Roy Smith
Mail: Roy Smith,
Department of Mathematics,
University of Georgia,
Athens, GA 30602.
Phone: (706)-542-2595 Fax: (706)-542-2573 Electronic mail: roy@math.uga.edu
I'm a professor of Mathematics
at 
My research interests are in Algebraic Geometry.
Here are my current preprints, all joint with Robert Varley:
1) A Torelli theorem for special divisor varieties associated to doubly covered
curves, sv1nr.ps
2) A Riemann singularities theorem for Prym theta divisors, with
applications, sv2rst.pdf
3) The curve of ``Prym canonical`` Gauss divisors on a Prym theta divisor, sv3pg.pdf
4) The Prym Torelli problem: an update and a reformulation as a question in
birational geometry, sv4ut.ps
5) A necessary and sufficient condition for Riemann's sigularity theorem to
hold on a Prym theta divisor, sv5rst2.pdf
6) The Pfaffian structure defining a Prym theta divisor, sv6pfaff.pdf
7) Deformations of isolated even double points of corank one, sv7cork1defs.pdf
8) A splitting criterion for an isolated singularity at x = 0 in a family of even hypersurfaces, sv8poscrkdefs.pdf
You may also view my current Vita and Publication List.
Here are some class notes. Take whatever you like.
1. (rev.lin.alg.pdf):
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!
2. (RRT.pdf): 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.
3. Algebra course notes
a. 843-1.pdf, Groups, and groups acting on sets.
b. 843-2.pdf, Why
some polynomials cannot be solved by radicals.
c. 844-1.pdf, Rings, fields, and the Galois correspondence between subgroups of automorphisms, and subfields of extensions.
d. 844-2.pdf, G(K/L) solvable implies (K/L) radical (in char.zero); examples where G = S(3), S(4), any finite abelian group.
e. 845-1.pdf,
Decomposing f.g. modules over a pid, by diagonalizing a presentation matrix;
e.g. min'l polynomial, "rat'l canonical" form.
f. 845-2.pdf, Existence
of "Jordan" form of a linear operator if min'l polynomial has all
factors linear; existence of "diagonal" form if min'l polynomial has
all factors linear and distinct, or if the matrix is "symmetric" /R
or "normal"/C.
g. 845-3.pdf, Hom functors,
duality, tensor products, exterior products.
4. Elementary Algebra course notes.
a. 4000.01-05.pdf
Well ordering, induction, binomial thm., Euclidean algorithm, gcd's, infinitude
of primes.
b. 4000.06-09.pdf
Modular arithmetic, solving congruences, Fermat little theorem, rings, domains,
fields
c. 4000.10-13.pdf
rational numbers, irrational numbers, rational roots theorem, real numbers,
infinite decimals, geometric series and repeating decimals, adjoining square
roots to Q
d. 4000.14-20.pdf
complex numbers and trigonometry, complex rationals, complex
("Gaussian") integers, Gaussian primes and Fermat's theorem on sums
of 2 squares.
e. 4000.21-24.pdf
polynomials, division algorithm, modular polynomial rings, connection with
fields obtained by adjoining roots.
f. 4000.25-30.pdf
vector dimension of field extensions, multiplicativity of dimension, dimension
of root fields, dimension of constructible extensions, impossibility proofs.
5. Algebraic Geometry Notes.
a. introAG.pdf Naive
introduction to algebraic geometry.
b. Riemann.pdf A
sketch of Riemann's approach to classifying convergent power series.
6. Review for PhD prelim preparation in algebra (100 pages)
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 > 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).
a.80006a.pdf course
outline,
b.80006b.pdf ab grps,
rings, modules,
c.80006c.pdf linear
algebra,
d.80006d.pdf grps,
fields galois,
e.80006e.pdf hw, tests
7. Math 4050. Advanced undergraduate linear algebra:
4050sum08.pdf
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.
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.
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 "spectral" 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
"locally" on spaces of polynomials, and a definition of the
derivative for any locally integrable function as an adjoint operator on
distributions.
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