Tuesday, October 13, 2015 at 1:30 PM in SAS 4201
Alexey Ovchinnikov, CUNY Queens College
Symbolic computation and systems of PDEs
We will discuss upper and lower bounds for the effective Nullstellensatz for systems of polynomial PDEs. These are uniform bounds for the number of differentiations to be applied to all equations of a system of PDEs in order to discover algebraically whether it is consistent (i.e., has a solution in a differential field extension). The bounds are functions of the degrees and orders of the equations of the system and the numbers of dependent and independent variables in them. Seidenberg was the first to address this problem in 1956. The first explicit bounds appeared in 2009, with the upper bound expressed in terms of the Ackermann function. In the case of one derivation, the first explicit bound is due to Grigoriev (1989). In 2014, another bound was obtained if restricted to the case of one derivation and constant coefficients. Our new result does not have these restrictions.
Tuesday, October 20, 2015 at 1:30 PM in SAS 4201
Daniel Brake, University of Notre Dame
Applications of Monodromy
Monodromy action plays an important role in a number of mathematical theories. Stemming from a fundamental principle in complex analysis, the Cauchy integral formula, monodromy loops give all sorts of information about the interior of a region given boundary data. The uses include computing whether a pole is contained in the interior, and determining the breakup of the sheets coming together at a pole. As a consequence, monodromy is used in numerical algebraic geometry to decompose a pure-dimensional set into its irreducible components.
This talk will give an overview of monodromy, and some new connections to algebraic geometry. In particular, we will discuss how to use it to compute some local properties of algebraic varieties, as in the Numerical Local Irreducible Decomposition, and a new method for computing real tropical curves.
Tuesday, October 27, 2015 at 1:30 PM in SAS 4201
Agnes Szanto, NC State
Subresultants and Symmetric Interpolation
In 1853, Sylvester introduced double sum expressions for two finite sets of indeterminates and subresultants for univariate polynomials,showing the relationship between both notions in several but not all cases. Here we show how Sylvesters double sums can be interpreted interms of symmetric multivariate Lagrange interpolation, allowing to recover in a natural way the full description of cases.We will also report on preliminary results on extensions to symmetric multivariate Hermite interpolation.
Tuesday, November 3, 2015 at 1:30 PM in SAS 4201
Hoon Hong, NC State
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Seminar Organizer: C. Vinzant