Francesca Lembo
Nationality: Italian
Host: Max Planck Society
| Background: Francesca completed her MSc in General Mathematics at the University of Genoa, where she also obtained her BSc in Mathematics in 2022. In the last semester of her Master’s, she spent some months as an Erasmus student at the Otto von Guericke Universität Magdeburg, where she started doing research work for her thesis and had the opportunity to attend the “5th Graduate Student Meeting in Applied Algebra and Combinatorics” at Freie Universität Berlin. During her studies, she also took part to initiatives of scientific divulgation in Italy as a speaker and as an organizer. In particular, she designed with some colleagues two mathematical workshops for the two last editions of the Genoa Science Festival, the largest science communication event in Italy. |
| Master thesis: SAGBI Bases of Algebras of Minors Abstract: The thesis explores the behavior of the minors of a matrix of indeterminates with respect to whether or not they are a SAGBI basis. It is known that the maximal minors are a universal Gröbner basis but, unfortunately, that is not true when switching to SAGBI bases: with the help of computer packages such as Macaulay2 and CoCoA we show a counterexample involving a lexicographic monomial order. However, it is true that the maximal minors are a SAGBI basis with respect to any diagonal monomial order. We discuss whether we can generalize this theorem to minors of arbitrary size, eventually showing that it’s not possible. Nevertheless, if we consider a square matrix of indeterminates of a certain size k, we prove that there exists a lexicographic monomial order for which the k-1-minors are a SAGBI basis. Starting from our proof of this result, we conjecture what could be a universal SAGBI basis for the special case of the algebra generated by the 2-minors of a 3×3 matrix of indeterminates and we support our conjecture with experimental results on both Macaulay2 and CoCoA. |
| Research interests: Her main research interests focus on Commutative Algebra, especially on its computational and algorithmic aspects. In addition, I am particularly interested in the interactions of Algebra with Discrete Mathematics, Combinatorics and Algebraic Geometry. |
| PhD Goals: As a PhD student in the TENORS Project, in addition, of course, to expand her knowledge, her goal is to effectively apply Algebraic Geometry and Combinatorics to approach Optimization problems. She hopes to grow both professionally and personally by cooperating with international researchers from different backgrounds and that this will enable her to become an increasingly multifaceted mathematician in the future. |
| Hobby: Her favorite thing to do is travel and experience local culture. Other than that, she really likes both listening and playing music: she can play guitar, bass and a bit of piano. She is also passionate about sports, especially football: she often play with my friends and she am a big supporter of the Genoa local team, Genoa CFC. |
