What are the criteria for associate professor habilitation in Italy in mathematics?
First of all, information about the Abilitazione Scientifica Nazionale procedure is available on https://abilitazione.miur.it/ , although in Italian only. There is an English version of that site, but it is a (sad) joke: it only contains outdated information from 2013. Unfortunately it seems impossible to find up-to-date information that is not in Italian. Anyway, here is an overview.

First thing, you need to figure out which 'branch' of academia (settore) you are applying for, because the requirements change depending on it. The branches for mathematics are:


01/A1 mathematical logic
01/A1-MAT/04 teaching/history of math
01/A2 geometry and algebra
01/A3 analysis
01/A3-MAT/06 probability and statistics
01/A4 mathematical physics
01/A5 numerical analysis
01/A6 optimization and operations research


A complete list of branches for each discipline, with their descriptions, is in the attachments at the end of http://attiministeriali.miur.it/anno-2015/ottobre/dm-30102015.aspx . Attachment D is a list in English.

For each branch, a committee (commissione) of five scholars (who where drawn at random among all eligible professors) evaluates all the applicants. The names of the members of this committee for each branch are published on https://abilitazione.miur.it/public/commissariSorteggiati_2018.php .

Then, the requirements are:


Passing certain thresholds (soglie) on 2 out of 3 of these indices: number of publications in the last 5 years, number of (Scopus or WoS) citations in the last 10 years, (Scopus or WoS) H-index restricted to the last 10 years. The thresholds for each branch and for the current version of the procedure are available on http://abilitazione.miur.it/public/documenti/2018/Tabelle_Valori_Soglia_ALLEGATI_DM_589_2018.pdf (in Italian; the right pane "II classe" is for habilitation to the associate professor role; the left pane "I classe" is for habilitation to full professor, where these intervals of 5 and 10 years are replaced by 10 and 15, respectively).
Having 3 out of 10 "titles" (titoli), which are the following fuzzily defined items:


Organization or participation as a speaker at scientific conferences in Italy or abroad
Direction or participation in the activities of a research group characterized by collaborations at national or international level
Responsibility for studies and scientific research entrusted by qualified public or private institutions
Scientific responsibility for international and national research projects, admitted to funding on the basis of competitive calls for peer review
Direction or participation in editorial committees of magazines, editorial series, encyclopedias and treaties of recognized prestige
Participation in the steering board and / or teaching assignments in  Phd programs [in Italy only] accredited by the Ministry
Teaching or research assignments (fellowships) at qualified universities or foreign or supranational research institutes
Prizes and awards for scientific activity, including affiliation to academies of recognized prestige in the sector
Results obtained in technology transfer in terms of participation in the creation of new businesses (spin-offs), development, use and marketing of patents
Specific professional experiences characterized by research activities related to the branch    


These are not binary, i.e., it is not sufficient to have n&gt;=1 conference talks to qualify for the first one, for instance. It is specified further what these criteria means in the context of each branch: this information is available on https://abilitazione.miur.it/public/pubblicacriteri_18.php . Even after reading these clarifications, it will not be immediately clear whether you qualify for one or not; it is up to the committee to decide if the documentation you have submitted is sufficient qualification to get one. To get an idea of how restrictively they are evaluated, it may help to check the past results and evaluations available on https://abilitazione.miur.it/public/pubblicarisultati_2018.php (and compare them to the submitted applications available on https://abilitazione.miur.it/public/pubblicacandidati_18.php).
Having your application evaluated positively by said committee: they get a chance to say no even if you have all the numbers and requirements.


One applies on the platform Loginmiur. You need an account there, which you get by sending information and a copy of your ID by e-mail to a certain address (instructions on that page). The application process is in Italian: there is an English version of the form, but in practice almost all text appears in Italian even if you select it.

The next deadline (and last published deadline) to apply is May 14, 2020. After that, new committees will be elected, possibly there will be adjustments to this system, and probably all the above links will be replaced by new ones. If you wish to go for the May deadline, I would recommend you to start your application immediately, because it may take some time to go through the bureaucracy (such as applying for this account). Also, you may have to submit corrections to Scopus, if needed, so that all your articles and citations are counted properly; the Scopus staff also take some time to process them.

Good luck! I understand that the procedure may look daunting for someone who does not speak Italian.

Everything I wrote above applies also to all other branches outside mathematics, with the exception of some branches, mostly liberal-art ones, that use a different method for requirement 1 (settori non bibliometrici).

One important note: habilitation does not guarantee you a job. It only grants you the right to apply for associate/full professor positions when a university has an opening for that role.
For an early-career researcher, the most common way to get a permanent job is via another kind of position, the tenure-track ricercatore di tipo B (RTD-B). You do not need a habilitation to apply for those; however, if you get a RTD-B position you will need to obtain habilitation within three years before your tenure evaluation. For this reason, often candidates who already have a habilitation are preferred. So it is a good idea to apply for habilitation as soon as you pass the thresholds in my point 1, if you plan to have a career in academia in Italy.