Ghent Algebra and Geometry Seminars

Weekly Research Seminars

2021

Ghent University

Welcome to the Ghent Algebra and Geometry Weekly Seminars taking place on Zoom. Please note that the talks will not be recorded.

Organisers: Yairon Cid-Ruiz and Fatemeh Mohammadi


Registration:

Please email the organisers if you would like to attend the talks and feel free to suggest speakers, volunteer to speak, and invite others to attend.


Schedule:


  • May 6, 2021 (Thursday, 11:00-12:00): Simon Rigby (Ghent)

  • May 12, 2021 (Wednesday, 11:00-12:00): Federico Castillo (MPI Leipzig)

  • May 20, 2021 (Thursday, 11:00-12:00): Oliver Clarke (Bristol and Ghent)

  • May 27, 2021 (Thursday, 11:00-12:00): Alessio Caminata (University of Genova)

  • June 3, 2021 (Thursday, 11:00-12:00): Guillem Blanco (KU Leuven)

  • June 17, 2021 (Thursday, 11:00-12:00): Roy Oste (Ghent)

  • June 24, 2021 (Thursday, 11:00-12:00): Hendrik Van Maldeghem (Ghent)

  • July 1, 2021 (Thursday, 11:00-12:00): Marina Garrote López (Universitat Politècnica de Catalunya)


Titles and Abstracts:


  • May 6, 2021 (Thursday, 11:00-12:00): Simon Rigby

Title: Cohomological invariants of quadratic forms

Abstract: A cohomological invariant is a function a : H1(k,G) → Hi(k) where H1(k,G) is the set of (isomorphism classes of) principal homogeneous spaces for an algebraic group G and Hi(k) is a cohomology group of the base field k (say, with coefficients in Z/2Z), such that a is natural with respect to field extensions. Wherever possible, we shall describe this situation in plainer language, for example: if G = On is an orthogonal group then a is just an isometry invariant of all n-dimensional quadratic forms. Cohomological invariants have applications to some hard questions concerning the existence (or non-existence) of certain k-defined algebraic objects. Invariants themselves are very rare and special discoveries, so it is interesting to pursue questions like “What is the set of all the cohomological invariants of G?”. In this talk, I will explain some old and new techniques for the classification of invariants, with examples related to quadratic form theory, orthogonal groups, and Spin groups.


Title: Effective linear restrictions on the spectrum of 1-reduced constrained operators

Abstract: In the 1960’s several authors characterized the set of 1-reduced density operators via a spectral characterization: the spectra must lie in a hypersimplex. We study a constrained version: we focus of 1-reduced density operators arising from density operators with a prescribed spectrum. By considering a convex relaxation, we provide explicit necessary linear inequalities one the spectra of these reduced operators.

Since these inequalities are essentially independent of the number of particles N and the dimension of the Hilbert space, they may be interpreted as generalized Pauli constraints. This is joint work with JP. Labbe, J. Liebert, A. Padrol, E. Philippe, and C. Schilling.


  • May 20, 2021 (Thursday, 11:00-12:00): Oliver Clarke (Bristol and Ghent)

Title: Combinatorial Mutations and Block Diagonal Polytopes

Abstract: Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes and more recently have been shown to give rise to toric degenerations of various families of varieties. Whenever a matching field gives rise to a toric degeneration of the Grassmannian, the associated polytope of the toric variety coincides with the matching field polytope. In this talk I will introduce combinatorial mutations of matching field polytopes. We will explore properties of polytopes which are preserved by mutation, in particular the property of giving rise to a toric degeneration is preserved by mutations. This gives us an easy way to generate new families of toric degenerations of the Grassmannian from old. This is a joint work with Akihiro Higashitani and Fatemeh Mohammadi.



Title: Point configurations, phylogenetic trees, and dissimilarity vectors


Abstract: In 2004 Pachter and Speyer introduced the dissimilarity maps for phylogenetic trees and asked two important questions about their relationship with tropical Grassmannian. Multiple authors answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. In this talk, we present a negative answer to this second question. Then, we introduce a weighted variant of the dissimilarity map and show that weighted dissimilarity vectors form a tropical subvariety of the tropical Grassmannian in exactly the way that Pachter and Speyer envisioned. This tropical variety has a geometric interpretation in terms of point configurations on rational normal curves. This is joint work with Noah Giansiracusa, Han-Bom Moon, and Luca Schaffler.


Title: Yano's conjecture

Abstract: In 1982, T. Yano proposed a conjecture about the generic b-exponents of an irreducible plane curve singularity. Given any holomorphic function f : (C^2, 0) --> (C, 0) defining an irreducible plane curve, the conjecture gives an explicit formula for the generic b-exponents of the singularity in terms of the resolution data of f. In this talk, we will present a proof of Yano's conjecture.


  • June 17, 2021 (Thursday, 11:00-12:00): Roy Oste (Ghent)

Title: Deformations of Howe dual pairs

Abstract: Howe duality relates the representation theories of a pair of Lie groups/algebras, or generalizations thereof. The relationship is manifest in the form of a multiplicity-free joint decomposition of a relevant space under the action of the pair. In this talk, we consider deformations of dual pairs involving Lie superalgebras to the context of rational Cherednik algebras, associated to finite reflection groups, and study in particular the algebraic structures appearing in this way.


Title: The geometric connectivity of the Freudenthal-Tits Magic Square

Abstract: The Freudenthal-Tits Magic square is a fantastic arrangement of 16 Lie algebras/varieties/geometries/algebraic groups/Tits indices in a 4x4 table, with remarkable properties. In this talk I concentrate on the geometric side and show how the relations between the cells give rise to interesting problems/conjectures/theories in the area of Tits-buildings in general, with unexpected applications outside. We highlight a few of those, among which one related to Severi varieties, one related to the density theorem for algebraic groups, and one improving on a result of Vavilov & Luzgarev who found an explicit set of quadrics describing the E7 geometry over an arbitrary field in a projective 55-space.


Title: Algebraic and semi-algebraic phylogenetic reconstruction

Abstract: Phylogenetic reconstruction aims to estimate the phylogenetic tree that best explains the evolutionary relationships of current biological entities using solely information from their genoma. To this end, one usually assumes that DNA sequences evolve according to a Markov process on a phylogenetic tree ruled by a model of nucleotide substitutions. Then, one can compute the distribution of nucleotide patterns at the leaves of a phylogenetic tree in terms of the model parameters. This joint distribution is represented by a vector whose entries can be expressed as polynomials on the model parameters and satisfy certain algebraic relationships. The study of these relationships and the geometry of the algebraic varieties defined by them have provided successful insight into the problem of phylogenetic reconstruction.

However, from a biological perspective we are not interested in the whole variety, but only in the region of points that arise from stochastic parameters. In this talk, we will discuss the importance of studying these regions and we will prove that, in some cases, considering them seems to be fundamental to cope with the phylogenetic reconstruction problem. Finally, we will present the phylogenetic quartet reconstruction method ASAQ which is based on the algebraic and semi-algebraic description of distributions that arise from the general Markov model on a quartet tree. Its performance on multiple types of data will be discussed and it will be compared with classical phylogenetic reconstruction methods.



Please also feel free to join our reading seminar on Toric Geomerty.

Here is the link to the previous semester's seminar talks.