Martino Garonzi

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Postdoctoral fellow in Mathematics in Padova (Italy).
Mathematics Department (Torre Archimede)
Via Trieste 63, 35121, Padova (Italy).
Office 534, Floor 5.
Phone number: 0039 049 827 1461.
E-mail address: mgaronzi@math.unipd.it.

My referent is Andrea Lucchini. My interests are in algebra, and especially in group theory. The topic of my Ph.D. thesis is group coverings. Given a finite group G, define the "covering number" of this group to be the smallest number of proper subgroups of G whose union equals G. The problem is to investigate the properties of this invariant.

Here are my five publications, in chronological order:

• Finite Groups that are union of at most 25 proper subgroups. Journal of Algebra and its Applications, ISSN: 02194988. It is about the results in my master thesis, which you can find here (in Italian).
• Covering certain wreath products with proper subgroups (joint work with Attila Maroti), J. Group Theory 14 (2011), no. 1, 103–125,
• Direct Products of Finite Groups as Union of Proper Subgroups (joint work with Andrea Lucchini), Arch. Math. (Basel) 95 (2010), no. 3, 201-206.
• Covering certain Monolithic Groups with Proper Subgroups, Communications in algebra, ID LAGB-2011-2732.R1 (preprint).
• Covering Monolithic Groups with Proper Subgroups, International Journal of Group Theory, ID 1115-IJGT (preprint, for the Prooceedings of the conference "Ischia Group Theory 2012").

Here are the Slides of the talk I gave at the 2nd Biennal International Group Theory Conference in Istanbul on February 7th 2013.

Here are the Slides of the talk I gave in Padova on February 27th 2013 (Seminario Dottorato). It is supposed to be a basic and understandable introduction to the topic of my Ph.D thesis.

At the end of my bachelor I wrote a short work about the simplicity of the projective special linear group and the symplectic group (nothing new).

I am writing a pdf file about all the algebra I learned in my life (obviously I'm far from being finished), you can find it here. Be aware of the mistakes (I find mistakes anytime I look at it). Ah, unfortunately it is in Italian. I will try to produce an english version of it.

TUTORATO ALGEBRA 2 (Anno Accademico 2012/2013).

• Soluzione del primo compitino 22/11/2012 (tema B).
• Testo del secondo compitino 31/01/2013 (tema A).
• Soluzione del secondo compitino 31/01/2013 (tema A).
• Testo del primo appello 4/02/2013 (tema A).
• Soluzione del primo appello 4/02/2013 (tema B).
• Ho raccolto QUI (versione 28)
  gli esercizi fatti al tutorato e ne ho aggiunti alcuni altri.
  Vi invito a scrivermi se ci sono alcuni svolgimenti che non vi convincono.
• Se cercate ispirazioni per cose a cui pensare vi segnalo questo compendio.
  Naturalmente cercate di non trattarlo come un libro di testo, ci possono essere
  errori e imprecisioni nelle cose che vengono dette dai vari utenti.
• Se cercate la pagina del prof. Menegazzo, eccola qui.
• Segnalo anche questa pagina di Gabriele Fusacchia, dove trovate i temi degli anni scorsi.

• I am a chess player. I have written something about chess, for example this, this and this (all in Italian, sorry). As you can notice, I like chess puzzles and endgames. This is my Fide Chess Profile.
• I am vegan. In other words, I don't eat anything containing animal products. Why? There are several reasons. The main reason is this.
• I like speaking French. I was in France for 10 months (in the university Paris 11 in Orsay) with the Erasmus (Algant) program.
• I was a moderator in this Italian forum of mathematics (www.matematicamente.it).
• I like the German language. I have written something (very short and simple) in German, for example this and this.
• I like Fabrizio De André's music. Very much. He was a genius. Here are the sentences of him that I like most:
  • "Le ombre lunghe dei sacerdoti costrinsero il sogno in un cerchio di voci" (Il Sogno di Maria).
  • "Uomini, poiché all'ultimo minuto non vi assalga il rimorso ormai tardivo per non aver pietà giammai avuto e non diventi rantolo il respiro, sappiate che la morte vi sorveglia gioir nei prati o fra i muri di calce, come crescere il gran guarda il villano finché non sia maturo per la falce." (Corale, Leggenda Del Re Infelice).
  • "Tu prova ad avere un mondo nel cuore, e non riesci ad esprimerlo con le parole. E la luce del giorno si divide la piazza tra un villaggio che ride e te, lo scemo che passa. E neppure la notte ti lascia da solo: gli altri sognan se stessi e tu sogni di loro." (Un Matto).
• I don't believe in any kind of god. I am an atheist. (Repetita Iuvant.) I need to talk about this property of mine because sometimes people (in Italy) think that if one doesn't express any religious attitude, he/she is catholic. This is somehow curious: why don't they automatically think that his/her unique god is a pumpkin? Or maybe a pomegranate?
• I like cats. I think that if there is some universal truth giving a sense to life, it has something to do with cats.
• I like reading, but I often don't manage to finish a book unless I like it very much. My favourite book is One Hundred Years Of Solitude (G. G. Marquez). Here is a fantastic extract:
  • El martes a las cinco de la mañana José Arcadio habia tomado el café y soltado los perros, cuando Rebeca cerró la ventana se agarró de la cabecera de la cama para no caer. «Ahí lo trae -suspiró-. Qué hermoso está». José Arcadio se asomó a la ventana, y lo vio, trémulo en la claridad del alba, con unos pantalones que habían sido suyos en la juventud. Estaba ya de espaldas al muro y tenía las manos apoyadas en la cintura porque los nudos ardientes de las axilas le impedían bajar los brazos. «Tanto joderse uno - urmuraba el coronel Aureliano Buendía-. Tanto joderse para que lo maten a uno seis maricas sin poder hacer nada,» Lo repetía con tanta rabia, que casi parece fervor, y el capitán Roque Carnicero se conmovió porque creyó que estaba rezando. Cuando el pelotón lo apuntó, la rabia se había materializado en una sustancia viscosa y amarga que le adormeció la lengua y lo obligó a cerrar los ojos. Entonces desapareció el resplandor de aluminio del amanecer, y volvió verse a sí mismo, muy niño, con pantalones cortos y un lazo en el cuello, y vio a su padre en una tarde espléndida conduciéndolo al interior de la carpa, y vio el hielo.
• I have a blog.

• The following is a great extract from the book of Robert Pirsig "Zen and the Art of Motorcycle Maintenance":
  
  "He'd been innovating extensively. He’d been having trouble with students who had nothing to say. At first he thought it was laziness but later it became apparent that it wasn’t. They just couldn’t think of anything to say. One of them, a girl with strong-lensed glasses, wanted to write a five-hundred-word essay about the United States. He was used to the sinking feeling that comes from statements like this, and suggested without disparagement that she narrow it down to just Bozeman.
  
  When the paper came due she didn’t have it and was quite upset. She had tried and tried but she just couldn’t think of anything to say. He had already discussed her with her previous instructors and they’d confirmed his impressions of her. She was very serious, disciplined and hardworking, but extremely dull. Not a spark of creativity in her anywhere. Her eyes, behind the thick-lensed glasses, were the eyes of a drudge. She wasn’t bluffing him, she really couldn’t think of anything to say, and was upset by her inability to do as she was told. It just stumped him. Now he couldn’t think of anything to say. A silence occurred, and then a peculiar answer: "Narrow it down to the main street of Bozeman." It was a stroke of insight.
  
  She nodded dutifully and went out. But just before her next class she came back in real distress, tears this time, distress that had obviously been there for a long time. She still couldn’t think of anything to say, and couldn’t understand why, if she couldn’t think of anything about all of Bozeman, she should be able to think of something about just one street. He was furious. "You’re not looking!" he said. A memory came back of his own dismissal from the University for having too much to say. For every fact there is an infinity of hypotheses. The more you look the more you see. She really wasn’t looking and yet somehow didn’t understand this. He told her angrily, "Narrow it down to the front of one building on the main street of Bozeman. The Opera House. Start with the upper left-hand brick."
  
  Her eyes, behind the thick-lensed glasses, opened wide. She came in the next class with a puzzled look and handed him a fivethousand-word essay on the front of the Opera House on the main street of Bozeman, Montana. "I sat in the hamburger stand across the street," she said, "and started writing about the first brick, and the second brick, and then by the third brick it all started to come and I couldn’t stop. They thought I was crazy, and they kept kidding me, but here it all is. I don’t understand it."
  
  Neither did he, but on long walks through the streets of town he thought about it and concluded she was evidently stopped with the same kind of blockage that had paralyzed him on his first day of teaching. She was blocked because she was trying to repeat, in her writing, things she had already heard, just as on the first day he had tried to repeat things he had already decided to say. She couldn’t think of anything to write about Bozeman because she couldn’t recall anything she had heard worth repeating. She was strangely unaware that she could look and see freshly for herself, as she wrote, without primary regard for what had been said before. The narrowing down to one brick destroyed the blockage because it was so obvious she had to do some original and direct seeing."