Membership

I have been a member of UMI (Italian Mathematical Union) since 1992.

I was a member of G.N.A.M.P.A. (National Group of Mathematical Analysis, Probability and Applications) of the Istituto Nazionale di Alta Matematica (until 1999 of the C.N.R.).

I was a member of the PRIN (National Research Projects) "Theory of measurement, integral geometry and theory of fixed points" (scientific coordinator: Paolo de Lucia, scientific director: Aljosa Volcic) starting from November 30, 2004.

I participated in the PRIN "Theory of measurement, real analysis and applications" (coordinator and scientific director: Aljosa Volcic) - 2007 call.

I have always been part of the former 60% annual research projects (both at the University of Bari and at the University of Calabria) in the field of Functional Analysis and Probability.

I was MathSciNet reviewer for Mathematical Reviews.

Organizational activity

I was a member of the board of the Department of Mathematics of the University of Calabria in the academic years 2002/2003, 2003/2004, 2004/2005.

I was in the board of the Ph. D. in Psychology of Programming and Artificial Intelligence of the Department of Educational Sciences of the University of Calabria.

I was delegated by the Dean of the Faculty of Economics for the initiative concerning the training internships "Learning on the job" object of the PON “Orientation, Counselling, Reception and Tutoring 2”.

I was President of the Scientific Library of the University of Calabria from February 2004 to November 2007.

Jury member

I was part of the Commission of selection for a post as a university researcher for the disciplinary sector MAT/05 at the Faculty of Economics of the University of Bari.

I was part of the jury for a research grant for the scientific-disciplinary sector MAT / 05 announced by Unical.

Colloquia

University of Warsaw - Department of Mathematics – October 12-19, 2009
Title: “A generalization of Kakutani’s splitting procedure I, II”.

Fields Institute for Research in Mathematical Sciences - Toronto – May 2, 2012
Title: “A new class of low discrepancy sequences of partitions and points”
(visibile sul web site del Fields Institute).

University of Linz – June 11, 2012
Title: “LS –sequences of partitions and points in dimensiono ne and two”

Graz University of Technology – June 15, 2012
Title: “LS-sequences of points in the unit interval” (visibile sul web site del Doctoral program in Discrete Mathematics).

Talks

XIX Congress of the Italian Mathematical Society (Bologna, Italy, September 2011)
Title: “LS-successioni di punti a bassa discrepanza”.

Second International Conference on Uniform Distribution (Strobl, Austria - July 2010)
Title: “Discrepancy of LS-sequences of partitions”

XVIII Congress of the Italian Mathematical Society (Bari, Italy, September 2007)
Title: “Successioni di partizioni e un teorema alla von Neumann”

Tulka Internet Seminar 2000 (Blaubeuren, Germany 2000)
Title: “Differential operators, semigroups and positive linear operators on unbounded intervals”.

S`eminaire Itinerant d’Equations Fonctionnelles, Approximation et Convexit´e (Cluj-Napoca, Romania - 1996)
Title: “Shape preserving properties of some positive linear operators on unbounded intervals”.

3rd International Conference on Functional Analysis and Approximation Theory (Maratea, Potenza, Italy 1996)
Title: “On a new sequence of positive linear operators on unbounded intervals”.

Analisi Funzionale e Applicazioni (Ancona, Italy - 1996)
Title: “On some degenerate differential operators on weighted function spaces”.

Functional Analysis: Methods and Applications (Maratea, Potenza, Italy 1995)
Title: “Approximation processes and diffusion equations on unbounded intervals”.

I took part to dozens of national and international conferences, in Italy and abroad (such as in Budapest, Erlangen, Leuven, Oberwolfach, Paris, Salzburg, Stockholm, Tel Aviv, Vienna, ...), in Measure Theory schools in Grado and in Mondello, at numerous Functional Analysis and Measure Theory conferences in Ischia, Ancona, Maratea, Cefalù, Gargnano, Varenna, at UMI congresses, at the UMI-DMV conference (Perugia, 2007) during which I also performed as a pianist during an Italian-German concert.

I also gave several talks at the Department of Mathematics of the University of Bari and Calabria.

Main research fields

At the beginning I was interested in complex analysis and, in particular, in Bergman spaces, and I wrote my first paper on this subject.

After this period, my interests have been devoted to positive operators, approximation theory and diffusion equations. The technique and the theory used in that researches belong to Functional Analysis, Approximation Theory, Semigroup Theory of Operators and Probability Theory. More precisely, the aim of these researches was to study some qualitative properties of the solutions of certain abstract Cauchy problems associated to degenerate differential operators of elliptic type on a class of function spaces of continuous functions defined on unbounded intervals. This analysis has been done with the tools of strongly continuous semigroups of operators, their representation and their approximation by iterates of positive linear operators defined ad hoc. In this way it was possible to generalize previous results obtained in the context of differential operators defined on bounded intervals. In this fields I wrote the six papers.

These researches have been generalized recently by a group of mathematicians of the Department of Mathematics of the University of Bari and of the University of Potenza.

Once I moved to the University of Calabria, and after a forced interruption of research due to the commitment made in the Presidency of the Scientific Library of the university, I dedicated myself to number theory and, in particular, to uniformly distributed sequences of partitions, reviving interest in an old Kakutani’s idea, which has not been sufficiently exploited. So, after two articles dedicated to the succession of partitions, also thanks to the ideas born from a master's thesis and a doctoral thesis in co-tutorship with the University of Graz of which I was co-supervisor (supervisor Aljosa Volcic), I wrote the 'article "Discrepancy of LS sequences of partitions and points": this article opened the way for a line of research that is still very active.

In this paper I analyze a family of uniformly distributed sequences of partitions of the interval [0,1], which is obtained by successive refinements of the interval [0,1]. The paper provides precise estimates of the discrepancy and identifies, among them, a subfamily with optimal discrepancy (i.e. of the order 1/n). Furthermore, I present an algorithm that associates a uniformly distributed suquence of points to each sequence of partitions. Finally, I provide estimates for the discrepancy of these sequences of points, and show that sequences of partitions with optimal discrepancy correspond to sequences of points with optimal discrepancy (that is, of the order log n / n). In the paper I also provide the first known explicit estimate for the discrepancy of a particular Kakutani sequence.

In this area of research I have published three other articles.

Articles

[1] Alcune osservazioni sulla norma essenziale dell'operatore di composizione interna nello spazio di Bergman, Istituto Lombardo (Rend. Sci.) A 128 (1994, pp. 19-29. ISSN: 0392-9523

[2] On a new sequence of positive linear operators on unbounded intervals (con F. Altomare), Rend. Circ. Mat. Palermo Serie II, Suppl. 40 (1996), 23-36. ISSN: 0009-725X

[3] On some degenerate differential operators on weighted function spaces (con F. Altomare), J. Math. Analysis Appl. 213 (1997), 308-333. ISSN: 0022-247X

[4] Shape preserving properties of some positive linear operators on unbounded intervals, J. Approx. Theory 93 No.1 (1998), 140-156. ISSN: 0021-9045

[5] Approximation properties of some sequences of positive linear operators on weighted function spaces, Rend. Circ. Mat. Palermo, Serie II, Suppl. 52 (1998), 305-336. ISSN: 0009-725X

[6] Markov processes and diffusion equations on unbounded intervals (con F. Altomare), Taiwanese J. of Math 5 No.1 (2001), 141-167. ISSN: 1027-5487

[7] Trotter's type theorems for generalized Szàsz-Mirakjan operators in polynomial weighted function spaces, Indian J. of Math. 44 No.1 (2002), 1-20. ISSN: 0019-5324

[8] Kakutani’s splitting procedure in higher dimension (con A. Volcic), Rend. Ist. Mat. Trieste 39 (2007), 119-126. ISSN: 0049-4704

[9] A von Numann theorem for uniformly distributed sequences of partitions (with A. Volcic). Rend. Cir. Mat. Palermo. vol. 60 n.1-2 (2011) p. 83-88, ISSN: 0009-725X

[10] Discepancy of LS-sequences of partitions and points, Ann. Mat. Pura Appl. vol. 191 (2012) p. 819-844, ISSN: 0373-3114

[11] A dynamical system approach to the Kakutani-Fibonacci sequence  (with Maria Rita Iacò and Aljosa Volcic). Erg. Theory & Dyn. Systems, vol. 34, p. 1794-1806 (2014), ISSN: 0143-3857

[12] Extension of van der Corput algorithm to LS-sequences.  App. Math. And Comp., vol. 255, p. 207-213 (2015), ISSN: 0096-3003

[13] Comparison between LS-Sequences and beta-adic van der Corput Sequences  In: Monte Carlo and Quasi-Monte Carlo Methods. vol. 163, p. 261-270 (2016), Lueven: Ronald Cools and Dirk Nuyens, ISBN: 978-3-319-33507-0

Other titles, activities, awards

I obtained a Diploma in Piano (equivalent to a master degree) at the age of 19 with full marks at the Conservatory of Cosenza, where I studied Composition for a few years.

I am a Bechstein pianist.

Award-winning in the two 2020 editions of the Global Music Awards international competition, I received three bronze medals: in March (the only Italian pianist awarded) for my CD on Schubert “L’Enchantement Retrouvé”, in December (the only classical musician to receive two awards for two works) for my CD dedicated to Liszt “Les Harmonies de l’Esprit” and for the recording of Liszt’s Consolation n. 2 performed in concert in Konstanz, Germany, on the occasion of the international photographic exhibition “Women of Mathematics throughout Europe – A Gallery of Portraits”.

In 2016 the New York IBLA Foundation awarded me the Scarlatti Special Mention and in 2017 the Piano Special Mention.

In 2020 the Conservatory of Music of Cosenza, on the occasion of the celebration of its 50th anniversary, selects me as one of the brightest and most successful students that the Conservatory has had, and awards me a plaque “for her artistic activity of great merit”.

I studied in Italy and abroad at prestigious academies such as the Internationale Sommerakademie – Universität Mozarteum in Salzburg and the Tel-Hai International Piano Master Classes in Israel and with internationally renowned pianists, including Lazar Berman, Cristiano Burato, Aquiles delle Vigne, Andrzej Pikul.

In addition to concert activity (I have played in Austria, China, Germany, Israel, Poland, Slovenia, Spain, Hungary), I am interested to spread musical and cultural knowledge through conversations - concerts and I am very ingaged in social issues.

Website: http://ingridcarbone.com/

Languages

English (advanced)
Russian (elementary)
Chinese (beginner)

Teaching

During the years 1994 and 1995 I had some teaching experiences with several courses I taught at the University of Calabria.

During the academic year 1996/97 I taught Mathematics and Medical Statistics in the Faculty of Pharmacy of the University of Calabria.

As assistant in Bari I taught Advanced Analysis, Analysis 1 and Analysis 2.

In the Faculty of Economics, Sciences and Engineering of the University of Calabria, I taught Mathematics, Calculus 1, Calculus 2, Calculus 3, Calculus 4, Analysis 1, Analysis 2, Geometry and Linear Algebra (and also Mathematics in the Faculty of Pharmacy).