Phd Coordinator: Prof. Alessandra Lunardi
Contact person for Ferrara: Prof. Lorenzo Pareschi
Administrative Office: Parma
Web site: https://smfi.unipr.it/it/didattica/dottoratodiricercamatematica
The Ph.D. degree in Mathematics is a joint program among three Universities: Modena/Reggio Emilia, Parma and Ferrara.
From 2016 to 2019, the administrative office was the University of Modena and Reggio Emilia (see http://www.mathphd.unimore.it/site/home.html for specific information).
Since 2019 the administrative office was the University of Parma ( https://smfi.unipr.it/it/didattica/dottoratodiricercamatematica)
The cycle is 3 years long, for 12 positions and 9 scholarships provided.
It is open to students who are interested in a career in academic research and teaching, as well as in the private and public sectors. Candidates to the program should have a background in hard sciences, including engineering, physics, chemistry, biology, as well as mathematics.
Doctoral students have the chance to attend courses in their areas of specialization. They also perform an extensive period of research, in collaboration with their supervisor. Finally they write and defend a doctoral thesis. Students will be placed in the research groups of the partner universities according to the seat of the supervisor (tutor) that will be assigned by the Faculty Board.
The Universities of ModenaReggio Emilia, Parma and Ferrara offer a wide training in mathematics and its applications. About 60 scholars in pure and applied mathematics are member of the PhD board, with an international scientific profile and a strong expertise in fields as diverse as partial differential equations, probability theory and stochastic processes, geometry and combinatorics, numerical analysis and scientific computation,number theory, mathematical physics.
The activities include lectures (especially during the first year), seminars, workshops and research periods in partner institutions, also abroad. With regard to the specific training objectives pursued, the PhD program in Mathematics aims particularly to the training of highly qualified researchers, able to carry out highprofile research. The PhD candidate will choose with his supervisor a research topic and will write a dissertation thesis containing original elements. Ph.D. recipients will be ready to work in universities, research institutes, industry, public administration as well as in independent commercial enterprises.
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Gli insegnamenti dedicati al dottorato, ripartiti per sede
Ferrara, ModenaReggio Emilia, Parma
Per i corsi di LM impartiti nei tre atenei si rimanda alle pagine web dei CdS e si prega di contattare il docente prima di frequentare il corso per concordare il programma
LM Ferrara, ModenaReggio Emilia, Parma
Ogni dottorando deve concordare con il proprio tutore/relatore un piano di studi che permetta di acquisire 60 cfu all'anno ripartiti nel modo seguente. La scheda da compilare per preventivo e consuntivo dell'attività del dottorando è qui.

Attività di formazione disciplinare 
Attività di formazione interdisciplinare 
Attività di ricerca 
Criteri di attribuzione crediti 
Primo anno 
Da un minimo di 40 crediti 
Fino ad un massimo di 20 crediti 
Fino ad un massimo di 20 crediti 
5 cfu per ogni attività seguita tra quelle promosse annualmente da IUSS Ferrara 1391 o da analoghe iniziative presso gli atenei di ModenaReggio Emilia e Parma. Frequenza ai seminari fino a 10 crediti. Frequenza a scuole intensive fino a 20 crediti. L'attività di ricerca è valutata in base al rapporto del tutore di tesi. 
Secondo anno 
Fino a un massimo di 20 crediti 
Entro il termine del secondo anno non meno di 20 crediti 
Da un minimo di 40 crediti 
5 cfu per ogni attività seguita tra quelle promosse annualmente da IUSS Ferrara 1391 o da analoghe iniziative presso gli atenei di ModenaReggio Emilia e Parma. Frequenza ai seminari fino a 10 crediti. Frequenza a scuole intensive fino a 20 crediti. L'attività di ricerca è valutata in base al rapporto del relatore 
Terzo anno 
Fino a un massimo di 10 crediti 

Da un minimo di 50 crediti. 
L'attività di ricerca è valutata in base al rapporto del relatore. Frequenza ai seminari fino a 10 crediti. Frequenza a scuole intensive fino a 20 crediti. 
Gli insegnamenti dedicati al dottorato, ripartiti per sede
Ferrara, ModenaReggio Emilia, Parma
Per i corsi di LM impartiti nei tre atenei si rimanda alle pagine web dei CdS e si prega di contattare il docente prima di frequentare il corso per concordare il programma
LM Ferrara, ModenaReggio Emilia, Parma
Ogni dottorando deve concordare con il proprio tutore/relatore un piano di studi che permetta di acquisire 60 cfu all'anno ripartiti nel modo seguente. La scheda da compilare per preventivo e consuntivo dell'attività del dottorando è qui.

Attività di formazione disciplinare 
Attività di formazione interdisciplinare 
Attività di ricerca 
Criteri di attribuzione crediti 
Primo anno 
Da un minimo di 40 crediti 
Fino ad un massimo di 20 crediti 
Fino ad un massimo di 20 crediti 
5 cfu per ogni attività seguita tra quelle promosse annualmente da IUSS Ferrara 1391 o da analoghe iniziative presso gli atenei di ModenaReggio Emilia e Parma. Frequenza ai seminari fino a 10 crediti. Frequenza a scuole intensive fino a 20 crediti. L'attività di ricerca è valutata in base al rapporto del tutore di tesi. 
Secondo anno 
Fino a un massimo di 20 crediti 
Entro il termine del secondo anno non meno di 20 crediti 
Da un minimo di 40 crediti 
5 cfu per ogni attività seguita tra quelle promosse annualmente da IUSS Ferrara 1391 o da analoghe iniziative presso gli atenei di ModenaReggio Emilia e Parma. Frequenza ai seminari fino a 10 crediti. Frequenza a scuole intensive fino a 20 crediti. L'attività di ricerca è valutata in base al rapporto del relatore 
Terzo anno 
Fino a un massimo di 10 crediti 

Da un minimo di 50 crediti. 
L'attività di ricerca è valutata in base al rapporto del relatore. Frequenza ai seminari fino a 10 crediti. Frequenza a scuole intensive fino a 20 crediti. 
The equation arises not by analyzing the individual positions and velocity of each particle in the fluid but by considering a probability distribution f(x,v,t) for the position and velocity of a typical particle—that is, the probability that the particle occupies a given very small region of space centered at the position x, and has velocity nearly equal to v (thus occupying a very small region of velocity space), at an instant of time t.
Ludwig Boltzmann (18441906)
Recently, kinetic equations have found novel applications in the study of emergent behaviors in complex systems characterized by the spontaneous formation of spatiotemporal structures as a result of simple local interactions between agents. These include the study of birds’ flocks, insects’ swarming, crowd dynamics, opinions’ formation, stock markets and wealth distributions. Clearly, the basic entities in these fields differ from physical particles in that they already have an intermediate complexity themselves and are commonly denoted as agents (see [6]).
Milling behavior in fishes (left) and flocking of starlings (right)
One of the major difficulties in applying the classical toolbox of kinetic theory to these new fields is the lack of fundamental principles which define the microscopic dynamic. A degree of uncertainty is therefore implicitly embedded in such models, since most modeling parameters can be assigned only as statistical information from experimental results (see [4] for recent surveys).
In spite of the vast amount of existing research, both theoretically and numerically (see [3,8]), the study of kinetic equations has mostly remained deterministic and ignored uncertainty. In reality, there are many sources of uncertainties that can arise in these equations:
Incomplete knowledge of the interaction mechanism between particles/agents.
Imprecise measurements of the initial and boundary data.
Other sources of uncertainty like forcing and geometry, etc.
Understanding the impact of these uncertainties is critical to the simulations of the complex kinetic systems to validate the kinetic models, and will allow scientists and engineers to obtain more reliable predictions and perform better risk assessment.
Simulation of a flock attacked by a predator using the method developed in [1]
The development of numerical methods for kinetic equations presents several difficulties due to the high dimensionality and the intrinsic structural properties of the solution. Nonnegativity of the distribution function, conservation of invariant quantities, entropy dissipation and steady states are essential in order to compute qualitatively correct solutions. Preservation of these structural properties is even more challenging in presence of uncertainties which contribute to curse of dimensionality.
Uncertainty quantification (UQ) in kinetic equations represents a computational challenge for many reasons. Simple UQ tasks such as the estimation of statistical properties of the solution typically require multiple calls to a deterministic solver. A single solver call is already very expensive for such complex mathematical models.
Convergence acceleration of MicroMacro Monte Carlo versus standard Monte Carlo
Researchers at DMCS have a recognized international experience in the numerical analysis of kinetic equations with fundamental contributions in the development of fast spectral methods, Monte Carlo methods and asymptotic preserving schemes (see the recent review in [3] for example). Recently they developed novel approaches to UQ of kinetic equations based on generalized Polynomial Chaos expansions at a particle level in order to reduce the problem dimension and maintain the main physical properties of the solution (see [2]) and on micromacro Monte Carlo techniques which using control variate estimators based on the local equilibrium are capable to accelerate the slow statistical convergence of Monte Carlo methods (see [4, Chapter 5]).
References
Albi G., Pareschi L. (2013), Binary interaction algorithms for the simulation of flocking and swarming dynamics. Multiscale Modeling & Simulation 11, 129.
Carrillo J.A., Pareschi L., Zanella M. (2017), Particle gPC methods for mean field models of swarming with uncertainties, arXiv:1712.01677.
Dimarco G., Pareschi L. (2015), Numerical methods for kinetic equations. Acta Numerica.
Jin S., Pareschi L., eds (2018), Uncertainty quantification in hyperbolic and kinetic equations. SEMASIMAI Series in Applied Mathematics, Springer.
Mouhot C., Pareschi L. (2006), Fast algorithms for computing the Boltzmann collision operator. Mathematics of Computation 75, 18331852.
Pareschi L., Toscani G. (2013), Interacting multiagents systems: kinetic equations and Monte Carlo methods. Oxford University Press, (2013)
Pareschi L., Toscani G., Villani C. (2003), Spectral methods for the non cutoff Boltzmann equation and numerical grazing collision limit. Numerische Mathematik 93, 527548.
Villani C. (2002), A survey of mathematical topics in kinetic theory. Handbook of fluid mechanics, S. Friedlander and D. Serre, Eds. Elsevier Publ. NorthHolland vol. I, 71305.
Contact
Prof. Lorenzo Pareschi
Department of Mathematics and Computer Science
University of Ferrara
Via Machiavelli 30, 44121 Ferrara
email: lorenzo.pareschi@unife.it
]]>The article is avaible in full version at this link: https://www.sciencedirect.com/science/article/pii/S0004370218305964#br0050
Keywords
Authors
Received 16 April 2017, Revised 7 September 2018, Accepted 20 September 2018, Available online 17 October 2018 on