Mini Real Algebraic Geometry Conference, Purdue, Mar 4, 2019.
Time and Location
9.30-11.20 AM, 2-4.20 PM: Rec 103
- Andrei Gabrielov (Purdue), 9.30-10.20 AM, Rec 103.
Title: Outer metric Lipschitz classification of definable surface singularities.
We consider the problem of Lipschitz classification with respect to the outer metric of singularities of real surfaces definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic). The problem is closely related to the problem of classification of definable functions with respect to Lipschitz contact equivalence. Invariants of Lipschitz contact equivalence presented in Birbrair et al. (2017) are used as building blocks for the complete invariant of bi-Lipschitz equivalence of definable surface singularities with respect to the outer metric. This is joint work with L. Birbrair and A. Fernandes (Fortaleza, Brazil).
- Marie-Francoise Roy (Universite de Rennes 1), 10.30 - 11.20 AM, Rec 103.
Title: Sylvester double sums, subresultants and symmetric multivariate Hermite interpolation
Sylvester doubles sums, introduced first by Sylvester are symmetric expressions of the roots of two polynomials P and Q. Sylvester's definition of double sums makes no sense if P an Q have multiple roots, since the definition involves denominators that vanish when there are multiple roots. The aims of this talk are to give a new definition for Sylvester double sums making sense if P and Q have multiple roots, which coincides with the definition by Sylvester in the case of simple roots, to prove the fundamental property of Sylvester double sums, i.e. that Sylvester double sums indexed by (k,l) are equal up to a constant if they share the same value for k+l, and to prove the relationship between double sums and subresultants, i.e. that they are equal up to a constant. In the simple root case, proofs of these properties are already known. Our more general proofs are using generalized Vandermonde determinants and a new symmetric multivariate Hermite interpolation as well as an induction on the length of the remainder sequence of P and Q.
This is joint work with Aviva Szpirglas.
- Nathanael Cox (Purdue), 2-2.40 PM, Rec 103.
Title: Semi-algebraic effective quotienting (work in progress).
- Negin Karisani (Purdue), 2.50-3.30 PM, Rec 103.
Title: Algorithms for computing spectral sequences and persistent homology for semi-algebraic filtrations (work in progress).
- Saugata Basu (Purdue), 3.30 - 4.20 PM, Rec 103.
Title: Vandermonde varieties, mirror spaces, and cohomology of symmetric semi-algebraic sets.
We prove new vanishing results on the multiplicities of the Specht modules in the cohomology groups of semi-algebraic
sets defined by symmetric polynomials. As a consequence we derive algorithms for computing the first $\ell$ Betti numbers
of such sets with complexity polynomial in the number of variables and the number of defining polynomials,
if the degrees of the defining polynomials, and $\ell$
are considered to be fixed.
(This is joint work with Cordian Reiner.)