The thesis considers two distinct strategies for algebraic computation with polynomials in high dimension. The first concerns ideals and varieties with symmetry, which often arise in applications from areas such as algebraic statistics and optimization. We explore the commutative algebra properties of such objects, and work towards classifying when symmetric ideals admit finite descriptions including equivariant Gröbner bases and generating sets. Several algorithms are given for computing such descriptions. Specific focus is given to the case of symmetric toric ideals. A second area of research is on problems in numerical algebraic geometry. Numerical algorithms such as homotopy continuation can efficiently compute the approximate solutions of systems of polynomials, but generally have trouble with multiplicity. We develop techniques to compute local information about the scheme structure of an ideal at approximate zeros. This is used to create a hybrid numeric-symbolic algorithm for computing a primary decomposition of the ideal.
Tuesday, May 19, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Umit Islak – University of Minnesota
For a nonnegative random variable Y with finite nonzero mean
\mu, we say that Y^s has the Y-size bias distribution if E[Yf(Y)] =
\mu E[f(Y^s)] for all bounded, measurable f. If Y can be coupled to
Y^s having the Y-size bias distribution such that for some constant C
we have Y^s \leq Y + C, then Y satisfies a 'Poisson tail' concentration
of measure inequality. This yields concentration results for examples
including urn occupancy statistics for multinomial allocation models and
Germ-Grain models in stochastic geometry, which are members of a class of
models with log concave marginals for which size bias couplings may be
constructed more generally. Similarly, concentration bounds can be shown
when one can construct a bounded zero bias coupling or a Stein pair for a
mean zero random variable Y. These latter couplings can be used to
demonstrate concentration in Hoeffding's permutation and doubly indexed
permutations statistics. The bounds produced, which have their origin in
Stein's method, offer improvements over those obtained by using other
methods available in the literature. This work is joint with J. Bartroff,
S. Ghosh and L. Goldstein.
Tuesday, May 19, 2015 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Annachiara Korchmaros – University of Basilicata
In the study of combinatorial aspects of symmetric groups, a major problem arising from applications to Genetics consists in finding a minimum factorization of any permutation with factors from a given generating set. The difficulty in developing an adequate theory as well as the hardness of the computational complexity may heavily vary depending on the generator set. In this talk, the generating set consists of all block transpositions introduced by Bafna and Pevzner in 1998 for the study of a particular ''genome rearrangement problem''. Results, open problems, and generalizations are discussed in terms of Cayley graphs and their automorphism groups.
Wednesday, May 6, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Guillermo Lopez – University of Madrid Carlos III
In the recent past multiple orthogonal polynomials have attracted great attention.
They appear in simultaneous rational approximation, simultaneous quadrature rules,
number theory, and more recently in the study of certain random matrix models.
These are sequences of polynomials which share orthogonality conditions with respect
to a system of measures. A central role in the development of this theory is played
by the so called Nikishin systems of measures for which many results of the standard
theory of orthogonal polynomials has been extended. In this regard, we present some
results on the convergence of type I and type II Hermite-Pade approximation for a
class of meromorphic functions obtained by adding vector rational functions with real
coefficients to a Nikishin system of functions (the Cauchy transforms of a Nikishin
system of measures).
Wednesday, April 29, 2015 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
J.-C. Breton – University of Rennes
In this talk, we propose moment identities for point processes. After
revisiting the case of Poisson point processes, we propose a direct approach to
derive (joint factorial) moment identities for point processes admitting Papangelou
intensities. Applications of such identities are given to random transformations of
point processes and to their distribution invariance properties.
Please Note: For Prof. Wickelgren's Stable Homotopy Theory class
The Steenrod algebra consists of all natural transformations of
cohomology over a prime field. I will present work of Milnor showing
that the Steenrod algebra also has a natural coalgebra structure and
giving an explicit description of the dual algebra.
Wednesday, April 29, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jason Mireles-James – University of Florida Atlantic
I will discuss a two dimensional spatial pattern formation
problem proposed by Doelman, Sandstede, Scheel, and Schneider in 2003 as
a phenomenological model of convective fluid flow . In the same work
the authors just mentioned use geometric singular perturbation theory to
show that the coexistence of certain spatial patterns is equivalent to
the existence of some heteroclinic orbits between equilibrium solutions
in a four dimensional vector field. More recently Andrea Deschenes,
Jean-Philippe Lessard, Jan Bouwe van den Berg and the speaker have
shown, via a computer assisted argument, that these heteroclinic orbits
exist. Taken together these arguments provide mathematical proof of the
existence of some non-trivial patterns in the original planar PDE. I
will present some of the ingredients of this computer assisted proof.
Tuesday, April 28, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Quansen Jiu – Capital Normal University, China
In this talk, we will present some results on global classical solution to the two-dimensional compressible Navier-Stokes equations with density-dependent of viscosity, which is the shear viscosity is a positive constant and the bulk viscosity is of the type $\r^\b$ with $\b>\frac43$. This model was first studied by Kazhikhov and Vaigant who proved the global well-posedness of the classical solution in periodic case with $\b> 3$ and the initial data is away from vacuum. Here we consider the Cauchy problem and the initial data may be large and vacuum is permmited. Weighted stimates are applied to prove the main results.