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Nicola Arcozzi

Full Professor

Department of Mathematics

Academic discipline: MAT/05 Mathematical Analysis

Curriculum vitae

Short CV
1963 Born on 05/31
1989 MB in Mathematics (Università di Milano)
1990-1995 PhD in Mathematics (Washington University in St. Louis)
1995-1998 Contract Professor.
1997 Postdoc, University di Genova
1998-2006 Assistant Professor (Università di Bologna)
2006-2017 Associate Professor (Università di Bologna)
2017- Full Professor (Università di Bologna)

Visiting professor and intensive research periods.
1999 and 2000 U. of Michigan in Ann Arbor (two months each visit)
2000 Mittag Leffler Institut (two months)
2004 Washington University in St. Louis (five months)
2009 CIRM in Trento: research in pairs (two weeks)
2010 Université de Bordeaux (one month, Prof. Invité)
2010 Oberwolfach (one week)

Short courses at advanced schools.
2011 MSRI in Berkeley (06/20-07/01): Summer Graduate Workshop
2014 Sevilla, 16-18 June;"Advanced Course in Operator Theory and Complex Analysis"
2015 St. Petersburg, 12-15 October; "Spaces of Analytic Functions and Singular Integrals (SAFSI2015)", Chebyschev Lab and Steklov Institut.
2017 Leganés (Madrid), 12-16 June, Summer School, organized by the Spanish Network in Complex Analysis and Operator Theory.

Invited speaker and colloquia (2016-2020 only, selected)
2016 "Probabilistic Harmonic Analysis and Spectral Theory", Mittag-Leffler Institut, July.
2016 IWOTA in St. Louis, July.
2016 INDAM meeting: Geometric Function Theory in Higher Dimension” in Cortona, September.
2016 INDAM meeting in Roma, workshop CFT, September.
2017 Probability and Analysis, Będlewo, Poland, May.
2018 Complex Analysis and Related Topics in S. Petersburg, April.
2018 Workshop: New Trends in Harmonic Analysis; Politecnico di Torino, May.
2018 WOTCA in Guimarães, Portugal, June.
2018 Conference on COMPLEX ANALYSIS, CIRM Trento, October.
2019 Workshop "Reproducing Kernels in Function Spaces and Their Applications", June, Euler Institute, St. Petersburg.
2019 The Second joint IMU-INdAM Conference in Analysis, September, Napoli.
2020 Invited talk for the (Online) Seminar in complex and Harmonic Analysis, Leonhard Euler International Mathematical Institute, St. Petersburg.

Selected research related activities.

In 2011, with others I organized the two-weeks Summer Graduate Workshop “The Dirichlet Space: Connections between Operator Theory, Function Theory, and Complex Analysis” at MSRI in Berkeley, and I was among the lecturers.

Each year, together with Eva Gallardo and Isabelle Chalendar, I organize the Advanced Courses in Operator Theory and Complex Analysis (Bologna 2015 and 2018, Lyons 2016, Madrid 2017, Paris 2019, Thessaloniki 2020, postponed to 2021). From 2004 to 2014 the courses were held in Seville. Among the novelties we introduced was eliminating the fees and supporting the participation of a number of PhD students and young researchers, the main target of the courses.

Together with Marco Peloso, Maura Salvatori, and others, I have organized some international conferences attended by some of most important researchers in Complex Analysis: Hilbert Function Spaces (Gargnano 2013 and 2017), INDAM workshop "Complex and Fourier Analysis, and Operator Theory (Roma 2019).

With the start of the Covid 2019 pandemic, I participated to the organization of several online initiatives aimed at keeping, even extending, the bonds within the research communities I belong to. In particular, I organized the 2020 edition of the annual Convegno Nazionale di Analisi Armonica (online).

With Marco Peloso and Filippo Bracci we organized in 2020 the (online) Seminar in Complex Analysis

Since 2017 I maintain the website Complex Analysis Lab, with seminars and lecture series which are then recorded and made available, together with the lecture notes.

I maintain regular relations with the Russian school in Complex Analysis. This has lead to the joint organization of workshops in Italy (Analysis days in Piedmont 2018 and 2019) and of a Summer School in St. Petersburg ("Operator Theory and Complex Analysis", postponed after the pandemic started). I have recently been co-organizer of the St.Petersburg Youth Conference on Probability and Mathematical Physics, 21-23/12/2020 (online), see

I have been in the organizing committee of some conferences in applied mathematics: the Summer school Mathematics in Imaging Science (2018, Bologna) and the workshop Intelligent Machines and Mathematics (2019, Bologna).

I have been in the organizing committee of several other conferences, in Italy and United States, and referee for several international grants.

I’m involved in a number of dissemination activities. Since 2017 I deliver each year a short course for high school students within the Progetto Lauree Scientifiche of the U. of Bologna.

Research interests.

My first interests were in harmonic analysis (sharp estimates for Riesz tranforms, probabilistic methods),

N. Arcozzi (1998). Riesz transforms ... ARK. MAT. 36,

but also, more recently,

N. Arcozzi, K. Domelevo, S. Petermichl (2020). Discrete Hilbert transform … PROC. AMS, 148,

then most of my research was in holomorphic spaces theory.

Mostly jointly with Richard Rochberg and Eric Sawyer, then Brett Wick, since 2002 I have been working on the classical Dirichlet space and some of its extensions and generalizations: Carleson measures, multipliers, interpolating sequences, Nehari theory. One of our main innovations was the systematic use of dyadic models. Much of that work found its synthesis in the book

N. Arcozzi, R. Rochberg, E. Sawyer, B. Wick (2019). The Dirichlet space and related function
spaces.. p. 1-536, AMS.

On weighted, analytic, Besov spaces the whole path from Carleson measures to multiplier interpolation and corona theorems is developed, using some ideas of Peter Jones, in

N. Arcozzi, D. Blasi, J. Pau (2009). Interpolating sequences ... INDIANA MATH. J., 58,

While investigating weighted Diriclet spaces in several dimensions on the unit ball, we became familiar with the Drury-Arveson space, which has a pivotal role in the operator theory of row contractions.
This lead to

N. Arcozzi, R. Rochberg, E. Sawyer (2008). Carleson Measures for the Drury-Arveson …. Adv. Math. 218,

Recently, with M. Peloso and other collaborators, we started an investigation of the Drury-Arveson space using tools from noncommutative harmonic analysis,

N. Arcozzi, A. Monguzzi, M. Peloso, M. Salvatori, (2019). Paley–Wiener Theorems … J. FOURIER ANAL. AND APPL., 25,

The analysis of the Drury-Arveson space requires a subtle analysis of the complex unit sphere’s subRiemannian structure. It was helpful working on it, some research I had carried out on this topic with collaborators from U. of Bologna, such as in

N. Arcozzi, F. Ferrari (2007). Metric normal and distance function in the Heisenberg group. MATH. ZEIT., vol. 256,

The notion of set capacity plays in Dirichlet theory a role analogous to that of arclength in Hardy theory. This way I became familiar with potential theory, which became one of my topics of interest. See e.g

N. Arcozzi (2012). Capacity of shrinking condensers in the plane. J. FUNCT. ANAL., vol. 263

or, of a more combinatorial nature,

N. Arcozzi, M. Levi, Equilibrium measures on trees, arXiv:1811.05194

For many years I tried to characterize the multipliers for the Dirichlet space on the bidisc, which presents difficulties comparable to those of two-parameter Hardy theory. In 2017 a group of young researchers and I finally managed to do that, in potential theoretic terms, in

N. Arcozzi, P. Mozolyako, K. Perfekt, G. Sarfatti, Bi-parameter Potential theory and Carleson measures for the Dirichlet space on the bidisc, arXiv preprint arXiv:1811.04990.

Soon Alexander Volberg became interested in the topic, and there is now a growing group of researchers working on related problems. For a non-capacitary approach see also

N. Arcozzi, I. Holmes, P. Mozolyako, A. Volberg, Bi-parameter embedding and measures with restriction energy condition, Math. Ann.

With the same coauthors we also have found a “Bellman proof” of the characterization of the Carleson measures for the Dirichlet space,

Bellman function sitting on a tree, arXiv:1809.03397, online on IMRN,

a nontrivial extension to the nonlinear case of which was proved by my PhD student M. Cavina.
On a different topic, Giulia Sarfatti and I developed some Hardy-Nehari theory in a quaternionic variable, for instance in

N. Arcozzi, G. Sarfatti, (2017). From Hankel operators to Carleson measures in a quaternionic variable. PROC. EDINB. MATH. SOC.

On the classical Hardy-Nehari theory from the viewpoint of signal analysis I have recently written an expository paper,

N. Arcozzi, R. Rochberg, The Hardy space from an engineer's perspective arXiv preprint arXiv:2009.12707

Another subject I came to be interested in, both via potential theory and subRiemannian geometry, is Geometric Measure Theory. A small contribution to fractal theory is in

N. Arcozzi, A. Monguzzi, M. Salvatori, Ahlfors regular spaces have regular subspaces of any dimension, arXiv:1912.02055

Most complex analysists work with the Poincaré metric, and I was curious about its origins. This lead to an article in history of maths,

N. Arcozzi (2012). Beltrami's models of non-Euclidean geometry, in Mathematicians in Bologna 1861--1960.

PhD students
MARIA ROSARIA TUPPUTI (PhD U. of Bologna 2012).
MATTEO LEVI (PhD U. of Bologna 2019), currently Postdoc at Politecnico di Torino.
NIKOLAOS CHALMOUKIS (PhD U. of Bologna 2020).
MICHELANGELO CAVINA (PhD student, 2019- ).
DANIEL BLASI (PhD Autonoma Barcelona), with a postdoc fellowship from the ISA of U. of Bologna (2009, six months), currently high school teacher in Catalunya;
GIULIA SARFATTI (PhD U. of Firenze), 2013-2015, two years; currently Assistant Professor U. Politecnica delle Marche.
PAVEL MOZOLYAKO (PhD State U. in St. Petersburg), 2017-2019, two years; currently Associate Professor at State U. of St. Petersburg.
MATTEO FIACCHI (PhD U. of Roma Tor Vergata) January 2021- .

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