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Series: Algebra Seminar

The Brill-Noether varieties of a curve C parametrize embeddings of C of prescribed degree into a projective space of prescribed dimension. When C is general in moduli, these varieties are well understood: they are smooth, irreducible, and have the “expected” dimension. As one ventures deeper into the moduli space of curves, past the locus of general curves, these varieties exhibit intricate, even pathological, behaviour: they can be highly singular and their dimensions are unknown. A first measure of the failure of a curve to be general is its gonality. I will present a generalization of the Brill—Noether theorem, which determines the dimensions of the Brill—Noether varieties on a general curve of fixed gonality, i.e. “general inside a chosen special locus". The proof blends a study of Berkovich skeletons of maps from curves to toric varieties with tropical linear series theory. The deformation theory of logarithmic stable maps acts as the bridge between these ideas. This is joint work with Dave Jensen.

Series: Algebra Seminar

We show that in many instances, at the heart of a problem in numerical computation sits a special 3-tensor, the structure tensor of the problem that uniquely determines its underlying algebraic structure. In matrix computations, a decomposition of the structure tensor into rank-1 terms gives an explicit algorithm for solving the problem, its tensor rank gives the speed of the fastest possible algorithm, and its nuclear norm gives the numerical stability of the stablest algorithm. We will determine the fastest algorithms for the basic operation underlying Krylov subspace methods --- the structured matrix-vector products for sparse, banded, triangular, symmetric, circulant, Toeplitz, Hankel, Toeplitz-plus-Hankel, BTTB matrices --- by analyzing their structure tensors. Our method is a vast generalization of the Cohn--Umans method, allowing for arbitrary bilinear operations in place of matrix-matrix product, and arbitrary algebras in place of group algebras. This talk contains joint work with Ke Ye and joint work Shmuel Friedland.

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).