"A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marl, resisting penetration… the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it… yet it finally surrounds the resistant substance. " \n—— A. Grothendieck

Biography

Paolo Dolce was born in Catanzaro, Italy in 1990. He obtained his Ph.D degree in Mathematics from the University of Nottingham in 2019. He then conducted postdoctoral research at the University of Udine from 2019-2022, and at Ben Gurion University from 2022-2024. He joined the Institute for Theoretical Sciences, Westlake University in February 2024.

Research

Some of Paolo Dolce's recent research interests include: Diophantine approximation and its generalisation to adelic curves; Arakelov geometry; Unlikely intersections.

Paolo Dolce is currently working in the field of Diophantine geometry which roughly speaking is the study of integral and rational solution of multivariable polynomial equations. These solutions can be seen as points of geometric objects (i.e. varieties), so the whole task is to locate such special points and establish which type of geometric/topological conditions on the varieties imply "the scarcity" of integral/rational points. Paolo Dolce uses several different approaches in my research, in particular techniques coming from: Arakelov geometry, Diophantine approximation and unlikely intersections.

Representative Publications

1) A note on some Diophantine inequalities over adelic curves. *Journal de Th*** éorie des Nombres de Bordeaux** (To appear).

2) (with P. Mercuri) Intersection matrices for the minimal regular model of X_0(N) and applications to the Arakelov canonical sheaf. Submitted Preprint.

3) (with F. Zucconi) On the generalisation of Roth's theorem. ** Kyoto Journal of Mathematics** (To appear).

4) (with R. Gualdi) Numerical equivalence of R-divisors and Shioda-Tate formula for arithmetic varieties. *Journal f*** ür die reine und angewandte Mathematik** (Crelle), 784 (2022), pp. 131--154.

5) Explicit Deligne pairing.** European Journal of Mathematics**, 8, Suppl. 1 (2022), pp. 101--129.

6) (with W. Czerniawska) Adelic geometry on arithmetic surfaces II: completed adeles and idelic Arakelov intersection theory. ** Journal of Number Theory**, 211 (2020), pp. 235--296.

7) Adelic geometry on arithmetic surfaces I: idelic and adelic interpretation of the Deligne pairing. ** Kyoto Journal of Mathematics**, 62, 2 (2022), pp. 433--470.

8) Fields of definition and Belyi type theorems for curves and surfaces. ** New York Journal of Mathematics**, 22 (2016), pp. 823--851.

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