Centro de Modelamiento Matemático - CMM
Universidad de Chile 2007
Our goal at CMM To establish meaningful and productive relationships
between advanced mathematics and all endeavors of modern society
Raúl Gouet, Alejandro Maass, Servet Martínez, Jaime San Martín,
Joaquín Fontbona, Michael Schraudner, Andrew Hart.
Current: Andrés Aravena.
Past: Pablo Moreno, Angélica Reyes, Marco Budinich, Rodrigo Vargas, Manuel Cano, Rodrigo Assar, Juan Ugalde, Daniel Maturana.
Current: Mahsa Allahbakshi (Iran), José Aliste (Chile), Alexis Bailler (France), Soonjo Hong (Korea), Nicolás Loira (Chile).
Past: Michael Schraudner (Germany), Yuki Yayama (Japan), Jean-François Jabir (France).
Alvaro Coronel, Guillermo Espinoza, José Aliste, Julien Broche, Olivier Lengeard.
J. Bertoin (France), F. Blanchard (France), M. Boyle (USA), X. Bressaud (France), P. Collet (France), M. Cortez (Chile), C. Dellacherie (France), T. Downarowicz (Poland), F. Durand (France), P. Ferrari (Brazil), J.M. Gambaudo (France), B. Host (France), W. Huang (China), P. Kurka (Czech Rep.), J. León (Mexico), J. Lepeltier (France), J. López (Spain), M. Lladser (USA), J. Ma (USA), S. Meleard (France), V. Maume (France), P. Ney (USA), Pécou (France), P. Protter (USA), G. Sanz (Spain), M. San Miguel (Spain), S. Shao (China), B. Ycart (France), X. Ye (China).
Randomness plays a key role in statistical mechanics, mathematical finance, computer science and bioinformatics, all of which are areas of our interest. We work on a variety of fundamental topics like killed processes, local entropy, thermodynamic formalism, spectral theory of dynamical systems, random fragmentation, and simulation of interacting systems. Applications range from systems biology to mining and forestry.
Some of our research achievements and future challenges are:
Key concepts in the study of killing process are the rate of killing and quasi-stationary distributions (qsd). We found necessary and sufficient conditions for existence of qsd for general Markov chains and classified them in the 1D case. We also studied the asymptotic behavior for the heat kernel on multidimensional unbounded domains with Dirichlet boundary conditions. Our main future challenges are: to extend the results to other multidimensional domains; in the 1D case when infinity is an entrance boundary as in some models in population ecology and finance; to study population extinction involving the trait location in a general setting.
Processes with infinite memory:
We study processes with summable infinite memory decay. The construction of regeneration times and the standardness property of the associated filtration (Vershick´s theory) are main achievements. These ideas were used to analyze the analog 2-3 Furstenberg´s problem in additive cellular automata (CA). Our goal is to use such techniques to study asymptotic randomization of infinite memory measures by the action of expansive CA.
Discrete random structures:
Discrete random structures are core tools in computer science and transport. Main challenges here are the study of urn models, probabilistic analysis of algorithms, the packing and parking problems, scenery reconstruction and bioinformatics. The theoretical tools we use are martingale theory, large deviations, generating functions and combinatorial techniques.
Ergodic theory and topological dynamics:
We work to understand the local dynamical properties that determine special low complexity factors. We look for topological nil-factors of order k analog to the Host-Kra factors in ergodic theory, by exploring combinatorial recurrence conditions of subsets of points. We seek to develop measure-theoretical complexity tuples to classify factors in between the distal and the maximal equicontinuous factor. For Cantor minimal systems we will characterize topological and measure-theoretical eigenvalues in terms of the recurrence structure of the system.
Backward stochastic differential equations:
BSDEs became widely used in the 90´s because its relations to game theory and control and the representation of the price process in classical finance. Our main contributions are the study of existence and uniqueness of solutions for these equations and their numerical approximation. We plan to investigate properties of semi-linear heat type equations through their BSDE representation.
Stochastic processes, ergodic theory and stochastic modeling
During the period of the Basal project (2008 – 2012) the main researchers of our group have been: Joaquín Fontbona, Raúl Gouet, Alejandro Maass, Servet Martínez, Jaime San Martín and Michael Schraudner; in addition Andrew Hart works as a researcher/engineer in stochastic modeling. M. Schraudner and A. Hart have been supported entirely by the Basal funds.
The main theoretical research topics of our group are Stochastic Processes, Ergodic Theory, Particle Systems in Mathematical Physics and Probability in Discrete Structures, and the most visited subjects in our research have been: Killed processes, Branching processes, Population dynamics, Low complexity systems, Recurrence in Dynamical Systems, and Stochastic Processes in Genomics and Fragmentation. The applications and our research in applied modeling is mainly concentrated on BioInformatics Applications in renewable and non-renewable resources, and Simulations for Engineering. Two of our main researchers are the directors of the Laboratories on BioInformatics and Mathematics of the Genome (A. Maass) and on Stochastic Simulation (J. Fontbona), maintaining contracts with the mining, aquiculture and agronomical industries, and attracting bright young scientists. The activities in both laboratories also inspired basic research, as can be seen in the lines Fragmentation and Stochastic modeling in Genomics.
Conferences.We have been main organizers of the well-recognized LatinAmerican Congress in Probability and Statistics CLAPEM, at Viña del Mar on March 2012, where there participated more than 200 researchers from all over the continent. On January 2012 we organized in France, the First Chilean-French Conference on Dynamics and Combinatorics thought to be a first of a regular meeting.We have given more than 30 lecturers as invited speakers in international conferences and minicourses in international Schools. Let us only mentioned a few of them: A Maass was invited speaker at Mathematical Sciences Research Institute, Berkeley: -Program in Ergodic Theory and Additive Combinatorics-, 2008, and -Trends in Dynamics-, Chicago, 2011. S. Martínez was plenary speaker at the 33rd Conference on Stochastic Processes and their Applications, Berlin, 2009; and gave a course at the Institut Henri Poicaré: -Workshop Interacting particle systems and percolation-, 2008. J. San Martín was invited speaker at IRMA, Strasbourg, -Congress on Filtrations-, 2011. M. Schraudner was invited speaker at -School on symbolic dynamics and homeomorphisms of the Cantor set-, Copenhagen, 2008.
At the CMM we have organized two editions of the School on Information and Randomness in 2008 and 2010; both instances being international meetings attracting around 70 researchers from all over the world. The first one took place in Santiago and the second one in Pucón. The proceedings of the school in 2010 were published in Probability Surveys. We have also organized workshops on applied areas addressed to local specialists and industries: Modeling in geobiometallurgy (2009), Trends in monitoring and surveillance of financial markets (2010), Workshop on metabolomics analysis of massive data sets (2011).
Training and placements. In the period of the Basal project we supervised 10 Ph.D. Students, 3 of them have been defended while 7 doctorates are still in course; 7 of all these theses are in -co-tutelle- with universities in France. The PhD that have defended their theses are in the following positions: José Aliste (defended 2009): postdoctoral CMM; A. Daniel Coronel (defended 2009) is finishing postdoctoral in PUC and is been hired by Universidad Andés Bello; Guillermo Espinoza (defended 2010): has a postdoctoral position at PUC.
During the 4 years period we have had 6 postdoctoral positions, two of them are finished and the other 4 are still in course. Those who have finished are: Yuki Yayama: arrived in 2008, at this moment she is a professor at U. Bio-Bio; Jean-François Jabir, arrived in 2009, at this moment he is a professor at U. Valparaíso. These postdocs activities show the importance of CMM and our group for attracting foreign researchers and inserting them in Chilean universities.
Long-term visitors. We hosted 2 CNRS long-term visitors: Professor Jérémi Bigot from U. Toulouse developed important relations to the laboratory on image processing (with R. Gouet) and strengthened the relations of CMM towards astronomers applying his methods in the analysis of large data bases. Professor Jean-Baptiste Bardet from U. Rouen worked with J. Fontbona on long-time behavior of continuous-time Markov processes. In collaboration with A. Christen (Ph.D. student) and others, both obtained rates of convergence to equilibrium for transmission control protocols used in the Internet.
Network.Our scientific network has been enlarged and reinforced in the last 4 years. Our main collaboration is with France, USA, R.P. China, Canada, Germany and Spain. Some of them: M. Boyle (U. Maryland), P. Collet (CNRS, E. Polytechnique), C. Dellacherie (CNRS, U. Rouen), T. Downarowicz (Wroclaw U.), F. Durand (CNRS, U. Picardie), B. Host (U. Marne la Vallée, IUF), T. Huillet (CNRS, U. Cergy), B. Kra (Northwestern U.), J. León (CINVESTAV, Nat. Polytech. Inst. Mexico), S. Méléard (CMAP, E. Polytechnique), W. Nagel (U. Jena), R. Pavlov (U. Denver). There are a variety of join activities in courses and schools with researchers from Brasil (IMPA, U. Sao Paulo), Argentina (U. Buenos Aires), thus one of us give a doctoral summer mini-course at IMPA in january 2012.
Some of our main scientific achievements published during the period are found in the following areas:
Killed processes. In population dynamics a killed process models extinction of populations. Key concepts in its study are the rate of killing and quasi-stationary distributions (qsd). The case ’inﬁnity is an entrance boundary’ is analyzed in (Ann. Prob. 2009) and we studied qsd in population ecology (Prob. Th. Relat. Fields (PTRF) 2011). Furthermore we have characterized completely the R-recurrence for nearest neighbor matrices (Stoch. Proc. Appl. 2010).
Discrete random structures. They appear in the core of new developments in Markov chains and in models in computer science, telecommunications and transport. In particular, we have studied the asymptotic search-cost of the move-to-front algorithm (Ann. Appl. Prob. 2010) and the duality in discrete Markov chains with applications in population dynamic models (Adv. Appl. Prob. 2011).
Interacting particle systems. We have obtained stochastic mean-field interacting particle approximations of the 3D Navier-Stokes equations (Ann. Appl. Prob. 2010), and have developed new coupling techniques based on optimal transport theory to construct simulatable mean-field particle approximations of the Fokker-Planck-Landau equation of plasma physics (PTRF 2009). We have studied diffusion processes and particle approximations of stochastic Lagrangian models for turbulent flows (PTRF 2010, Math. Model. Numer. Anal 2012, Stoch. Proc. Appl. 2012).
Recurrence and local complexity in ergodic theory. Recurrence phenomena influenced ergodic theory as well as number theory. Recent developments in additive combinatorics and ergodic theory motivated us to produce a topological structure theorem allowing us to characterize nilfactors of a topological dynamical system (Adv. Math. 2010). In addition, we gave necessary and sufficient conditions for a topological system to be measurable isomorphic to the inverse limit of all its nilfactors and computed its topological complexity (ETDS 2012).
Dynamical systems: low complexity systems and expanding maps. We have contributed to the spectral theory of low complexity symbolic systems and models of quasicrystals in physics. Our techniques allowed to study cohomological equations and rotation numbers in these systems (three articles in Erg. Th. Dyn. Syst. (ETDS) 2010), as well as expansivity for finite rank systems (ETDS 2008). These results motivated the proof of a long standing conjecture on symbolic extensions of interval maps (Inventiones Math. 2009).
Ultrametric Potential. In the 90’s we proved that finite ultrametric matrices are potential matrices, so these matrices play the same role as the Green potential in classical potential theory. In the period to report on we studied the countable infinite case. In particular the influence of the boundary of the tree and a new representation theorem for positive harmonic functions on trees have been proven (J. Theor. Prob. 2009).
Backward Stochastic Differential Equations (BSDE). We constructed a new algorithm to solve a reflected BSDE, which consists in finding a solution to the equation that is maintained between two barriers, in a minimal way (Stoch. Anal. Appl. 2011). We were invited to contribute a survey article on the topic of numerical solution of BSDE to the four volume Encyclopedia of Quantitative Finance (John Wiley & Sons 2010).
Extreme value theory. In this area we have studied the behavior of exceptional subsequences of observations, mainly records and near-records. Results include both theoretical (asymptotic behavior of counting processes) and applied (inference based on near-records) findings (Adv. Appl. Prob. 2011).
Multidimensional symbolic dynamics. We have investigated mixing properties, separation and the (non-)existence of certain factors of Zd sofic shifts (ETDS 2008, J. Appl. Diff. Eq. 2009, Trans. AMS 2010). We also introduced the novel notion of projective subdynamics of Zd subshifts for which we obtained various results (Discr. Cont. Dyn. Syst. 2010), including a complete classification both in the case where the projective subdynamics is a one-dimensional sofic system and where the Zd shift of finite type satisfies some kind of uniform mixing condition.
Thermodynamic formalism in symbolic dynamics. We studied the thermodynamic formalism in two contexts: For a subadditive potential we applied the results to problems in factor maps between subshifts and Hausdorff dimension of compact invariant sets for non-conformal expanding maps (ETDS 2011, Stoch. & Dyn. 2011) and for an almost additive potential on a countable state Markov shift we applied the results to dimension problems (Nonlinearity, 2012).
Branching processes and Fragmentation. We showed hat the prolific individual in a super-critical continuous state branching process is a continuous-time discrete-space Galton-Watson process embedded in the continuous-state one (J. Appl. Prob. 2008). In fragmentation processes we studied the energy efficiency of the consecutive use of two fragmentation processes (J. Appl. Prob. 2010), questions motivated by problems arising from the mining industry.
Stochastic Modeling in Genomics. We studied Chargaff-s second parity rule, a phenomenon which equates the frequencies of k-mers and their reversed complements within a DNA strand. Using a multidimensional Dirichlet approach we developed statistical hypothesis tests and gave an explanation of the heuristic rule using Gibbs states. We have applied all these techniques to a large number of bacteria (Stoch. Models 2010, J. Stat. Phys. 2011).
Connection with laboratories.As already noticed, these last two basic research lines are inspired directly from the activity of our laboratories in stochastic simulation and in mathematics of the genome.
Editorial work. Some of our responsibilities activities in scientific associations and journals during the period are: S. Martínez and J. San Martín are Members Editorial Board ALEA LatinAmerican Journal of Probability and Mathematical Statistics; A. Maass is member of the Internation Scientific Committee of the ICSASG consortium to produce a reference genomics sequence for the Atlantic salmon, formed by Canada, Chile and Norway.