# my personal website!

## about me...

### About me

I was born in Italy in 1985; 13 years ago I decided to study Mathematics: after a Bachelor in Bologna, I got a Master and a Ph.D. in Triest; then I worked for a few years as a Mathematics researcher (in the fields of algebraic and complex geometry, algebraic topology and category theory).

#### Currently

Since July 2015 I live in Vienna, where I've learned German from scratch, reaching a C1 level in March 2017 (apart from that, I speak Italian, English and French). In the last year I've been also studying some programming languages in order to work with Big Data and Machine Learning.

#### Currently

Since July 2015 I live in Vienna, where I've learned German from scratch, reaching a C1 level in March 2017 (apart from that, I speak Italian, English and French). In the last year I've been also studying some programming languages in order to work with Big Data and Machine Learning.

From June 2017 I work as Scientific Manager of the SFB Project Taming Complexity in partial differential systems with nodes at the University of Vienna (UniWien), the Vienna University of Technology (TU Wien), and the Institute of Science and Technology Austria (IST).

#### Previously

In April and May 2015 I was a teacher of mathematics at the Upper Secondary School Serpieri in Bologna.

From April 2013 until March 2015 I worked as post-doc at the Mathematics Research Unit of the University of Luxembourg (with an AFR grant by FNR – cofunded by a Marie Curie grant).

From November 2012 until February 2013 I was a postdoc at the Riemann Center for Geometry and PhysicsInstitute of Algebraic Geometry of Leibniz Universität Hannover.

I got my Ph.D. in Geometry at the International School of Advanced Studies (SISSA-ISAS) in Trieste in October 2012 under the supervision of Professor Peter Newstead (University of Liverpool).

From November 2012 until February 2013 I was a postdoc at the Riemann Center for Geometry and PhysicsInstitute of Algebraic Geometry of Leibniz Universität Hannover.

I got my Ph.D. in Geometry at the International School of Advanced Studies (SISSA-ISAS) in Trieste in October 2012 under the supervision of Professor Peter Newstead (University of Liverpool).

If you are interested in a short version (2 pages) of my CV, please follow one of the following links:

A longer version (with talks, teaching and published papers) is available at the following links:

### Publications, preprints and theses

If you want more info, simply click on any title below.

#### Publications and papers accepted for publication

##### Orbifolds and groupoids

Published on "Topology and its Applications" (Elsevier), Vol. 159, Issue 3, pp. 756-786, 15 February 2012.

Download the published version from here and the preprint (with some more details) from arXiv.

We define a $2$-category structure $(\textbf{Pre-Orb})$ on the category of reduced complex orbifold atlases. We construct a $2$-functor $F$ from $(\textbf{Pre-Orb})$ to the $2$-category $(\textbf{Grp})$ of proper étale effective groupoid objects over the complex manifolds.

Both on $(\textbf{Pre-Orb})$ and on $(\textbf{Grp})$ there are natural equivalence relations on objects: (a natural extension of) equivalence of orbifold atlases in $(\textbf{Pre-Orb})$ and Morita equivalences in $(\textbf{Grp})$. We prove that $F$ induces a bijection between the equivalence classes of its source and target.

##### Some insights on bicategories of fractions: comparison results, associators and compositions of 2-morphisms

Published on "Theory and Applications of Categories", Vol. 31, No. 10, pp. 257-329, 19 April 2016.

Download the published version from here and the preprint from arXiv.

In this paper we investigate the construction of bicategories of fractions originally described by D. Pronk: given any bicategory $\mathbf{\mathscr{C}}$ together with a suitable class of morphisms $\mathbf{W},$ one can construct a bicategory $\mathbf{\mathscr{C}}\left[\mathbf{W}^{-1}\right],$ where all the morphisms of $\mathbf{W}$ are turned into internal equivalences$,$ and that is universal with respect to this property.

Most of the descriptions leading to this construction were long and heavily based on the axiom of choice. In this paper we considerably simplify the description of the equivalence relation on $2$-morphisms and the constructions of associators, vertical and horizontal compositions in $\mathbf{\mathscr{C}}\left[\mathbf{W}^{-1}\right],$ thus proving that the axiom of choice is not needed under certain conditions. The simplified description of associators and compositions will also play a crucial role in two forthcoming papers about pseudofunctors and equivalences between bicategories of fractions.

##### A bicategory of reduced orbifolds from the point of view of differential geometry

Published on "Journal of Geometry and Physics" (Elsevier), Vol. 108 (2016), pp. 117 - 137.

Download the published version from here and the preprint from arXiv.

We describe a bicategory $(\mathcal{R}\mathbf{ed}\,\mathcal{O}\mathbf{rb})$ of reduced orbifolds in the framework of differential geometry (i.e. without any explicit reference to the notions of Lie groupoids or differentiable stacks$,$ but only using orbifold atlases$,$ local lifts and changes of charts).

In order to construct such a bicategory$,$ we firstly define a $2$-category $(\mathcal{R}\mathbf{ed}\,\mathcal{A}\mathbf{tl})$ whose objects are reduced orbifold atlases (on any paracompact$,$ second countable$,$ Hausdorff topological space). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl$,$ while the definition of $2$-morphisms and compositions of them is new in this setup. Using the bicalculus of fractions described by D. Pronk$,$ we are able to construct the bicategory $(\mathcal{R}\mathbf{ed}\,\mathcal{O}\mathbf{rb})$ from the $2$-category $(\mathcal{R}\mathbf{ed}\,\mathcal{A}\mathbf{tl})$.

We prove that $(\mathcal{R}\mathbf{ed}\,\mathcal{O}\mathbf{rb})$ is equivalent to the bicategory of reduced orbifolds described in terms of proper$,$ effective$,$ étale Lie groupoids by D. Pronk and I. Moerdijk$,$ and to the well-known $2$-category of reduced orbifolds constructed from a suitable class of differentiable Deligne-Mumford stacks.

##### Right saturations and induced pseudofunctors between bicategories of fractions

Accepted for publication by "Journal of Pure and Applied Algebra" (Elsevier).

The preprint can be found on arXiv (October 2014).

We fix any bicategory $\mathbf{\mathscr{A}}$ together with a class of morphisms $\mathbf{W}_{\mathbf{\mathscr{A}}},$ such that there is a bicategory of fractions $\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]$ (as described by D. Pronk).

Given another such pair $(\mathbf{\mathscr{B}},\mathbf{W}_{\mathbf{\mathscr{B}}})$ and any pseudofunctor $\mathcal{F}:\mathbf{\mathscr{A}}\rightarrow\mathbf{\mathscr{B}},$ we find necessary and sufficient conditions in order to have an induced pseudofunctor $\mathcal{G}:\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]\rightarrow\mathbf{\mathscr{B}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{B}}}\right]$.

Moreover$,$ we give a simple description of $\mathcal{G}$ in the case when the class $\mathbf{W}_{\mathbf{\mathscr{B}}}$ is "right saturated".

The interested reader can download an additional Appendix from here (from here you can also download the TeX file).

#### Papers submitted

##### Universal families of extensions of coherent systems

The preprint can be found on arXiv (December 2012).

We prove a result of cohomology and base change for families of coherent systems over a curve. We use that in order to prove the existence of (non-split$,$ non-degenerate) universal families of extensions of coherent systems (in the spirit of the paper Universal families of extensions by H. Lange). Such results will be applied in subsequent papers in order to describe the wallcrossing for some moduli spaces of coherent systems.

##### Equivalences of bicategories of fractions

The preprint can be found on arXiv (October 2014).

We fix any bicategory $\mathbf{\mathscr{A}}$ together with a class of morphisms $\mathbf{W}_{\mathbf{\mathscr{A}}},$ such that there is a bicategory of fractions $\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]$. Given another such pair $(\mathbf{\mathscr{B}},\mathbf{W}_{\mathbf{\mathscr{B}}})$ and any pseudofunctor $\mathcal{F}:\mathbf{\mathscr{A}}\rightarrow\mathbf{\mathscr{B}},$ we find necessary and sufficient conditions in order to have an induced equivalence of bicategories from $\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]$ to $\mathbf{\mathscr{B}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{B}}}\right]$.

In the special case when $\mathbf{W}_{\mathbf{\mathscr{B}}}$ consists only of internal equivalences of $\mathbf{\mathscr{B}},$ the bicategories $\mathbf{\mathscr{B}}$ and $\mathbf{\mathscr{B}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{B}}}\right]$ are equivalent. So the previous result gives necessary and sufficient conditions in order to have an equivalence from any bicategory of fractions $\mathbf{\mathscr{A}}\left[\mathbf{W}^{-1}_{\mathbf{\mathscr{A}}}\right]$ to any given bicategory $\mathbf{\mathscr{B}}$.

##### Weak fiber products in bicategories of fractions

The preprint can be found on arXiv (December 2014).

We fix any pair $(\mathbf{\mathscr{C}},\mathbf{W})$ consisting of a bicategory and a class of morphisms in it, admitting a bicalculus of fractions$,$ i.e. a "localization" of $\mathbf{\mathscr{C}}$ with respect to the class $\mathbf{W}.$ In the resulting bicategory of fractions$,$ we identify necessary and sufficient conditions for the existence of weak fiber products.

##### Atlases for ineffective orbifolds
(in collaboration with Dorette Pronk and Laura Scull)

The preprint can be found on arXiv (June 2016).

We give a definition of atlases for ineffective orbifolds$,$ and prove that this definition leads to the same notion of orbifold as that defined via topological groupoids.

#### Theses

##### The Hodge-Deligne polynomials of some moduli spaces of coherent systems

Ph.D. thesis in Algebraic Geometry at SISSA

Advisor: Prof. Peter Newstead

Date: October 25th, 2012

Download from here

##### Orbifolds and groupoids

Master Degree at the University of Trieste in collaboration with SISSA

Advisors: Prof. Emilia Mezzetti and Prof. Barbara Fantechi

Date: September 24th, 2009

Download from here (introduction in Italian, thesis in English)

##### Cubiche ellittiche
(elliptic curves)

Bachelor Degree at the University of Bologna

Advisor: Prof. Monica Idà

Date: July 20th, 2007

Download from here (only in Italian)

#### Preprints available on request and papers in preparation

Contact me for more details about these papers.

### Teaching

This is a list of the courses that I taught (click on each of them for more details).

##### Mathematics

at the Upper Secondary School "Serpieri" (Bologna, 2015)

In April and May 2015 I taught Mathematics at the Upper Secondary School Istituto Tecnico Agrario Arrigo Serpieri (Bologna - Italy) for the classes 1D, 2D, 3B, 4B and 5B. The topics varied from basic set theory to (pre)Calculus I (limits, derivatives and integrals).

Below you will find the classworks (for all the classes except 5B) and the solutions to each of them (in Italian since the lectures were in Italian).

##### Analysis 3B

at the University of Luxembourg (exercise sessions, 2014)

In Autumn 2014 I gave exercise sessions for the course Analyse 3B (Bachelor in Physics and Engineering) at the University of Luxembourg (lectures in French).

You can find the Exercise Sheets and the Lecture Notes (by Prof. Schlenker) on the Moodle website of the University of Luxembourg.

Recommended books (first 3 in French, last 3 in English):

• M. Hulin and M.F. Quinton, Outils mathématiques pour la physique: premier cycle universitaire et formation permanente, classes préparatoires (Collection U. Armand Colin, 1986)

• F. Liret and D. Martinais. Mathématiques pour le DEUG: Analyse 2ème année (DEUG MIAS, MASS et SM. Cours de mathématiques. Dunod, 1998)

• J. E. Marsden and A. Weinstein, Calculus: 1-3, volume 2 (Springer, 1985), available at cds.caltech.edu/~marsden/volume/Calculus/

• M. Spivak, Calculus (Cambridge University Press, 2006)

• J. Stewart, Calculus (Available 2010 Titles Enhanced Web Assign Series. Thomson Brooks/Cole, 2008)

• G. Strang, Calculus (Number Bd. 1. Wellesley-Cambridge Press, 1991)

Below you can find the solutions of almost all the Exercises listed in the lecture notes (lecture notes available on Moodle). All solutions are in French, since the course was given in French.

##### Euclidean, non euclidean and projective geometry

at the University of Luxembourg (exercise sessions, 2014)

In Spring 2014 I gave exercise sessions for the course "Géométrie euclidienne, non euclidienne et projective" (Bachelor in Mathematics) at the University of Luxembourg (lectures in English).

You can find the Exercise Sheets and the Lecture Notes (by Prof. Merkulov) on the Moodle website of the University of Luxembourg.

Recommended books:

• Miles Reid, Balazs Szendroi, Geometry and Topology (Cambridge University Press, 2005)

• Patrick J. Ryan, Euclidean, Non-Euclidean Geometry. An analytic approach (Cambridge University Press, 1986)

Below you find a series of PDFs with all the exercises solved. Sheets of type A contain all the exercises that we solved in class (with more details in some cases). Sheets of type B contain all the exercises that we had no time to cover in class.

##### Various mini-courses on stacks, orbifolds and prime numbers and cryptography

at the University of Luxembourg (2013), SISSA (2013) and Upper Secondary School "Alighieri" of Ravenna (2007)

• Mini-course for doctorands: Orbifolds and 2-categories - 13/05, 23/05 and 03/06/2013 - University of Luxembourg

• Mini-course for doctorands: Introduction to (algebraic) Deligne-Mumford stacks - 17/10, 24/10, 31/10, 07/11 and 14/11/2013 - University of Luxembourg

• Mini-course for doctorands: Introduction to algebraic (Deligne-Mumford) stacks - 07/12, 14/12/2011 and 07/02/2012 - SISSA (Trieste)

• Mini-course for high school: Prime numbers and cryptography - March - May 2007 - Secondary High School Dante Alighieri (Ravenna, Italy)

### Talks and seminars

#### Invited talks

20/02/2012
Hodge-Deligne polynomials of some moduli spaces of coherent systems
30/05/2012
Hodge-Deligne polynomials of some moduli spaces of coherent systems
24/10/2012
Hodge-Deligne polynomials of some moduli spaces of coherent systems

#### Talks at schools and conferences

26/01/2015
Wallcrossings for moduli spaces of coherent systems of type (2,d,2)
20/06/2013
The Hodge-Deligne polynomials of some moduli spaces of coherent systems

18/02/2013

#### Seminars/lectures at the International School of Advanced Studies (SISSA-ISAS), Trieste, Italy

22/07/2011
05-06/07/2010
03/03/2010
Some basics concepts of algebraic quantum field theory
03/03/2009
Elliptic curves: classification and group laws
18/12/2008

### Some links

#### Blogs and other websites

My blognote (used mainly to re-post articles, videos and photos that I found interesting, Italian mixed with English)

### Contacts

skype: matteo.tomm

skype: matteo.tomm