single; } p { font-size:10.0pt; font-family:Times; } pre { tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt; font-size:10.0pt; font-family:Times; } span.HTMLPreformattedChar { font-family:Courier; } span.Heading1Char { font-family:Calibri; color:#345A8A; font-weight:bold; } span.Heading2Char { font-family:Calibri; color:#4F81BD; font-weight:bold; } .MsoChpDefault { font-size:10.0pt; } --> </style> </head> <body bgcolor=white lang=EN-US link=blue vlink=blue> <div class=WordSection1> <h1><span style='"Times New Roman"'>Roy Smith</span></h1> <h2><span style='"Times New Roman"'><a href="http://www.math.uga.edu/~roy/Images/Smith.jpg"><span style='color:windowtext;text-decoration:none;text-underline:none'><img border=0 width=215 height=318 id="_x0000_i1025" src="http://www.math.uga.edu/~roy/Images/Smith.jpg" align=middle></span></a></span></h2> <pre><b>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Mail:&nbsp;</b>&nbsp;&nbsp;&nbsp;<i>&nbsp; Roy Smith,<br>  </i></pre> <pre><i>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Department of Mathematics,<br>  </i></pre> <pre><i>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; University of Georgia,<br>  </i></pre> <pre><i>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Athens, GA 30602.&nbsp;</i></pre> <p><b><span style='"Times New Roman"'>Phone: </span></b><i><span style='"Times New Roman"'>(706)-542-2595</span></i><span style='"Times New Roman"'> <b>Fax: </b><i>(706)-542-2573 </i><b>Electronic mail: </b><i>roy@math.uga.edu</i> </span></p> <p>I'm a professor of <a href="http://www.math.uga.edu/">Mathematics</a> at&nbsp;<a href="http://www.uga.edu"><span style='text-decoration:none;text-underline: none'><img border=0 width=26 height=26 id="_x0000_i1026" src="ugasmall.gif"align=middle></span></a> </p> <p>My research interests are in Algebraic Geometry. </p> <p>Here are my current preprints, all joint with Robert Varley: <br> 1) A Torelli theorem for special divisor varieties associated to doubly covered curves, <a href="http://www.math.uga.edu/%7Eroy/sv1nr.ps">sv1nr.ps</a> </p> <p>2) A Riemann singularities theorem for Prym theta divisors, with applications, <a href="http://www.math.uga.edu/%7Eroy/sv2rst.pdf">sv2rst.pdf</a> </p> <p>3) The curve of ``Prym canonical`` Gauss divisors on a Prym theta divisor, <a href="http://www.math.uga.edu/%7Eroy/sv3pg.pdf">sv3pg.pdf</a> </p> <p>4) The Prym Torelli problem: an update and a reformulation as a question in birational geometry,&nbsp; <a href="http://www.math.uga.edu/%7Eroy/sv4ut.ps">sv4ut.ps</a></p> <p>5) A necessary and sufficient condition for Riemann's sigularity theorem to hold on a Prym theta divisor, <a href="http://www.math.uga.edu/%7Eroy/sv5rst2.pdf">sv5rst2.pdf</a></p> <p>6) The Pfaffian structure defining a Prym theta divisor, <a href="http://www.math.uga.edu/%7Eroy/sv6pfaff.pdf">sv6pfaff.pdf</a></p> <p>7) Deformations of isolated even double points of corank one, <a href="http://www.math.uga.edu/%7Eroy/sv7cork1defs.pdf">sv7cork1defs.pdf</a></p> <p>8) A splitting criterion for an isolated singularity at x = 0 in a family of even hypersurfaces, <a href="http://www.math.uga.edu/%7Eroy/sv8poscrkdefs.pdf">sv8poscrkdefs.pdf</a></p> <p>&nbsp;You may also view my current <a href="http://www.math.uga.edu/%7Eroy/Roy.pdf">Vita and Publication List</a>. </p> <div align=center style=' text-align:center;'><span style='"Times New Roman"'> <hr size=2 width="100%" align=center> </span></div> <p><span style='"Times New Roman"'><a href="classes.htm">Classes I'm Teaching<br> </a>&nbsp; </span></p> <p>Here are some class notes. Take whatever you like.</p> <p>1. (<a href="http://www.math.uga.edu/%7Eroy/rev.lin.alg.pdf">rev.lin.alg.pdf</a>): &nbsp;Linear algebra notes, including spectral theorem for symetric operators, jordan form, rational canonical form, minimal and characteristic polynomials, and Cayley Hamilton, all in 15 pages!</p> <p>2. (<a href="http://www.math.uga.edu/%7Eroy/rrt.pdf">RRT.pdf</a>): &nbsp;A discussion of the easy aspects of the Riemann Roch theorem for curves, surfaces, and n dimensional smooth manifolds. We give Riemann's classical proof for curves, assuming his results on the existence of meromorphic differentials of first and second kinds. Then we reprove from scratch the Hirzebruch version for curves, using (and recalling) sheaf cohomology, but only sketching Serre's proof of the duality theorem. Then we prove similarly Hirzebruch's version for smooth surfaces embedded in projective three space. Finally we sketch the formalism of chern roots and their use in defining Todd classes and in stating the general HRR. This is aimed at a grad student who has had complex analysis of one variable, and a little topology.</p> <p>3. Algebra course notes</p> <p><span style='"Times New Roman"'>a. <a href="http://www.math.uga.edu/%7Eroy/843-1.pdf">843-1.pdf</a>,&nbsp; Groups, and groups acting on sets.</span></p> <p><span style='"Times New Roman"'><br> b. <a href="http://www.math.uga.edu/%7Eroy/843-2.pdf">843-2.pdf</a>,&nbsp; Why some polynomials cannot be solved by radicals.</span></p> <p><span style='"Times New Roman"'>c. <a href="http://www.math.uga.edu/%7Eroy/844-1.pdf">844-1.pdf</a>, Rings, fields, and the Galois correspondence between subgroups of automorphisms, and subfields of extensions.</span></p> <p><span style='"Times New Roman"'>d. <a href="http://www.math.uga.edu/%7Eroy/844-2.pdf">844-2.pdf</a>, G(K/L) solvable implies (K/L) radical (in char.zero); examples where G = S(3), S(4), any finite abelian group.</span></p> <p><span style='"Times New Roman"'><br> e. <a href="http://www.math.uga.edu/%7Eroy/845-1.pdf">845-1.pdf</a>, Decomposing f.g. modules over a pid, by diagonalizing a presentation matrix; e.g. min'l polynomial, &quot;rat'l canonical&quot; form.</span></p> <p><span style='"Times New Roman"'><br> f. <a href="http://www.math.uga.edu/%7Eroy/845-2.pdf">845-2.pdf</a>, Existence of &quot;Jordan&quot; form of a linear operator if min'l polynomial has all factors linear; existence of &quot;diagonal&quot; form if min'l polynomial has all factors linear and distinct, or if the matrix is &quot;symmetric&quot; /R or &quot;normal&quot;/C.</span></p> <p>g. <a href="http://www.math.uga.edu/%7Eroy/845-3.pdf">845-3.pdf</a>, Hom functors, duality, tensor products, exterior products.</p> <p><span style='"Times New Roman"'>4. Elementary Algebra course notes.</span></p> <p><span style='"Times New Roman"'><br> a. <a href="http://www.math.uga.edu/%7Eroy/4000.01-05.pdf">4000.01-05.pdf</a>&nbsp;&nbsp;&nbsp; Well ordering, induction, binomial thm., Euclidean algorithm, gcd's, infinitude of primes.</span></p> <p><span style='"Times New Roman"'>b. <a href="http://www.math.uga.edu/%7Eroy/4000.06-09.pdf">4000.06-09.pdf</a>&nbsp;&nbsp; Modular arithmetic, solving congruences, Fermat little theorem, rings, domains, fields<br> <br> c. <a href="http://www.math.uga.edu/%7Eroy/4000.10-13.pdf">4000.10-13.pdf</a>&nbsp; rational numbers, irrational numbers, rational roots theorem, real numbers, infinite decimals, geometric series and repeating decimals, adjoining square roots to Q<br> <br> d. <a href="http://www.math.uga.edu/%7Eroy/4000.14-20.pdf">4000.14-20.pdf</a>&nbsp;&nbsp; complex numbers and trigonometry, complex rationals, complex (&quot;Gaussian&quot;) integers, Gaussian primes and Fermat's theorem on sums of 2 squares.<br> <br> e. <a href="http://www.math.uga.edu/%7Eroy/4000.21-24.pdf">4000.21-24.pdf</a>&nbsp;&nbsp; polynomials, division algorithm, modular polynomial rings, connection with fields obtained by adjoining roots.<br> <br> f. <a href="http://www.math.uga.edu/%7Eroy/4000.25-30.pdf">4000.25-30.pdf</a>&nbsp; vector dimension of field extensions, multiplicativity of dimension, dimension of root fields, dimension of constructible extensions, impossibility proofs.</span></p> <p>5. Algebraic Geometry Notes. </p> <p><span style='"Times New Roman"'>a. <a href="http://www.math.uga.edu/%7Eroy/introAG.pdf">introAG.pdf</a> Naive introduction to algebraic geometry. <br> <br> b. <a href="http://www.math.uga.edu/%7Eroy/Riemann.pdf">Riemann.pdf</a> A sketch of Riemann's approach to classifying convergent power series. </span></p> <p>6. Review for PhD prelim preparation in algebra (100 pages) </p> <p><span style='"Times New Roman"'>These condensed notes include basic theorems about pid: uniqueness of factorization and decomposition of finitely generated modules, application to Jordan and rational canonical forms of matrices; also Gausstheory of content and unique factorization in Z[X], Dumas Eisenstein irreducibility, Noetherian rings, Sylow theorems, Jordan - Holder, the fundamental theorem of Galois theory, Zorn lemma, existence of algebraic closures of fields, normality, separability, cyclotomic polynomials, insolvability of general polynomials of degree &gt; 4, duality and spectral theorems. These notes are most useful for someone reviewing the material a second time. Consult the 843-4-5 course notes for more details, examples and omitted topics (tensors). <br> <br> a.<a href="http://www.math.uga.edu/%7Eroy/80006a.pdf">80006a.pdf</a> course outline, <br> <br> b.<a href="http://www.math.uga.edu/%7Eroy/80006b.pdf">80006b.pdf</a> ab grps, rings, modules, <br> <br> c.<a href="http://www.math.uga.edu/%7Eroy/80006c.pdf">80006c.pdf</a> linear algebra, <br> <br> d.<a href="http://www.math.uga.edu/%7Eroy/80006d.pdf">80006d.pdf</a> grps, fields galois, <br> <br> e.<a href="http://www.math.uga.edu/%7Eroy/80006e.pdf">80006e.pdf</a> hw, tests </span></p> <p>7. Math 4050. Advanced undergraduate linear algebra: </p> <p><span style='"Times New Roman"'><a href="http://www.math.uga.edu/%7Eroy/4050sum08.pdf">4050sum08.pdf</a> <br> <br> Review of basic definitions, dimension theory, statements of basic facts about row reduction, and detailed proofs of existence and uniqueness of jordan forms for split minimal polynomials, as well as generalized jordan forms (rational canonical forms) for all minimal polynomials. <br> <br> Every decomposition theorem is proved three times, in increasing degree of complexity: i) reduced minimal polynomial, i.e. maximal number of cyclic factors, or diagonal case, ii) minimal polynomial equals characteristic polynomial, i.e. minimal number of factors, or cyclic case, iii) general case. <br> <br> There is also a sketch of existence of cyclic product decomposition for finite abelian groups, a complete treatment of determinants and the cayley- hamilton theorem. a complete treatment of &quot;spectral&quot; i.e. structure theorems for normal operators in finite dimensional real and complex spaces, some discussion of duality, solving homogeneous ode's with constant coefficients to motivate jordan form, of inverting linear constant coefficient operators &quot;locally&quot; on spaces of polynomials, and a definition of the derivative for any locally integrable function as an adjoint operator on distributions.</span></p> <p>8.On teaching. </p> <p><span style='"Times New Roman"'>a. <a href="http://www.math.uga.edu/~roy/onteaching.pdf">onteaching.pdf</a> </span></p> <p><span style='"Times New Roman"'>9. Math 5200: metric Euclidean geometry</span></p> <p><span style='"Times New Roman"'>               a. <a href="http://www.math.uga.edu/~roy/5200/5200.1.pdf">5200.1 axioms, congruence fa 07.pdf</a></span></p> <p><span style='"Times New Roman"'>               b. <a href="http://www.math.uga.edu/~roy/5200/5200.2.pdf">5200.2 parallel lines.pdf</a></span></p> <p><span style='"Times New Roman"'>               c. <a href="http://www.math.uga.edu/~roy/5200/5200.3.pdf">5200.3 fa07 similarity 1.pdf</a></span></p> <p><span style='"Times New Roman"'>               d. <a href="http://www.math.uga.edu/~roy/5200/5200.4.pdf">5200.4 fa07 euclid's content 1.pdf</a></span></p> <p><span style='"Times New Roman"'>               e. <a href="http://www.math.uga.edu/~roy/5200/5200.5.pdf">5200.5  fa07 law of cosines.pdf</a></span></p> <p><span style='"Times New Roman"'>               f. <a href="http://www.math.uga.edu/~roy/5200/5200.6.pdf">5200.6 fa07 concurrence.pdf</a></span></p> <p><span style="Times New Roman&quot;">10. <a href="http://www.math.uga.edu/~roy/camp2011/10.pdf">Epsilon camp notes on Euclidean Geometry</a></span></p> <div align=center style=' text-align:center;'><span style='"Times New Roman"'> <hr size=2 width="100%" align=center> </span></div> <p><span style='font-size:7.5pt;"Times New Roman"'>Woods Hole conference 1964<br> <br> Some people have expressed a wish to have a copy of the proceedings of the 1964 conference in algebraic geometry at Woods Hole, Massachusetts.&nbsp; <br> Here is an apparently complete set, courtesy of my friend Doug Clark, who attended the meeting.<br> &nbsp; <br> <a href="http://www.math.uga.edu/%7Eroy/woodshole1.pdf">woods hole 1</a>.&nbsp; Theory of singularities: Abhyankar, Hironaka, Zariski;<br> Classification of surfaces and moduli: Kodaira, Matsusaka, Mumford, Nagata, Rosenlicht, Igusa.<br> <br> <a href="http://www.math.uga.edu/%7Eroy/woodshole2.pdf">woods hole 2</a>. Grothendieck cohomology: Artin, Verdier, Tate.<br> Zeta functions and arithmetic of abelian varieties: Cassels, Dwork, Shimura, Serre.<br> <br> <a href="http://www.math.uga.edu/%7Eroy/woodshole3.pdf">woods hole 3</a>. Seminar on singularities: Abhyankar, Hironaka, Zariski.&nbsp; <br> Moduli questions: Ehrenpreis, Kodaira, Mayer, Mumford, Rauch.<br> Seminar on etale cohomology of number fields: Artin, Verdier.<br> Families of abelian varieties and number theory: Kuga, Shimura.<br> Seminar on commutative algebra: Auslander, Greenleaf, Lichtenbaum, Rim, Samuel, Schlessinger.<br> Etale cohomology: Hartshorne.<br> Fixed point theorem seminar: Atiyah, Bott.</span></p> <div align=center style=' text-align:center;'><span style='"Times New Roman"'> <hr size=2 width="100%" align=center> </span></div> <p>The content and opinions expressed on this Web page do not necessarily reflect the views of nor are they endorsed by the University of Georgia or the University System of Georgia. </p> </div> </body> </html>