# Algebra @ Ghent University

## Members of our research group

### Tenured

### Postdocs

- Wouter Castryck (in Leuven, 10% voluntary postdoctoral researcher in Ghent)
- Yairon Cid Ruiz

### PhD students

- Jens Bossaert
- Jeroen Meulewaeter
- Harshit Jitendra Motwani
- Simon Rigby

### Master students

- Jari Desmet
- Michiel Smet
- Mathias Stout

## Current research topics

Our main research topics are group theory, non-associative algebraic structures, and combinatorial and applied algebraic geometry.

### Structurable algebras, linear algebraic groups and related structures in geometry and group theory

We investigate the connection between a class of non-associative algebras known as *structurable algebras*
and linear algebraic groups (over arbitrary fields) and related group-theoretical structures such as Moufang sets
(this is a class of doubly transitive permutation groups, introduced by Jacques Tits)
and buildings (also introduced by Jacques Tits) and related point-line geometries.
In particular, we make use of Jordan algebras, quadratic forms, and other algebraic objects related to linear algebraic groups.
We are particularly interested in exceptional linear algebraic groups and in the corresponding cohomological invariants.

- Tom De Medts, Richard M. Weiss, Moufang sets and Jordan division algebras, Math. Ann. 335 (2006), no. 2, 415–433.
- Lien Boelaert, Tom De Medts, Anastasia Stavrova, Moufang sets and structurable division algebras, Mem. Amer. Math. Soc. 259 (2019), no. 1245, 90 pp.

### Finite groups, axial algebras and decomposition algebras

An *axial algebra* is a commutative non-associative algebra generated by certain idempotents such that the multiplication
of eigenspaces with respect to each idempotent satisfies a certain fusion law.
The example after which these algebras have been modeled, is the Griess algebra, the automorphism group of which is the Monster group.
We have been investigating connections between axial algebras and 3-transposition groups and we have developed
a theory of modules for axial algebras.
We have recently introduced the more general notion of *decomposition algebras*, allowing for a wider variety of interesting examples.
We are working on constructing explicit examples of such algebras for Chevalley groups of simply laced type.
For E8, for instance, this provides an explicit construction of a commutative non-associative 3875-dimensional algebra.

- Tom De Medts, Michiel Van Couwenberghe, Modules over axial algebras, Algebr. Represent. Theory 23 (2020), no. 1, 209-227.
- Tom De Medts, Simon Peacock, Sergey Shpectorov, Michiel Van Couwenberghe, Decomposition algebras and axial algebras, 21 pp., submitted.

### Totally disconnected locally compact groups

We have been studying automorphism groups of locally finite trees, in particular the so-called universal groups (with respect to a prescribed local action). We have extended this to the study of universal groups for locally finite right-angled buildings. We are currently extending this to right-angled buildings that are not necessarily locally finite, leading to an interesting interplay between local vs. global topological properties.

- Pierre-Emmanuel Caprace, Tom De Medts, Trees, contraction groups, and Moufang sets, Duke Math. J. 162 (2013), no. 13, 2413–2449.
- Tom De Medts, Ana C. Silva, Koen Struyve, Universal groups for right-angled buildings, Groups Geom. Dyn. 12 (2018), 231–287.

### Combinatorial algebraic geometry

We study varieties with combinatorial structures which can be initially given or obtained as degeneration of other algebraic varieties. Therefore, the study of these objects encompasses toric and tropical geometry, which have proven to be ubiquitously occurring in algebraic geometry, commutative algebra, representation theory, mathematical physics, and many other fields. The research interests of our group at Ghent University include tropical and toric geometry, polyhedral geometry as well as matroid theory. The main idea is to read the algebraic invariants of varieties in terms of their combinatorial/geometric counterparts like graphs, matroids and polytopes.

- Jürgen Herzog, Fatemeh Mohammadi,Janet Page, Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings, J. Algebra 540 (2019), 78-99.
- Fatemeh Mohammadi, Kristin Shaw, Toric degenerations of Grassmannians from matching fields, Algebraic Combinatorics 2, no. 6 (2019), 1109-1124.

### Applied algebraic geometry and algebraic statistics

Applied algebra lies at the intersection of mathematical statistics, combinatorics, multilinear algebra, and computational algebraic geometry. The main idea is to investigate mathematical structures in problems motivated by statistics and probability theory. Examples include (but are not limited to) system reliability theory, graphical models, conditional independence models and models with hidden variables.

Here is an example from system reliability theory. A system in mathematical language is a network with nodes and edges between some of the nodes. Assume that the nodes are reliable but each edge may fail with a fixed probability. A popular question in system reliability theory is to compute the probability of having a path which connects two specific nodes (a source and a target). In the algebraic approach to this problem, we associate a variety to such a network whose Hilbert series encodes the reliability of the system. The algebraic approach, besides unifying and simplifying the results in the literature, gives a clearer insight into the structure of the system.

- Amir Kafshdar Goharshady, Fatemeh Mohammadi, An Efficient Algorithm for Computing Network Reliability in Small Treewidth, Reliability Engineering and System Safety 193 (2020), 106665.
- Fatemeh Mohammadi, Caroline Uhler, Charles Wang, Josephine Yu, Generalized permutohedra from probabilistic graphical models, SIAM J. Discrete Math. 32, no. 1 (2018), 64-93.