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
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
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necessarily
reflect the views of nor are they endorsed by the University of Georgia
or
the University System of Georgia.