April 2, 2002
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Prof. Gary Kennedy (Ohio State University)
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Title: Exploiting the recursive structure of moduli space
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Date, Time, Place: Tuesday, April 2, 2002, 4:00 PM, Boyd Graduate
Studies Room 304
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Rating:
G
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abstract
An elementary dimension count shows that, in the complex projective plane,
the number of rational curves of degree d passing through
3d-1 specified points is finite, but until the last decade these
numbers had been computed only through dimension d = 5. Now
all such numbers are known, thanks to a remarkable recursion discovered
by Konstevich. After looking at a naive proof, we delve more deeply
and see that Kontsevich's formula is the numerical aspect of a deeper recursive
structure: the moduli space for such curves can be compactified in
such a way that the compactifying matter is built out of simpler instances
of the same sort of moduli space. Using the moduli space, we can
even define a new "quantum product" extending the ordinary cup product
on the projective plane, and observe that Kontsevich's formula is exactly
the statement that this product is associative.
April 4, 2002
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Prof. Ernie Croot (U. C. Berkeley)
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Time, Place: 2:00 PM, Room 304
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Title: Sign Oscillations of Multiplicative Functions
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Rating:
G
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abstract
Suppose that f(n) is a completely multiplicative function taking
on values +1 and -1. It is easy to show that
f(n) = f(n+1) for more than c x integers
n < x (where c is some constant):
By a pigeonhole argument, there exist integers i, j with
0 <= i < j <= 2, such that f(2n+i) = f(2n+j)
for at least c x values of n with 2n + 2
< s. If i = 0, j = 2, then f(n)
= f((2n+i)/2) = f(n+1); and the other possibilities for i,
j immediately yield c x consecutive numbers 2n+i,
2n+i+1 = 2n+j, such that f(2n+i) = f(2n+i+1).
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The following `complementary' problem, however, is not
so easy to solve. For how many integers n < x do we
have f(n) = -f(n+1)? That is, how many sign changes does
f(n) go through as n goes from 1 to
x ? I will give some of the history of this problem, including partial
results and conjectures due to G. Harman, Pintz, Wolke, and Hildebrand,
R. Heath-Brown, Rusza, and Balog, as well as a sketch of the following
recent result due to myself: There are at least x/exp((log(x))^{1+o(1)})
values n < x such that f(n) = -f(n+1) .
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***
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Coffee and Cookies at 3:00
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***
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Prof. Juan Carlos Alvarez
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Time, Place: 3:30 PM, Room 304
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Title: Jewels and Open Problems in Finsler Geometry
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Rating:
G
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abstract
Finsler Manifolds - manifolds provided with a norm in each tangent space
- have long been poor cousins of their Riemannian counterparts. However,
recent researches relating their study to convex, metric, integral, and
symplectic geometry suggest that Finsler geometry may very well develop
to be one of the most exciting branches of global differential geometry.
In this talk I will survey some of the most attractive results and open
problems in the field.
April 11, 2002
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Prof. Chris Phillips (University of Oregon)
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Title: The structure of C* algebras associated with minimal diffeomorphisms
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Rating:
Faculty, Advanced Graduate Students
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abstract
C*-algebras are a particularly tractable class of Banach algebras, and
are moreover important in the unitary representation theory of groups,
in mathematical physics, and in parts of geometry and topology such as
index theorems and foliations. The Elliot classification program
seeks to determine up to isomorphism all simple separable nuclear
C*-algebras in terms of K-theory (essentially algebraic topology) and traces
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The easiest case, the purely infinite case, is done up to
one technical problem. The next case is the stably finite case with
projections. Here, much is known about C*-algebras obtained as direct
limits of more elementary C*-algebras. However, C*-algebraists are
more interested in crossed products, and they are also closer to the applications
listed above. Qing Lin and I have proved a direct limit decomposition
theorem for crossed products by minimal diffeomorphisms of compact manifolds.
If the diffeomorphism is uniquely ergodic and the image of the K-theory
under the trace is non-trivial (both are often satisfied), then the crossed
product belongs to the class of simple separable nuclear C*-algebras known
to be determined up to isomorphism by the Elliott invariant.
April 23, 2002
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Prof. Andrew Pollington (BYU)
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Title: Irregularities of Distribution in the d-dimensional unit cube.
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Time, Place: Tuesday, 3:30 PM, Room 304, Boyd Graduate Studies Building
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Rating:
Advanced Undergraduates
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abstract
Let P = {p_1,...,p_N} be a set of N points
in the rectangular box [0,1]^d. Let Z(P,x_1,...,x_d)
be the number of these points in the rectangle [0,x_1]x[0,x_2]x...x[0,x_d].
Define the "discrepancy" D(P,x_1,...,x_d) to be Z(P,x_1,...,x_d)-N x_1...x_d.
We will use wavelet-theoretic methods and Riesz products to obtain lower
bounds for the L_p norm of this discrepancy function for various
values of p. These functions appear as the error term in certain
pseudo Monte-Carlo methods of integration. The methods that we develop
are extensions and reformulations of techniques introduced into the subject
by Roth, Schmidt, and Halasz.
April 25, 2002
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Prof. R. Q. Jia (University of Alberta)
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Title: Spectral Analysis of the Transition Operator and its Applications
to Wavelet Analysis
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Rating:
G
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abstract
Wavelets are generated from refinable functions, which are solutions of
refinement equations. The properties of refinable functions and wavelets
can be analyzed through the subdivision and transition operators associated
with the refinement mask. In this talk we will discuss some recent
results on the spectral analysis of the transition operator and its applications
to the approximation and smoothness properties of refinable functions and
wavelets. In particular, we will give a simple algorithm to calculuate
the regularity (smoothness order) of a refinable function in terms of the
corresponding mask.
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