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Being Italian, I have a biased view on my homeland's mathematical impact in the world, so I would like to get some impartial opinion on the topic.

I would measure the mathematical relevance in terms, for example, of: -contemporary importance -historical importance

Contemporary importance is about the mathematical productivity of new papers, the quantity and quality of resourceful and respected mathematicians of a nation, and the international prestige of math universities.

Historical importance is almost the same but in the broader landscape of history.

I would also like some thoughts about perks and defects of Italian mathematics, again, historically and contemporarily.

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    $\begingroup$ This question is too broad. What do you expect as an answer? A history of Italian mathematics? Italy was the first country in Europe where mathematics started during the Renaissance. It always was and still is one of the principal contributors to mathematics. Look at some history of math book. $\endgroup$ – Alexandre Eremenko Feb 5 '18 at 1:03
  • $\begingroup$ Last two sentences could be made more specific as to the time frame and what "importance", "respect" and "defects" mean, and perhaps even asked separately. $\endgroup$ – Conifold Feb 5 '18 at 4:32
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I do not believe that the question can be answered here in its sweeping generality so I will only offer some remarks.

US News and World Report has a list of Best Global Universities for Mathematics, although one can expect bias in their listings. Still, in the top 10 we have US, UK, France and Switzerland. The highest Italian one is #32, University of Milan, with Sapienza University of Rome close second, #36. The highest German one is University of Bonn, #26. Population of Italy is not far behind UK and France, but it is 20 million less than Germany's.

As for "prestige" of mathematical journals, one can get some idea using SJR index, a "prestige" sensitive version of the impact factor. Unlike the impact factor, SJR assigns different values to citations depending on the "prestige" of the journals they came from, citations from the more "prestigious" journals are valued higher. The top ten "prestigious" mathematics journals have SJR indices between 4.063 and 9.989 and come from the US (3), Germany (3), Sweden, UK, Russia and Switzerland (1 each). The highest ranking Italian journal is Rivista di Matematica, #33, with SJR index 1.708. Only two others have it above 1, Annali della Scuola normale superiore di Pisa (1.456, #41) and Atti della Accademia Nazionale dei Lincei (1.083, #68). These are the only three in the top 100. I am not sure how to measure output of mathematical papers from a country, but perhaps one can extract something from MathSciNet.

Wikipedia gives a list of Italian mathematicians by century, but I am not quite sure how to measure "prestige", "importance", etc., in centuries past either. Here is a shorter list of "famous" Italian mathematicians. If one insists on subjective impressions, I'd say that Italy is not quite at the epicenter of world's mathematics, where Cardano and Ferrari, or Cavalieri and Toricelli, or Peano, Ricci and Levi-Civita were. Peano's axiomatization of arithmetic, Volterra's integral equations and Ricci-Curbastro's tensor calculus and the Ricci flow, date to the turn of the 20th century. Fewer names come to mind after 1950s, one is Calabi of the Calabi-Yau manifolds prominent in applications to string theory. Guerraggio writes that "Renato Caccioppoli is probably the most “storied” Italian mathematician, the one who has been most been talked and written about, even beyond the circle of specialists", but I am not sure that many US mathematicians would recognize the name. His student De Giorgi, who solved Hilbert's 19th problem, might be better known. Bombieri, who collaborated with him on the Bernstein's problem for minimal surfaces, is probably better known for winning the 1974 Fields medal for applications of the large sieve in number theory.

Perhaps the most recognizable (recent) Italian names, other than the aforementioned, come from algebraic geometry: Castelnuovo, Fano, Enriques, del Pezzo, Veronese, Segre, Severi. As for "defects", Italian school of algebraic geometry again comes to mind, unfortunately. There was a progressive decline in rigor standards from Castelnuovo to Enriques to Severi, right at the time when the general trend went the other way. Eventually, the results became too uncertain to be usable and had to be reworked in Zariski-Weil framework, see The Italian School of Algebraic Geometry and Abel’s Legacy by Brigaglia et al. Here is Mumford:

"The Italian school, and notably Severi, Todd, Eger, and B. Segre, developed a general theory of Chern classes in the algebraic case... In the period 1935 to 1950, Severi published many papers on series of equivalence and its generalizations to higher dimensions. It is hard to untangle everywhere what he conjectured and what he proved and, unfortunately, some of his conclusions are incorrect.

Of other relevant resources I'll mention Guerraggio-Nastasi's book Italian Mathematics Between the Two World Wars and proceedings of 2014 History of Mathematics in Italy and Spain: State of the Art and Future Developments, abstracts are available online.

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    $\begingroup$ To the list of "most recognizable Italian names" I'll add two remarks: 1st in between 19th and 20th century there was an Italian school also in differential geometry, considering the work of Beltrami, Betti, Bianchi with this last name which deserves special attention: his "Lezioni sulla teoria dei gruppi continui di trasformazioni" is one of the few text in Italian which still quite often makes its way through bibliographies... $\endgroup$ – Nicola Ciccoli Feb 5 '18 at 7:06
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    $\begingroup$ And then there is Vito Volterra. At his times he was certainly considered one of the best in mathematical physics throughout the world. Four times plenary speaker at the ten emerging ICM congresses, his work in pure and applied math had a profound impact within his contemporaries. The comparison with the level of post WWII Italian math is certainly saddening; the effect of 20 years of fascism was deep. $\endgroup$ – Nicola Ciccoli Feb 5 '18 at 7:09
  • $\begingroup$ If you learn real analysis today, you will hear the names Fubini, Tonelli, Vitali, etc. Another strength of Italian mathematics in the early 1900s. $\endgroup$ – Gerald Edgar Feb 6 '18 at 13:23
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The very top of the mathematics profession is shown by the Fields Medal.

The Italian mathematician Enrico Bombieri received the Fields Medal in 1974. So there is an Italian in that august list. (He is the only one.) Most countries have no recipients of the Fields Medal at all, so this is an argument that Italy has high international standing in mathematics.

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    $\begingroup$ The list of Fields medalists is too small sample to make any conclusions. $\endgroup$ – Alexandre Eremenko Feb 5 '18 at 14:01

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