69 research outputs found
Plethysm and fast matrix multiplication
Motivated by the symmetric version of matrix multiplication we study the
plethysm of the adjoint representation
of the Lie group . In particular, we describe the decomposition of this
representation into irreducible components for , and find highest weight
vectors for all irreducible components. Relations to fast matrix
multiplication, in particular the Coppersmith-Winograd tensor are presented.Comment: 5 page
Algebraic geometry for tensor networks, matrix multiplication, and flag matroids
This thesis is divided into two parts, each part exploring a different topic within
the general area of nonlinear algebra. In the first part, we study several applications of tensors. First, we study tensor networks, and more specifically: uniform
matrix product states. We use methods from nonlinear algebra and algebraic geometry to answer questions about topology, defining equations, and identifiability
of uniform matrix product states. By an interplay of theorems from algebra, geometry, and quantum physics we answer several questions and conjectures posed
by Critch, Morton and Hackbusch. In addition, we prove a tensor version of the
so-called quantum Wielandt inequality, solving an open problem regarding the
higher-dimensional version of matrix product states.
Second, we present new contributions to the study of fast matrix multiplication. Motivated by the symmetric version of matrix multiplication we study the
plethysm S^k(sl_n) of the adjoint representation sl_n of the Lie group SL_n . Moreover, we discuss two algebraic approaches for constructing new tensors which
could potentially be used to prove new upper bounds on the complexity of matrix
multiplication. One approach is based on the highest weight vectors of the aforementioned plethysm. The other approach uses smoothable finite-dimensional
algebras.
Finally, we study the Hessian discriminant of a cubic surface, a recently introduced invariant defined in terms of the Waring rank. We express the Hessian
discriminant in terms of fundamental invariants. This answers Question 15 of the
27 questions on the cubic surface posed by Bernd Sturmfels.
In the second part of this thesis, we apply algebro-geometric methods to
study matroids and flag matroids. We review a geometric interpretation of the
Tutte polynomial in terms of the equivariant K-theory of the Grassmannian. By
generalizing Grassmannians to partial flag varieties, we obtain a new invariant of
flag matroids: the flag-geometric Tutte polynomial. We study this invariant in
detail, and prove several interesting combinatorial properties
Matrix product states, geometry, and invariant theory
Matrix product states play an important role in quantum information theory to
represent states of many-body systems. They can be seen as low-dimensional
subvarieties of a high-dimensional tensor space. In these notes, we consider
two variants: homogeneous matrix product states and uniform matrix product
states. Studying the linear spans of these varieties leads to a natural
connection with invariant theory of matrices. For homogeneous matrix product
states, a classical result on polynomial identities of matrices leads to a
formula for the dimension of the linear span, in the case of 2x2 matrices.
These notes are based partially on a talk given by the author at the
University of Warsaw during the thematic semester "AGATES: Algebraic Geometry
with Applications to TEnsors and Secants", and partially on further research
done during the semester. This is still a preliminary version; an updated
version will be uploaded over the course of 2023.Comment: 10 pages; comments welcome
The complexity of a hybrid life: Female immigrants in France and Germany in search of their own identity
Female Muslim immigrants in France and Germany are increasingly experiencing challenges integrating into Western lifestyle. They have to adjust to occidental communities in which they try to incorporate their own traditions and values. They experience feelings of simultaneous belonging and rejection at the same time. My research concentrated on the adaptation of Muslim Women in Germany and France. Although many similarities exist between the French and German immigrant groups, it was necessary to examine each one separately. In order to understand Islamic female immigrants\u27 quest for a new identity, it was beneficial not only to address the sociocultural aspects of their lives, but also to examine the great influence immigrant literature and mass media have on them. I hope academic endeavors such as mine will be a step in the direction of a society that will be more aware of racism and sexism
The Hessian discriminant
We express the Hessian discriminant of a cubic surface in terms of fundamental invariants. This answers Question 15 from the 27 questions on the cubic surface
The Hessian Discriminant
We express the Hessian discriminant of a cubic surface in terms of
fundamental invariants. This answers Question 15 from the \emph{27 questions on
the cubic surface}. We also explain how to compute the fundamental invariants
for smooth cubics of rank 6
K-theoretic Tutte polynomials of morphisms of matroids
We generalize the Tutte polynomial of a matroid to a morphism of matroids via
the K-theory of flag varieties. We introduce two different generalizations, and
demonstrate that each has its own merits, where the trade-off is between the
ease of combinatorics and geometry. One generalization recovers the Las Vergnas
Tutte polynomial of a morphism of matroids, which admits a corank-nullity
formula and a deletion-contraction recursion. The other generalization does
not, but better reflects the geometry of flag varieties.Comment: 27 pages; minor revisions. To appear in JCT
Preface
The aim of this volume is to advance the understanding of linear spaces of symmetric matrices. These seemingly simple objects play many different roles across several fields of mathematics.
For instance, in algebraic statistics these spaces appear as linear Gaussian covariance or concentration models, while in enumerative algebraic geometry they classically represent spaces of smooth quadrics satisfying certain tangency conditions. In semidefinite programming, linear spaces of symmetric matrices define the spectrahedra on which optimization problems are considered, and in nonlinear algebra they encode partially symmetric tensors.
It is often the case that one of the above-mentioned fields inspires or pro- vides tools for the advancement of the others. In the articles that follow, the reader will find several examples where this has happened through the common link of linear spaces of symmetric matrices.
This volume is the culmination of a collaboration project with the same name, which began at MPI Leipzig in June 2020. Over the course of several months, about 40 researchers gathered on-line to work on the ideas and projects that eventually became the articles of this special issue.
We are grateful to Bernd Sturmfels for initiating the project and for being its driving force, in particular for presenting the list of open problems that served as a starting point for the working groups that formed. Many of his conjectures became theorems in this volume.
We thank Biagio Ricceri and the editorial team of Le Matematiche for co- ordinating the publication of this volume. Finally, thanks to all participants for their contributions to the talks, discussions, and articles around the project
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