## Anneleen De Schepper

PhD Student in Mathematics

## About Me

picture by Laura Andriessen

My name is Anneleen De Schepper. I'm doing a PhD in Mathematics at Ghent University, funded by the FWO-Flanders and supervised by prof. Hendrik Van Maldeghem. I'm part of the research group Incidence Geometry.

Contact info: Anneleen.DeSchepper AT UGent.be

## Research

My main research interest is geometry, in particular, buildings. Currently I work on singular Mazzocca-Melone sets. These are families of degenerate quadrics satisfying three axioms. The non-singular version is already neatly classified and I aim for a classification of the singular ones. Prominent examples are the Veronese varieties. More information on this can be found in the slides below, or on this poster.

Travel directions: It all starts in the Cayley-Dickson Factory (upper left corner, in blue). Given an arbitrary field, it produces 16 quadratic alternative algebras (even more in fact, if the characteristic is 2). Each of those algebras A is then brought to the Veronese Trans. (the yellow one). The input is A, tripled: we consider point-line geometries where the points have the shape (x,y,z), with x,y,z elements of A (and some other restrictions, of course), likewise for the lines. Once this goes through the Veronese Transformer (which applies a Veronese mapping), the lines are blown up to quadrics, which shapes differ depending on A. The resulting geometry is called a Veronese variety. For the algebras in yellow and green, there are three nice properties satisfied by each of their Veronese varieties. These properties even characterise these varieties, because even when we allow quadrics of the same kind but of arbitrary dimension, we only obtain those 7 varieties. These properties are called Mazzocca-Melone Axioms (as the first appearance of such axioms, in a more specific context, is due to those two Italians). One would expect the same thing holds for the blue and red geometries; helas, they lack structure. This can be seen by a nice representation of these geometries, as in the copy center. Indeed, it happens that each Veronese variety is the result of gluing together two dual copies of their preceder (the one on top of them).

# Work

### Publications

• A. De Schepper, H. Van Maldeghem, Graphs, defined by Weyl distance or incidence, that determine a vector space, Linear Algebra and its Applications, 449 (2014), 435-464.
• A. De Schepper, N.S.N. Sastry, H. Van Maldeghem, Split buildings of type F4 in buildings of type E6, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 88 (2017), 97-160.
• A. De Schepper, H. Van Maldeghem, Maps of polar spaces preserving a certain Weyl distance or intersection and projection properties , Journal of Comb. Theory, series A, 160 (2018), 332 - 408
• A. De Schepper, O. Krauss, J. Schillewaert, H. Van Maldeghem, Veronesean representations of projective spaces over quadratic associative algebras, submitted
• A. De Schepper, J. Schillewaert, H. Van Maldeghem, M. Victoor, On exceptional Lie geometries, submitted
• A. De Schepper, H. Van Maldeghem, Veronese representation of Hjelmslev planes over Cayley-Dickson algebras, in progress

• ### Talks

October 2018 • Buildings (Muenster, Germany)
Holy geometries
April 2018 • Seminar on Algebra and Combinatorics (Auckland, New Zealand)
A geometric counterpart to the Cayley Dickson Doubling Process
October 2017 • Buildings (Muenster, Germany)
Characterising singular Veronese varieties
September 2017 • Buildings and Symmetry (Perth, Australia)
Between the split and nonsplit Magic Square
May 2017 • 9th Shanghai Conference on Combinatorics (Shanghai, China)
Singular Veronese varieties
March 2017 • Seminar on Incidence Geometry and Buildings (Brussels, Belgium)
Singular Mazzocca-Melone sets
September 2015 • Mini-composium on Octonions (Hamburg, Germany)
Degenerate Cayley-Dickson algebras
September 2015 • Geornate di Ceometria (Caserta, Italy)
A new dimension in the Magic Square
November 2014 • 50 years of finite geometry: a conference on the occasion of Jef Thas's 70th birthday (Ghent, Belgium)
Maps related to polar spaces preserving a Weyl distance
September 2014 • Buildings (Muenster, Germany)
Veronesean representations of projective spaces over associative quadratic algebras
July 2014 • Seminar (Londen, United Kingdom)
Graphs related to classical groups and geometries
September 2013 • Buildings (Muenster, Germany)
Variations on the fundamental theorem of projective geometry

View all slides

## A geometric counterpart to the Cayley Dickson Doubling Process

#### Auckland, April 2018

View all slides

## Between the split and non-split Magic Square

#### Perth, October 2017

View all slides

## Incidence Geometry and Buildings

### Attended conferences

October 2018 • Buildings (Muenster, Germany)
January 2018 • Baer Colloquium (Braunschweig, Germany)
October 2017 • Buildings (Muenster, Germany)
September 2017 • Buildings and Symmetry (Perth, Australia)
August 2017 • Symmetries of Discrete Structures in Geometry (Oaxaca, Mexico)
May 2017 • 9th Shanghai Conference on Combinatorics (Shanghai, China)
January 2017 • Baer Colloquium (Berlin, Germany)
May 2016 • Groups, geometry and the influence of Jacques Tits - A conference in honour of Richard Weiss (Karlsruhe, Germany)
March 2016 • Sheds (Muenster, Germany)
September 2015 • Buildings (Muenster, Germany)
September 2015 • Mini-composium on Octonions (Hamburg, Germany)
September 2015 • Geornate di Geometria (Caserta, Italy)
January 2015 • Winter meeting on Bruhat-Tits buildings (London, United Kingdom)
November 2014 • 50 years of finite geometry: a conference on the occasion of Jef Thas's 70th birthday (Ghent, Belgium)
September 2014 • Buildings (Muenster, Germany)
May 2014 • Baer Colloquium (Ghent, Belgium
November 2013 • Baer Colloquium (Kaiserslautern, Germany)
September 2013 • Buildings (Muenster, Germany)

# Non-Related

Apart from mathematics, I like swimming, biking and running. One day I might call it triatlon.