Welcome to the Ghent Algebra and Geometry Weekly Seminars taking place on Zoom. Please note that the talks will not be recorded.
Note: Here is the updated link to the seminar webpage for 2021.
Organizer: Fatemeh Mohammadi
Registration:
Please fill in this form to get updates on the meetings. We will email you the link/password on the day of the meeting. Please feel free to suggest speakers, volunteer to speak, and invite others to attend.
Tentative Schedule:
October 26, 2020 (Monday, 12:00-13:00): Tom De Medts
October 30, 2020 (Friday, 13:30-14:30): Anneleen De Schepper
November 9, 2020 (Monday, 12:00-13:00): Yairon Cid Ruiz
November 16, 2020 (Monday, 12:00-13:00): Job Rock (HIM Bonn)
November 23, 2020 (Monday, 12:00-13:00): Oliver Clarke (Bristol)
November 30, 2020 (Monday, 12:00-13:00): Jeroen Meulewaeter
December 7, 2020 (Monday, 12:00-13:00): Johannes Roth
December 15, 2020 (Tuesday, 13:00-14:00): Koen Thas
Titles and Abstracts:
October 26, 2020 (Monday, 12:00-13:00): Tom De Medts
Structurable algebras and Hopf algebras. Slides
Abstract: Structurable algebras form a class of non-associative algebras with involution that play an important role in understanding the exceptional linear algebraic groups over arbitrary fields. However, their definition looks strange and does not make sense over fields of characteristic 2 or 3. We explore a connection with Hopf algebras and we have strong indications that this setup is better suited for allowing those bad characteristics, as I will illustrate with concrete examples. We hope that this will eventually lead to a new definition and a deeper understanding of structurable algebras, but at this point, this is very much work in progress.
October 30, 2020 (Friday, 13:00-14:00): Anneleen De Schepper
Local recognition theorems in parapolar spaces. Slides
Abstract: Parapolar spaces are axiomatically defined point-line geometries, introduced by Cooperstein in the 70ies to capture the behaviour of the Grassmannians of spherical buildings; and indeed, if not a projective space or a polar space, these Grassmannians are instances of parapolar spaces. There exists no complete classification of parapolar spaces but the interesting known examples arise from buildings (not necessarily spherical). After an introduction to parapolar spaces, I will present ways to recognise a parapolar space based on local information (i.e., what happens in one or more points). To conclude I will demonstrate the use of such results in a characterisation theorem related to the third row of the Freudenthal-Tits magic square.
November 9, 2020 (Monday, 12:00-13:00): Yairon Cid Ruiz
When are multidegrees positive?
Abstract: The concept of multidegree provides the right generalization of degree to a multiprojective setting, and its study goes back to seminal work by van der Waerden in 1929. In the first part, I will give a basic introduction to the notion of multidegrees of a multiprojective variety. Then, I will present a complete characterization for the positivity of multidegrees. I will also present several applications that derive from this characterization. This is based on joint work with Federico Castillo, Binglin Li, Jonathan Montaño and Naizhen Zhang.
November 16, 2020 (Monday, 12:00-13:00): Job Rock (MPI Bonn)
Continuous Generalized Associahedra of Type A
Abstract: Fomin and Zelevinsky introduced generalized associahedra as a way of capturing the cluster structure of their cluster algebras. Recently, Bazier-Matte, Douville, Mousavand, et al showed how to construct a generalized associahedra from the representation theory of quivers. Arkani-Hamed, He, Salvatori, and Thomas explored a continuum limit of such constructions in type A. We present some parts of a direct approach at constructing a continuous generalized associahedron in type A. This is work-in-progress with Maitreyee Kulkarni, Jacob Matherne, and Kaveh Mousavand.
November 23, 2020 (Monday, 12:00-13:00): Oliver Clarke (Bristol)
Conditional probabilities via line arrangements and point configurations. Paper
Abstract: We will explore the connection between probability distributions satisfying certain conditional independence (CI) constraints, arising from algebraic statistics, and point and line arrangements in incidence geometry. For each collection of CI statements, we associate an algebraic variety whose irreducible decomposition gives a characterisation of the distributions satisfying the original CI statements. Classically, the defining equations of these varieties are 2-minors of a matrix of variables, however in the presence of hidden variables these equations may be of higher degree. This leads to the study of the structure of determinantal hypergraph varieties which naturally include CI varieties and whose decomposition can be understood in terms of point configurations and matroids. We provide a general framework for working with hypergraph varieties and give a number of applications of our method, yielding new families of results and unifying some of existing results. This presentation is based on two joint works: one with Kevin Grace, Fatemeh Mohammadi and Harshit Motwani, and another with Fatemeh Mohammadi and Harshit Motwani as follows:
Oliver Clarke, Fatemeh Mohammadi, and Harshit J. Motwani. "Conditional probabilities via line arrangements and point configurations." arXiv preprint arXiv:2011.02450 (2020).
Oliver Clarke, Kevin Grace, Fatemeh Mohammadi, Harshit J Motwani. "Matroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models." arXiv preprint arXiv:2103.16550
November 30, 2020 (Monday, 12:00-13:00): Jeroen Meulewaeter
Geometries from inner ideals and structurable algebras
Abstract: Structurable algebras are a class of non-associative algebras introduced by Bruce Allison, which includes the class of Jordan algebras. Allison also introduced a procedure to associate a 5-graded Lie algebra to any structurable algebra, generalizing the Tits-Kantor-Koecher construction for Jordan algebras. In earlier work of Tom De Medts and co-authors on low rank geometries it became clear that structurable algebras play an important role in their description. We will show how to recover some of those geometries directly from the structurable algebras and their associated Tits-Kantor-Koecher Lie algebra, using the notion of an inner ideal.
December 7, 2020 (Monday, 12:00-13:00): Johannes Roth
Buildings of type F4
Abstract: Buildings were introduced by J. Tits to provide a geometric/combinatorial description of groups of Lie type. Via the notion of a BN-pair, the theory turned out to apply to simple algebraic groups over arbitrary fields. The spherical Moufang buildings of rank at least 2 were classified by J. Tits and R. Weiss, and the result of this classification is that most of these buildings are the buildings associated to a simple algebraic group. Among the exceptions are the mixed buildings of type F4 and the Moufang quadrangles of type F4. In this talk I first want to give a brief introduction to buildings and then continue with a construction of both the mixed buildings of type F4 and the Moufang quadrangles of type F4.
December 15, 2020 (Tuesday, 13:00-14:00): Koen Thas
Frohardt's Theorem, 30 years later
Please also feel free to join our Algebra Reading Seminar
NEW reading seminar on Toric Geomerty: First meeting in January 04, 2021