15 research outputs found
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
Reconstruction of Orthogonal Polyhedra
In this thesis I study reconstruction of orthogonal polyhedral surfaces
and orthogonal polyhedra from partial information about their
boundaries. There are three main questions for which I provide novel
results. The first question is "Given the dual graph, facial angles and
edge lengths of an orthogonal polyhedral surface or polyhedron, is it
possible to reconstruct the dihedral angles?" The second question is
"Given the dual graph, dihedral angles and edge lengths of an
orthogonal polyhedral surface or polyhedron, is it possible to
reconstruct the facial angles?" The third question is "Given the
vertex coordinates of an orthogonal polyhedral surface or polyhedron, is
it possible to reconstruct the edges and faces, possibly after
rotating?"
For the first two questions, I show that the answer is "yes" for
genus-0 orthogonal polyhedra and polyhedral surfaces under some
restrictions, and provide linear time algorithms. For the third
question, I provide results and algorithms for orthogonally convex
polyhedra. Many related problems are studied as well
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Teaching informatics to novices: big ideas and the necessity of optimal guidance
This thesis reports on the two main areas of our research: introductory programming as the traditional way of accessing informatics and cultural teaching informatics through unconventional pathways.
The research on introductory programming aims to overcome challenges in traditional programming education, thus increasing participation in informatics. Improving access to informatics enables individuals to pursue more and better professional opportunities and contribute to informatics advancements. We aimed to balance active, student-centered activities and provide optimal support to novices at their level. Inspired by Productive Failure and exploring the concept of notional machine, our work focused on developing Necessity Learning Design, a design to help novices tackle new programming concepts. Using this design, we implemented a learning sequence to introduce arrays and evaluated it in a real high-school context. The subsequent chapters discuss our experiences teaching CS1 in a remote-only scenario during the COVID-19 pandemic and our collaborative effort with primary school teachers to develop a learning module for teaching iteration using a visual programming environment.
The research on teaching informatics principles through unconventional pathways, such as cryptography, aims to introduce informatics to a broader audience, particularly younger individuals that are less technical and professional-oriented. It emphasizes the importance of understanding informatics's cultural and scientific aspects to focus on the informatics societal value and its principles for active citizenship. After reflecting on computational thinking and inspired by the big ideas of science and informatics, we describe our hands-on approach to teaching cryptography in high school, which leverages its key scientific elements to emphasize its social aspects. Additionally, we present an activity for teaching public-key cryptography using graphs to explore fundamental concepts and methods in informatics and mathematics and their interdisciplinarity. In broadening the understanding of informatics, these research initiatives also aim to foster motivation and prime for more professional learning of informatics
Play Among Books
How does coding change the way we think about architecture? Miro Roman and his AI Alice_ch3n81 develop a playful scenario in which they propose coding as the new literacy of information. They convey knowledge in the form of a project model that links the fields of architecture and information through two interwoven narrative strands in an “infinite flow” of real books