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Titolo	Docente	Periodo	Ore
Branching Processes and their Applications to Population Dynamics	Giacomo Ascione	February-March	20
Chain Conditions in Group Theory	Fausto De Mari	January	20
Splitting methods for the time integration of Parabolic PDEs of Advection Diffusion Reaction type	Severiano González Pinto	September	16
Perron’s method in abstract harmonic spaces and applications to linear second order PDEs with non-negative characteristic form	Alessia E. Kogoj	September	10
Arguing, demonstrating, explaining	Maria Alessandra Mariotti	January-February	10
An introduction of groups of automorphisms of rooted trees	Marialaura Noce	May or September	10
The Poincaré problem for linear elliptic equations	Dian Palagachev	February-June	10
PFG groups and subgroup growth	Matteo Vannacci	March	10

Branching Processes and their Applications to Population Dynamics

Prerequisites

Some basics on Probability Theory, Stochastic Processes, Ordinary and Partial Differential Equations. In any case, we will recall the necessary tools whenever needed.

Contents

Branching Processes are a sort of building block in stochastic population dynamics, as they are a fundamental tool in such a context. The aim of the cycle of lectures is to describe such processes, some particular cases (such as Birth-Death processes) and some generalizations (as multitype branching processes) and then show some exemplary models in population dynamic that make use of them. Here is a draft of the main topics of the lectures:

1. Examples and applications of stochastic processes in population dynamics (Queueing theory, Epidemiological models, Mutations). Introduction to Probability Generating Functions.
2. Compound distributions and their probability generating functions. Example: the compound Poisson distribution. The Galton-Watson Process and its basic properties.
3. Asymptotic classes of the Galton-Watson process. Example: Family Trees and the extinction of Family Names.
4. Continuous-time Markov Chains. Birth-Death processes: basic properties and an introduction to their spectral theory. Examples: M/M/1 and M/M/∞ queueing systems.
5. Phase-type distributions: Hypoexponential and Hyperexponential. Quasi-Birth-Death processes. Example: M/Ek/1 queueing system.
6. Large Population Limits. Examples: Lotka-Volterra model ad SIR model.
7. Continuous-time Branching processes. Birth-Death processes as continuous-time Branching processes. Example: Cancel cells, chemotherapy and mutations.
8. Multitype Branching Processes: Luria-Delbruke and multitype Galton-Watson processes. Example: SEIR model with spatial dispersion.
9. Infinite Type Branching Processes. Example: Yule’s evolution problem.
10. Age-dependent models. Semi-Markov Birth-Death processes. Bellman-Harris Processes. Example: stathmokinetic test.

References

Main Bibliography

• Harris. The theory of branching processes. Vol. 6. Berlin: Springer, 1963.
• Kimmel and Axelrod. Branching Processes in Biology. Springer, New York, NY, 2015.

Other References

• Feller. An introduction to probability theory and its applications, Volume 1, 3rd Edition, Wiley Series in Probability and Mathematical Statistics, 1963
• Karlin and McGregor. "The differential equations of birth-and-death processes, and the Stieltjes moment problem." Transactions of the American Mathematical Society 85.2 (1957): 489-546.
• Karlin and McGregor. "The classification of birth and death processes." Transactions of the American Mathematical Society 86.2 (1957): 366-400.
• Schoutens. Stochastic processes and orthogonal polynomials. Vol. 146. Springer Science & Business Media, 2012.
• Griffiths, Leonenko and Williams. "The transient solution to M/Ek/1 queue." Operations Research Letters 34.3 (2006): 349-354.
• Allen. Stochastic Population and Epidemic Models Persistence and Extinction, Springer, 2015
• Saaty. Elements of queueing theory: with applications. Vol. 34203. New York: McGraw-Hill, 1961.
• Daley and Gani. Epidemic modelling: an introduction. No. 15. Cambridge University Press, 2001.
• Kurtz. "Solutions of ordinary differential equations as limits of pure jump Markov processes." Journal of applied Probability 7.1 (1970): 49-58.
• Notes provided by the teacher.

Chain Conditions in Group Theory

Prerequisites

The basic knowledge of Group Theory.

Contents

A finiteness condition is a property satisfied by finite groups. The origins of the study of groups satisfying a finiteness condition date back to the last century and are associated with the names of R. Baer, S.N. Chernikov, K.A. Hirsch, D.I. Zaicev and many others.

Among finiteness conditions there are chain conditions. In these lectures the minimal and maximal conditions on (all) subgroups, on normal subgroups and on subnormal subgroups will be presented, some properties of these conditions will be highlighted and the links between them will be discussed. The structure of solvable groups satisfying a chain condition will be described. Furthermore, weak chain conditions will be introduced and the link between them and the class of minimax groups will be shown. Finally, we will present some possible applications of chain conditions that guarantee the presence of "many" subgroups with a certain property in a group, and which have been the object of research also in recent times.

References

[1] C. Casolo, Torsion-free groups in which every subgroup is subnormal. Rend. Circ. Mat. Palermo 50 (2001), 321-324.
[2] C. Casolo, Groups with all subgroups subnormal. Note Mat. 28 (2008), 1- 149.
[3] M.R. Celentani, A. Leone, Groups with restrictions on non-quasinormal sub- groups. Boll. Un. Mat. Ital. A (7) 11 (1997), 139-147.
[4] G. Cutolo, On groups satisfying the maximal condition on non-normal sub- groups. Riv. Mat. Pura App. 9 (1991), 49-59.
[5] G. Cutolo, L.A. Kurdachenko, Weak chain conditions for non-almost normal subgroups. In: Groups ’93 Galway/St. Andrews, 120-130 (Galway, 1993).
[6] G. Cutolo, L.A. Kurdachenko, Groups with a maximality condition for some non-normal subgroups. Geom. Dedicata 55 (1995), 279-292.
[7] U. Dardano, F. De Mari, S. Rinauro, The weak minimal condition on sub- groups which fail to be close to normal subgroups. J. Algebra 560 (2020), 371-382.
[8] M.R. Dixon, I.Y. Subbotin, Groups with finiteness conditions on some sub- group systems: a contemporary stage. Algebra Discrete Math. 4 (2009), 29- 54.
[9] M. De Falco, C. Musella, Groups satisfying the maximal condition on non- modular subgroups. Algebra Colloq. 12 (2005), 449-460.
[10] F. De Mari, Groups satisfying weak chain conditions on certain non-normal subgroups. Boll. Un. Mat. Italiana (8) 10–B (2007), 853-866.
[11] F. De Mari, Groups satisfying weak chain conditions on non-modular sub- groups. Comm. Algebra 46 (2018), 1709-1715.
[12] F. De Mari, Groups with the weak minimal condition on non-normal non- abelian subgroups. Beitr. Algebra Geom. 61 (2020), 1-7.
[13] F. De Mari, Groups with many modular or self-normalizing subgroups. Comm. Algebra 49 (2021), 2356-2369.
[14] F. De Mari, On groups whose subgroups are either modular or contranormal. Bull. Austr. Math. Soc. 105 (2022), 286-295.
[15] F. De Mari, Groups with many abelian or self-normalizing subgroups. Archiv Math. (Basel) 119 (2022), 225-235.
[16] S. Franciosi, F. de Giovanni, Groups satisfying the minimal condition on non- subnormal subgroups. In: Infinite groups 1994 (Ravello), 63-72 (de Gruyter, Berlin 1996)
[17] F. de Giovanni, G. Vincenzi, Pronormality in infinite groups, Math. Proc. R. Ir. Acad. 100A (2000), 189-203.
[18] H. Heineken, I.J. Mohamed, A group with trivial cenre satisfying the nor- malizer condition. J. Algebra 10 (1968), 368-376.
[19] N.S. Hekster, H.W. Lenstra, Groups with finitely many non-normal sub- groups. Arch. Math. (Basel) 54 (1990), 225-231.
[20] M.J. Karbe, Groups satisfying the weak chain conditions for normal sub- groups. Rocky Mountain J. Math. 17 (1987), 41-47.
[21] M.J. Karbe, L.A. Kurdachenko, Just infinite modules over locally soluble groups. Arch. Math. (Basel) 51 (1988), 401-411.
[22] L.A. Kurdachenko, Groups that satisfy weak minimality and maximality conditions for normal subgroups. Sibirsk. Mat. Zh. 20 (1979), 1068-1076, 1167.
[23] L.A. Kurdachenko, Groups satisfying weak minimality and maximality con- ditions for subnormal subgroups. Math. Notes 29 (1981), 11-16.
[24] L.A. Kurdachenko and V.E. Goretski ̆ı, Groups with weak minimality and maximality conditions for subgroups that are not normal. Ukrainian Math. J. 41 (1989), 1474-1477 (1990).
[25] L.A. Kurdachenko, H. Smith, Groups with the maximal condition on non- subnormal subgroups. Boll. Un. Mat. Ital. B 10 (1996), 441-460.
[26] L.A. Kurdachenko, H. Smith, Groups with the weak minimal condition for non-subnormal subgroups. Ann. Mat. Pura Appl. 173 (1997), 299-312.
[27] L.A. Kurdachenko, H. Smith, Groups with the weak maximal condition for non-subnormal subgroups. Ricerche Mat. 47 (1998), 29-49.
[28] L.A. Kurdachenko, H. Smith, Groups with the weak minimal condition for non-subnormal subgroups. II. Comment. Math. Univ. Carolin. 46 (2005), 601-605.
[29] J.C. Lennox, D.J.S. Robinson, The theory of infinite soluble groups (Clare- don Press, Oxford 2004)
[30] J.C. Lennox, S.E. Stonehewer, Subnormal subgroups of groups (Claredon Press, Oxford 1987)
[31] J.C. Lennox, J.S. Wilson, A note on permutable subgroups. Arch. Math. (Basel) 28 (1977), 113-116.
[32] W. Mo ̈hres, Auflosbarkeit von Gruppen deren Untergruppen alle subnormal sind. Arch. Math. (Basel) 54 )(1990), 232-235.
[33] R.E. Phillips, J.S. Wilson, On certain minimal conditions for infinite groups. J. Algebra 51 (1978), 41-68.
[34] D.J.S. Robinson, Finiteness conditions and generalized soluble groups (Springer, Berlin 1972)
[35] D.J.S. Robinson, A course in the theory of groups (Springer, Berlin 1996)
[36] V.P. Shunkov, On locally finite groups with a minimality condition for abelian subgroups, Algebra and Logic 9 (1970), 350-370.
[37] H. Smith, Hypercentral groups with all subgroups subnormal. Bull. London. Math. Soc. 15 (1983), 229-234.
[38] H. Smith, Torsion-free groups with all subgroups subnormal. Arch. Math. (Basel) 76 (2001), 1-6.
[39] D.I. Za ̆ıcev, On the theory of minimax groups. Ukrainian Math. J. 23 (1971),

Splitting methods for the time integration of Parabolic PDEs of Advection Diffusion Reaction type

Prerequisites

Linear Algebra, Ordinary Differential Equations, Partial Differential Equations. Numerical and Mathematical Analysis. Programming in Matlab.

Contenuti

1. Time integration of Ordinary Differential Equations (3 hours)
1. One-step methods: Runge-Kutta methods. Rosenbrock methods. Or- der conditions. Stiff problems. Linear stability properties. Variable step-size implementations. Integration of some stiff models: Nonlin- ear Reaction problems from Chemistry. (2 hours)
2. Linear Multistep methods. Order conditions. Linear stability Barriers. Practical methods for non-stiff and stiff problems. (1 hour)
2. Spatial Discretizations of Advection-Diffusion Reactions PDEs (3 hours)
1. Basic spacial discretizations for the 1D Advection-diffusion equations. Convergence of spatial discretizations with finite differences. Discrete Fourier Analysis. Boundary conditions and Spatial accuracy. Time stepping discretization. The Method of Lines (MoL). Stability and convergence of the MoL approach. Von Neumann stability analysis. Monotonicity properties. Positivity of time stepping methods. Mul- tidimensional aspects: Cartesian grid discretization for diffusion and advection. (1.5 hours)
2. Space discretizations of some time dependent models in PDEs: Advection- Diffusion equations. Pollulant Transport-Chemistry. 2D-Chemo-taxis problems. Angiogenesis model. Gray-Scott model. 3D-Combustion model. 2D Radiation-Diffusion Model. 2D Heston Model in Finance. (1.5 hours)
3. Splitting Methods for parabolic PDEs of advection dif- fusion reaction type (6 hours)
1. Operator Splitting: Lie-Trotter splitting and Strang splitting. (1 hour) 1
2. Locally One Direction methods (LOD). Trapezoidal Splitting Method. Local and global error in 2D. (1 hour)
3. Alternating Direction Implicit (ADI)-methods. Peaceman and Rach- ford method. Douglas Method. Stability. Local and global error analy- sis in several dimensions. (1 hour)
4. Rosenbrock type AMF methods. Stability aspects. Some interesting methods for 2D and 3D problems. IMEX methods (One-step and Mul- tistep cases). Stability and convergence of some methods. (2 hours)
5. Numerical experiments with: 2D-Angiogenesis model, 2D-Tumour In- vasion model, 3D-Combustion, 2D Radiation-Diffusion models. (1 hour)
4. AMF-W methods for parabolic PDEs. Applications in Finance (4 hours)
1. AMF-W- methods and AMFR-W-methods. A few interesting methods in the recent literature. Boundary corrections. (1 hour)
2. New results on stability and convergence in the Euclidean and inn the Maximum norms. (2 hours)
3. Applications to PDEs with mixed derivatives. Integration of the 2D Heston Model in Finance. (1 hour)

Remark: The main reference for the course is [5]

Riferimenti

[1] S. Gonzalez-Pinto, E. Hairer, D. Hernandez-Abreu, Convergence in l2 and l∞ norms of one-stage AMF-W methods for parabolic problems, SIAM J. Numer. Anal. 58, 2020, pp. 1117–1137.
[2] S. Gonzalez-Pinto, D. Hernandez-Abreu, S. Perez Rodriguez, AMFR- W-methods for parabolic problems with mixed derivatives. Application to the Heston Model, J. Comput. Appl. Math. 387 (2021) 112518.
[3] E. Hairer, S.P. Norsett, G. Wanner, Solving Ordinary Differential Equa- tions I – Nonstiff problems, Springer-Verlag (1987, 1993).
[4] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II – Stiff and Differential Algebraic Problems, Springer-Verlag (1991, 1996).
[5] W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection Diffusion Reaction Equations, Springer (2003, 2007).

Perron’s method in abstract harmonic spaces and applications to linear second order PDEs with non-negative characteristic form

Prerequisites

Competences of a master graduate student in Mathematics are sufficient.

Contents

1. Perron's method for the heat operator.
   Harmonic functions, Gauss mean value formula, superharmonic functions, solution to the Dirichlet problem in Euclidean balls, Poisson kernel, generalized solution in the sense of Perron-Wiener of the Dirichlet problem, boundary behavior of the Perron-Wiener solution, Bouligand theorem.
2. Abstract harmonic spaces.
   Subharmonic and superharmonic functions, the Perron-Wiener-Brelot operator, the generalized solution of the Dirichlet problem, boundary behavior of the generalized solution: Bouligand type theorem.
3. Application of the potential theory in abstract harmonic spaces to the study of the first boundary value problem for linear second order operators with non-negative characteristic form. In particular : Kolmogorov- Fokker- Planck operators.

References

• Axler, Sheldon; Bourdon, Paul; Ramey, Wade, Harmonic Function Theory. Second edition. Graduate Texts in Mathematics, 137. Springer-Verlag, New York, 2001. xii+259 pp.
• Armitage, David H.; Gardiner, Stephen J., Classical Potential Theory. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2001. xvi+333 pp.

Arguing, demonstrating, explaining

Prerequisites

A good knowledge of 'elementary' mathematics; basic elements of Mathematics Education.

Contents

Introduction to the problematic of the relationship between argumentation and proof as an area of research in mathematics education. The course aims to present some of the theoretical approaches introduced and some results achieved by research in this field. It will also illustrate some open problems and some avenues to follow for those interested in the problem.

References

• Balacheff N. (2002/2004). The researcher epistemology: a deadlock from educational research on proof. Fou Lai Lin (Ed.) 2002 International Conference on Mathematics - "Understanding proving and proving to understand". Taipei: NSC and NTNU (pp. 23-44). Reprinted in Les cahiers du laboratoire Leibniz, 109, http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/
• Bartolini Bussi, M. G., and Mariotti, M. A. (2008), Semiotic mediation in the mathematics classroom: artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, and D. Tirosh, eds. Handbook of International Research in Mathematics Education, second revised edition (pp. 746-805). Lawrence Erlbaum, Mahwah, NJ.
• Duval, R. (1991) Structure du raisonnement déductif et apprentissage de la démonstration. In Educational Studies in Mathematics, 22(3), 233-263.
• Fischbein, E. (1982) Intuition and proof. In For the learning of mathematics 3 (2), 8-24.
• Lolli, G. (1988) Capire una dimostrazione. Il ruolo della logica nella matematica. il Mulino, Bologna.
• Mariotti M.A. (2006) Proof and proving in mathematics education. A. Gutiérrez & P. Boero (eds) Handbook of Research on the Psychology of Mathematics Education, (pp. 173-204) Sense Publishers, Rotterdam, The Netherlands. ISBN: 9077874194 , 173-204.
• Mariotti M.A., Durand-Guerrier V. and Stylianides, G. (2018). Argumentation and proof. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger & K. Ruthven (Eds.), Developing research in mathematics education - twenty years of communication, cooperation and collaboration in Europe (pp. 75-99). London and New York: Routledge - New Perspectives on Research in Mathematics Education - ERME series, Vol. 1.
• Mariotti M. A. (2022) Argomentare e dimostrare come problema didattico. Ed. D Scuola.
• Reid, D. A., Knipping, C. (2010). Proof in mathematics education: Research, learning and teaching. Rotterdam, The Netherlands, Sense Publisher.
• Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutierrez, G. Leder, P. Boero (eds) The second handbook of research on the psychology of mathematics education: The journey continues. Rotterdam, The Netherlands, Sense Publisher, 315-351.

An introduction of groups of automorphisms of rooted trees

Prerequisites

Algebra undergraduate courses.

Contents

Groups of automorphisms of regular rooted trees have been studied for years as an important source of examples in group theory. Some families of subgroups of the group of automorphisms of a regular rooted tree provide a counterexample to the famous General Burnside Problem. Furthermore, these groups have many applications in other areas of mathematics such as dynamics, probability, and cryptography. In this course, we will give an introduction to groups of automorphisms of rooted trees, explaining their connections with different branches of mathematics. Then, we will discuss remarkable examples, important recent developments of this theory, and open problems.

References

L. Bartholdi, R. I. Grigorchuk, and Z. Sunik, Branch groups, Handbook of Algebra, Volume 3, North-Holland pp. 989-1112 (2003)

The Poincaré problem for linear elliptic equations

Prerequisites

Functional Analysis, Partial Differential Equations.

Contents

The Poincaré problem is a variant of the oblique derivative problem in the case when the vector field is tangent to the boundary of the domain considered. This fact makes the problem degenerate and gives rise to new effects such as loss of regularity of the solution, infinite dimension of the kernel or co-kernel of the problem and many others. The mini-course is aimed to present a direct approach to the solvability of the Poincaré problem in the context of the Hoelder spaces.

References

Popivanov, Peter R.; Palagachev, Dian K. The degenerate oblique derivative problem for elliptic and parabolic equations, Mathematical Research. 93. Berlin: Akademie Verlag. 253 p. (1997).

PFG groups and subgroup growth

Prerequisites

Algebra undergraduate.

Contents

The goal of this PhD course is to prove the Theorem of Mann-Shalev (Theorem 11.1) which provides a connection between PFG groups and the growth of maximal subgroups of a group. To this end, we will present several tools used in different areas of modern group theory, such as the Haar measure and the Zeta functions. If time permits, we will present some new and open results.

References

Lubotzky and Segal, Subgroup Growth, Chapter 1 e 11.