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Series: Algebra Seminar

It is well-known, that any univariate polynomial matrix A over the complex numbers that takes only positive semidefinite values on the real line, can be factored as A=B^*B for a polynomial square matrix B. For real A, in general, one cannot choose B to be also a real square matrix. However, if A is of size nxn, then a factorization A=B^tB exists, where B is a real rectangular matrix of size (n+1)xn. We will see, how these correspond to the factorizations of the Smith normal form of A, an invariant not usually associated with symmetric matrices in their role as quadratic forms. A consequence is, that the factorizations canusually be easily counted, which in turn has an interesting application to minimal length sums of squares of linear forms on varieties of minimal degree.

Series: Algebra Seminar

We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.

Series: Algebra Seminar

Conductors and minimal discriminants are two measures of degeneracy of the singular fiber in a family of hyperelliptic curves. In the case of elliptic curves, the Ogg-Saito formula shows that (the negative of) the Artin conductor equals the minimal discriminant. In the case of genus two curves, equality no longer holds in general, but the two invariants are related by an inequality. We investigate the relation between these two invariants for hyperelliptic curves of arbitrary genus.

Series: Algebra Seminar

Recent work of Jeff and Noah Giansiracusa exhibits a scheme theoretic structure for tropicalizations of classical varieties in terms of so-called semiring schemes. This works well in the framework of closed subvarieties of toric varieties, and Maclagan and Rincon recover the structure of a weighted polyhedral complex from the scheme theoretic tropicalization of a variety embedded into a torus.In this talk, I will review these ideas and show how these results can be extended by using blue schemes. This leads to an intrinsic notion of a tropicalization, independent from an embedding into an ambient space, and generalizes the above mentioned results to the broader context of log-schemes.

Series: Algebra Seminar

In this talk, I will refine the concept of the symmetry group of a
geometric object through its symmetry groupoid, which incorporates both
global and local symmetries in a common framework. The symmetry
groupoid is related to the weighted differential invariant signature of a
submanifold, that is introduced to capture its fine grain equivalence
and symmetry properties. The groupoid/signature approach will be
connected to recent developments in signature-based recognition and
symmetry detection of objects in digital images, including jigsaw puzzle
assembly.

Series: Algebra Seminar

Many varieties of interest in algebraic geometry and applications

are given as images of regular maps, i.e. via a parametrization.

Implicitization is the process of converting a parametric description of a

variety into an intrinsic (i.e. implicit) one. Theoretically,

implicitization is done by computing (a Grobner basis for) the kernel of a

ring map, but this can be extremely time-consuming -- even so, one would

often like to know basic information about the image variety. The purpose

of the NumericalImplicitization package is to allow for user-friendly

computation of the basic numerical invariants of a parametrized variety,

such as dimension, degree, and Hilbert function values, especially when

Grobner basis methods take prohibitively long.

Series: Algebra Seminar

This talks presents two projects at the interface of computer vision
and algebraic geometry. Work with Zuzana Kukelova, Tomas
Pajdla and Bernd Sturmfels introduces the distortion varieties
of a given projective variety. These are parametrized by
duplicating coordinates and multiplying them with monomials.
We study their degrees and defining equations. Exact formulas
are obtained for the case of one-parameter distortions, the
case of most interest for modeling cameras with image distortion.
Single-authored work determines the algebraic degree of
minimal problems for the calibrated trifocal variety. Our techniques
rely on numerical algebraic geometry, and the homotopy
continuation software Bertini.

Series: Algebra Seminar

Real sub-varieties and more generally semi-algebraic subsets of $\mathbb{R}^n$
that are stable under the action of the symmetric group on $n$ elements acting
on $\mathbb{R}^n$ by permuting coordinates, are expected to be topologically
better behaved than arbitrary semi-algebraic sets. In this talk I will
quantify this statement by showing polynomial upper bounds on the
multiplicities of the irreducible $\mathfrak{S}_n$-representations that
appear in the rational cohomology groups of such sets.
I will also discuss some algorithmic results on the complexity
of computing the equivariant Betti numbers of such sets and sketch some
possible connectios with the recently developed theory of FI-modules.
(Joint work with Cordian Riener).

Series: Algebra Seminar

In Multiview Geometry, a field of Computer Vision one is interested in reconstructing 3-dimensional scenes from 2-dimensional images. I will review the basic concepts in this area from an algebraic viewpoint, in particular I'll discuss epipolar geometry, fundamental matrices, and trifocal and quadrifocal tensors. I'll also highlight some in open problems about the algebraic geometry that arise.This will be an introductory talk, and only a background in basic linear algebra should be necessary to follow.

Series: Algebra Seminar

The Macaulay dual space offers information about a polynomial ideal localized at a point such as initial ideal and values of the Hilbertfunction, and can be computed with linear algebra. Unlike Gr\"obner basis methods, it is compatible with floating point arithmetic making it anatural fit for the toolbox of numerical algebraic geometry. I willpresent an algorithm using the Macaulay dual space for computing theregularity index of the local Hilbert function.