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CRM > English > Activities > Curs 2017-2018 > 4th Barcelona Summer School on Stochastic Analysis: A 2018 EMS Summer School
4th Barcelona Summer School on Stochastic Analysis: A 2018 EMS Summer School
General information
 
Dates: July 9th to 13th, 2018
Description
 
The Barcelona Summer School on Stochastic Analysis is a one-week scientific activity consisting mainly of courses addressed to PhD students and young researchers on current research topics in Stochastic Analysis. Selected participants are also given the opportunity to deliver short talks or to display posters. You can check the past edition here​​.  
 
The courses in 2018 will be the following. For a detailed description see below
    Some stochastic models in eco-evolution, by Sylvie Méléard (École Polytechnique, Palaiseau, France)
    Zero sets of random functions, by Mikhail Sodin (Tel Aviv University, Israel)

Courses
 
The Summer School will consist of the following two courses:
 
Some stochastic models in eco-evolution
Sylvie Méléard, École Polytechnique, Palaiseau, France
 
We are interested in the modeling of population dynamics based on individual behaviors and on the main role played by the parameters scales. We will begin by the study of rescaled birth and death processes, looking for extinction conditions, asymptotic limits and existence of quasi-stationary distributions. The scale is measured in terms of a 'carrying capacity' K and allow us to quantify the limiting behavior of the process. Then we will assume that the individuals are characterized by phenotypic heritable parameters. We consider birth and death processes with mutation and competition. Mutations can occur during the reproduction events and the competition between individuals will select the successive successful mutations. These mechanisms of inheritance, mutation and selection are the basis of the theory of Darwinian evolution. We will show how to model such dynamics. Several approximations can be derived, either deterministic or stochastic, depending on the renormalization they assume. We explain how these models can lead to more concrete applications.
 
The individual-based probabilistic models play a main role to understand the evolution mechanisms. Biologists can now obtain data at the individual level, which makes these models very relevant. Applications are numerous, from environmental challenges to medicine.

References 

Eco-evolutionary stochastic models and  Adaptive biology

Unifying evolutionary dynamics : from individual stochastic processes to macroscopic models, N. Champagnat, R. Ferrière and S. Méléard, Theoretical
Population Biology 69 (2006) 297–321.

Stochastic Models for Structured Populations,  V. Bansaye and S. Méléard, MBI
Lecture Series 1.4, Springer, 2015.
 
 

Polymorphic evolution sequence and evolutionary branching, N.
Champagnat and S. Méléard, Probab. Theory Related Fields, Volume 151, Issue 1
(2011), 45–94.

Quasi-stationary distributions 

Quasi-stationary distributions and population processes, S. Méléard and  D. Villemonais.
Probability Surveys, Vol. 9 (2012) 340–410.

Champagnat, N., Villemonais, D. Exponential convergence to quasi-stationary distribution and Q-process. 
Probability Theory and Related Fields, 164, no. 1, 243-283 (2016).

Sharp asymptotics for the quasi-stationary distribution of birth-anddeath processes,  Jean-René Chazottes,Pierre Collet and S. Méléard. Probab.
Theory Related Fields 164 (2016), no. 1-2, 285–332.

Micro-organisms eco-evolutionary stochastic models

The effect of competition and horizontal trait hesitance on invasion, fixation and polymorphism,  S. Billiard, P. Collet, R. Ferrière, S. Méléard,
V.C. Tran, J. Theoret. Biol. 411 (2016), 48–58.
 
Stochastic dynamics for adaptation and evolution of microorganisms, S. Billiard, P. Collet, R. Ferrière, S. Méléard, V.C. Tran, to appear in 7ECM
Proceedings (EMS Publishing House).

   
Zero sets of random functions
Mikhail Sodin, Tel Aviv University, Israel
  
Zero sets of random functions is a rapidly growing area that lies at the crossroads of analysis and probability theory. The lectures will aim to introduce the audience to this fascinating area. Here is a sample of questions we plan to discuss:
 
- How many real zeroes has a random polynomial of large degree? (from Mark Kac, John Edensor Littlewood and Cyril Offord to Ken Soze)
- How many connected components has the zero set of a random Gaussian polynomial of several real variables of large degree?
- Asymptotic behaviour of Taylor series with correlated coefficients and spectral measures of finitely-valued stationary sequences.
- Non-trivial translation invariant probability measures on the space of entire functions (``Benjy Weiss phenomenon'').
 
The lectures will not require any advanced background either in analysis, or in probability theory.
  
References for each topic

- How many real zeroes has a random polynomial of large degree?
(from Mark Kac, John Edensor Littlewood and Cyril Offord to Ken Soze)

Mark Kac, On the average number of real roots of a random algebraic equation.
Bull. Amer. Math. Soc. 49, (1943), 314-320.

Alan Edelman, Eric Kostlan, How many zeros of a random polynomial are real?
Bull. Amer. Math. Soc. (N.S.) 32 (1995), 1–37.

Ken Söze, Real zeroes of random polynomials.
Part I. Flip-invariance, Turán’s lemma, and the Newton-Hadamard polygon.
Israel J. Math. 220 (2017), 817-836, arXiv:1601.04850
Part II. Descartes' rule of signs and anti-concentration on the symmetric group. Ibid, 837-872, arXiv:1601.04858


- How many connected components has the zero set of a random Gaussian polynomial of several real variables of large degree?

Eugene Bogomolny and Charles Schmit, Percolation Model for Nodal Domains of Chaotic Wave Functions.
Phys. Rev. Lett. 88 (2002), 114102, arXiv:nlin/0110019

Fedor Nazarov, Mikhail Sodin,
On the number of nodal domains of random spherical harmonics.
Amer. J. Math. 131 (2009), 1337-1357, arXiv:0706.2409
Asymptotic laws for the spatial distribution and the number of
connected components of zero sets of Gaussian random functions.
Zh. Mat. Fiz. Anal. Geom. 12 (2016), 205-278, arXiv:1507.02017

Mikhail Sodin, Lectures on Random Nodal Portraits.
In: Probability and Statistical Physics in St. Petersburg.
Proc. Symp. Pure Math. 91. Amer. Math. Soc., Providencs RI, 2016.
Available online http://www.math.tau.ac.il/sodin/SPB-Lecture-Notes.pdf.


- Asymptotic behaviour of Taylor series with correlated coefficients and spectral measures of finitely-valued stationary sequences.

Alexander Borichev, Mikhail Sodin, Benjamin Weiss, Spectra of stationary processes on Z, arXiv:1701.03407

Alexander Borichev, Alon Nishry, Mikhail Sodin, Entire functions of exponential type represented by pseudo-random and
random Taylor series. J. Anal. Math. 133 (2017), 361-396.
arXiv:1409.2736


- Non-trivial translation invariant probability measures on the space of entire functions (''Benjy Weiss phenomenon'')

B. Weiss, Measurable entire functions.
In: The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin.  Ann. Numer. Math. 4 (1997), 599-605.

Lev Buhovsky, Adi Glucksam, Alexander Logunov, Mikhail Sodin
Translation-invariant probability measures on entire functions, arXiv:1703.08101 (if time permits)
Scientifc Committee
 
Maria Jolis, Universitat Autònoma de Barcelona, Spain.
Davar Khoshnevisan, University of Utah, USA.
Giovanni Peccati, Université du Luxembourg, Luxembourg.
Marta Sanz-Solé, Universitat de Barcelona, Spain.
 
Organising Committee
 
Xavier Bardina, Universitat Autònoma de Barcelona
Lluís Quer-Sardanyons, Universitat Autònoma de Barcelona
Marta Sanz-Solé, Universitat de Barcelona
Josep Vives, Universitat de Barcelona ​

Contributed talks and poster presentations

If you wish to give a talk or present a poster, submit the following form before May 1st, 2018.

Resolutions will be send before May 20th, 2018.

Application form (closed)

 

Registration​
 
Registration fee: 180 €

 
Registration deadline: June 10th, 2018

Registration includes: attendance to the courses, documentation package, a copy of the lecture notes of each course, lunch tickets, a guided tour in Barcelona, the social dinner, and the coffee breaks.

 

Refund policy:

Cancellations received 1 month before the start of the activity will incur in an administrative fee of 50% of the total amount.

Cancellations received less than one month prior to the start of the activity are not refundable.

Grants

* Only applications from pre-registered participants will be accepted 

In order to increase the number of young researchers participating in this activity, the CRM announces a call for full grants (registration+lodging) for those participants interested in taking part in this activity. Accommodation for granted participants will be in a shared apartment on campus from July 8th to 14th, 2018.  

Application form(closed)

 

Application deadline for grants: April 1st, 2018

 
EMS:
 
The EMS offers some travel grants to young mathematicians from less-favoured regions within the geographical area of EMS membership for presenting results at conferences or attending courses, or for research stays in foreign countries, normally up to a maximum of 900 euros in each case or 500 euros for trips within Europe.
 
Eligible researchers should use this online form​ in order to apply for travel grants.
Lodging information
 
For lodging in the area please click here​
 
For off-campus and family accommodation click here​
 
Acknowledgements
 
This Summer School has been partially supported by the EMS, by the IMUB, by the Facultat de Matemàtiques of the Universitat de Barcelona and by the MINECO (Reference numbers: MTM2015-67802-P and MTM2015-65092-P).
 
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Contact information
 
If you have any questions please contact Ms. Núria Hernandez  (nhernandez@crm.cat​​​)​​
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