UW Algebra Seminar
Abstracts
Speaker:
J.M. Bois, Universite de Reims
Title:
Higher level transcendence degree of division rings
Date:
April 5
Abstract:
Let A be an associative algebra. For all positive integers q,
Petrogradsky
defines a dimension of level q of A, denoted Dim_q(A), generalising
Gelfand-Kirillov dimension (q=2). Using these we define a
transcendence
degree
of level q for A, denoted Tdeg_q(A), generalising the
Gelfand-Kirillov
transcendence degree. We will survey the basic properties of
Tdeg_q. We
proceed to show: if U is the enveloping algebra of a
Lie
algebra
and has a quotient ring Frac(U), then Tdeg_q Frac(U) = Dim_q U. This
is done using the natural Poisson structure on the associated graded
algebra of U (with respect to the canonical filtration).
Speaker:
Jenia Tevelev, University of Texas, Austin
Title:
Tropical Compactifications
Date:
April 12
Abstract:
Many interesting varieties arising in algebraic geometry
are not compact. I will describe a new method for compactifying
varieties that generalizes the construction of a toric variety
in the following way. Compactifications of an algebraic torus
correspond to complete fans. Similarly, our compactifications
correspond to polyhedral structures on the non-archemedean
amoeba of a non-compact variety. I will also explain how these
compactifications relate to known compactifications such as
log canonical models, Chow quotients, etc in a few understood
cases such as complements to hyperplane arrangements and moduli
spaces of hyperplane arrangements.
Speaker:
Sasha Polishchuk, University of Oregon
Title:
Tautological cycles on Jacobians
Date:
May 3
Abstract:
In this talk I will describe an action of a certain large Lie
algebra on the Chow group of the Jacobian J of a curve. Using
this action
one can get a number of interesting relations between
"tautological classes" on J (associated with the curve embedded
into J).
Speaker:
Sandor Kovacs, University of Washington
Title:
Families of varieties of general type: the Shafarevich conjecture
and related problems
Date:
May 17
Abstract:
At the 1962 ICM Shafarevich announced a conjecture regarding
finiteness
properties of families of smooth projective curves. It was
confirmed in the
geometric case by Parshin (1968) and Arakelov (1971), and in the
arithmetic
case by Faltings (1983). This conjecture is related to many other
problems,
perhaps the most famous one is the Mordell conjecture: by a very
nice
argument, now know as "Parshin's covering trick" the Mordell
conjecture
follows from the Shafarevich conjecture.
In recent years many results have been obtained with regard to
higher
dimensional generalizations of Shafarevich's conjecture in the
geometric case. In this talk I will review the original conjecture,
it's possible generalizations, and the current knowledge in the
field.
Speaker:
Sangjib Kim, Yale University
Title:
Standard monomial theory for flag algebras
Date:
May 31
Abstract:
Let G be the general linear group or the symplectic group over the
complex number field, and U be its maximal unipotent subgroup. We
study standard monomial theory of the ring of regular functions on
G/U,
called the flag algebra, using the philosophy of Groebner bases and
SAGBI bases combined with classical invariant theory. We describe
the
flag algebra in terms of Gelfand-Tsetlin patterns. In particular,
we
show that the flag algebra is a flat deformation of a toric
variety.
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