Winter School and 9th ICOR on PDE and Mathematical Biology

2009-10-23T02:40:00.000-07:00

The goal of the Winter School and of the Sessions on PDE and Mathematical Biology is to promote research interplays between Cuba and other countries on recent developments on PDE mathematical investigations, with special interest on applications to Biology and other modelization areas.

The Winter School is specially intended for graduate and PhD students, although it of course welcomes senior scientists as well.

The Sessions on PDE and Mathematical Biology are organized in the frame of the 9th ICOR International Conference.

One can consult the program, the list of talks, the list of invited speakers, and the website of the previous edition. A detailed schedule will be posted here when available.

Here is the page for the location and practical informations and here is the poster of the conference.

For further informations, please contact the organizers (on the right).

Organizers and contact :

Lucilla Corrias (Laboratoire d'Analyse et Probabilité, Université d'Evry, France)

Stéphane Mischler (CEREMADE, Université Paris-Dauphine, France)

Clément Mouhot (DMA, ÉNS Paris, France)

Mariano Rodriguez Ricard (Departamento de Matematicas, Universidad de La Habana, Cuba)

Sponsors :

CEREMADE, Paris-Dauphine, France

Laboratoire d'Analyse et Probabilité, Université d'Evry, France

2009-06-28T02:26:00.000-07:00

Program

WINTER SCHOOL on MATHEMATICAL BIOLOGY

Monday 15, Tuesday 16, Wednesday 17, Thursday 18

Courses:

1. José Antonio Carrillo (Barcelona, Spain)

"Macroscopic and Kinetic Equations in Swarming and Chemotaxis"

''Ecuaciones cinéticas y macroscópicas en "Swarming" y quimiotaxis''

2. Benoît Perthame (Paris, France)

"Adaptive evolution: a population view"

3. Marc Thiriet (Paris, France)

"Blood and air flows: from biomechanical simulations (patient-specific macroscopic scale investigations) to biomathematical approaches (coupling of short-term cell reactions at nano- and microscopic scales to flow-induced stresses)"

SESSION on MATHEMATICAL BIOLOGY

Friday 19, Monday 22, Tuesday 23

The talks of the Mathematical Biology session will be intended to present recent advances on topics such as :

Structured Population Dynamic

Fluid Mechanic in Human Environment

Chemotaxis

Modelling of Biological Phenomena

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WINTER SCHOOL on PDE and applications

Courses:

1. David Lannes (Paris, France)

"Stability/instability issues for two-fluid interfaces"

2. François Murat (Paris, France)

"Basic homogenization"

SESSIONS on PDE and applications

Opening Plenary Lecture, Monday 22 :

P. L. Lions (Paris, France)

"An Introduction To Mean Field Games"

Plenary Lecture :

I. Gamba (Austin, USA)

"Non-conservative models of multi-linear particle interactions"

Tuesday 23, Wednesday 24, Thursday 25, Friday 26

Fluid Mechanic and Coastal Dynamic
Mean Field Games
Classical and Quantum Kinetic Equations
Homogenization, Inverse Problems and Control

Sessions and Talks

2009-06-20T04:36:00.000-07:00

A. Mathematical Biology

D. Ambrosi (Milano, Italia) "The electromechanical coupling in cardiac mechanics"

V. Calvez (Lyon, France) ''The geometry of one-dimensional self-attracting particles" (pdf file)

M. R. D'Orsogna (Los Angeles, USA) "Modelling viral entry dynamics"

M. Doumic-Jauffret (INRIA, France) "Aggregation/fragmentation models for protein polymerization" (pdf file)

L. Dumas (Paris, France) "A numerical simulation of the human arterial network based on non invasive measurements" (pdf file)

K. Fellner (Cambridge, UK) "Stability of stationary states of non-local evolution equations" (pdf file)

T. Hillen (University of Alberta, Canada) "Pointwise weak steady states of transport equations for cell movement in network tissue"

P.-E. Jabin (Nice, France) "Selection-mutation dynamics for the evolution of traits in a population" (pdf file)

D. Levy (Maryland, USA) "Group dynamics in phototaxis"

S. Martin (Orsay, France) "Modelling of air flows and gas exchange in the human lung" (pdf file)

N. Meunier (Paris, France) "Analysis of self-organization systems for cell polarization"

J. Mitchell (Wisconsin-Madison, USA) "Using clustering and optimization in flexible protein docking"

G. Raoul (Cachan, France) "Kimura's model : an integro-differential model to study evolution" (pdf file)

C. Schmeiser (Vienna, Austria) "Nonlinear friction in the cytoskeleton as a macroscopic limit of the activity of cytoskeletal proteins" (pdf file)

A. Vidal (Evry, France) ''A model of the Gonadotropin Releasing Hormone secretion by hypothalamic neuron clusters''

B. Fluid Mechanic and Coastal Dynamic

A. Alvarez Cruz (La Habana, Cuba) - "Solving Mild Slope equation in Matlab with Domain decomposition method"

T. Kano (Kyoto, Japon) - "Tunamis on a deep open sea and on a gentle sloaping beach"

A. Nachbin (Rio de Janeiro, Brazil) - "Asymptotic issues regarding nonlinear wave/topography interaction"

C. Turner (Cordoba, Argentina) - "Mathematical model for stratified fluids"

B. Valiño Alonso (La Habana, Cuba) - "Infinitely narrow solitons in shallow water equations"

C. Mean Field Games

I. Capuzzo Dolcetta (Roma, Italia) "A numerical approach to some mean field games problems"

M. P. Gualdani (Austin, USA) "Global Existence of a Free Boundary Problem with Non-Standard Sources"

Aimé Lachapelle (Paris, France) "Mean Field Games: a numerical methodology with application to a stylised model of technology switch"

P. Markowich (Cambridge, UK and Vienna, Austria) ''On PDE Models for (some) Socio-Economic Problems: Price Formation, Opinion Formation and Crowd Modeling''

D. Classical and quantum kinetic equations

M.J. Caceres (Granada, España) - "A numerical solver for a nonlinear Fokker-Planck equation in Neuroscience"

S. Calogero (Granada, España) - "Self-similar behavior in relativistic galactic d ynamics"

J.A. Cañizo (Barcelona, España) - "Entropy inequalities and speed of convergence to equilibrium for the growth-fragmentation equation"

M. Escobedo (Bilbao, España) - "Post Gel solutions for coagulation equations"

R. .Galeano Andrades (Cartagena, Colombia) "Variational methods in stationary Boltzmann equation"

E. Homogenization, Inverse Problems and Control

F. Dickstein (Rio de Janeiro, Brazil) - "Automatic History Matching in Oil Reservoir Simulation"
O. Kavian (Versailles, France) - "Remarks on Inverse Problems arising from Electrical Impedance"
N. Ghoussoub - "Homogenization of maximal monotone vector fields via selfdual variational calculus"

2008-12-15T10:03:00.000-08:00

Davide Ambrosi

(Politecnico di Milano, Italia)

The electromechanical coupling in cardiac mechanics

The coupling between cardiac mechanics and electric signalling can be addressed in a two possible alternative framework: the electrical potential dictates the active stress or the active strain. In this talk the explore the second (less popular) strategy, obtained thanks a factorization of the deformation tensor. After rewriting the equations in a deforming domain, we prove that neglecting the deformation of the substrate induces a non-negligible error in predicting celerity and width of the pulse, while the amplitude and the stability conditions are not affected by the electro--mechanical decoupling. Exact solutions of travelling front type can be found with explicit form of the celerity and steepness. The role of visco-elasticity and the characterization of unstable initial conditions are discussed with the support of numerical simulations, both in one and two spatial dimensions.

2008-10-23T11:52:00.000-07:00

Schedule

Winter School on Mathematical Biology

Monday 15, Tuesday 16, Wednesday 17, Thursday 18

The Winter School on Mathematical Biology will be held at the "Salón 250 Aniversario de la UH" in the Varona Building, located at one side of the Faculty of Mathematics and Computer Science inside the Campus.

Click here to see the complete schedule of the Winter School on mathematical Biology.

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Session on Mathematical Biology

Friday 19, Monday 22, Tuesday 23

The session on Mathematical Biology on Friday 19 and Monday 22 will be held at the "Salón 250 Aniversario de la UH" in the Varona Building, located at one side of the Faculty of Mathematics and Computer Science inside the Campus.

On Tuesday 23, the session will be held at the Hotel Parque Central, in the Habana Vieja/Centro Habana district.

Click here to see the complete schedule of the Mathematical Biology session.

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Sessions on PDE

Opening lecture of Pierre-Louis Lions on monday 22 and Tuesday 23, Wednesday 24, Thursday 25, Friday 26.

The sessions on PDEs will be held at the Hotel Parque Central, in the Habana Vieja/Centro Habana district.

Click here to see the complete schedule of the PDE session.

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Abredged schedule

Monday, February 15 (Winter School)
1:30-1:45pm: Welcome by the organizers
1:45-3:15pm: J. A. Carrillo
Coffee break
3:30-5:00pm: J. A. Carrillo

Tuesday, February 16 (Winter School)
9:00-10:30am: M. Thiriet
Coffee break
10:45-12:15am: M. Thiriet
Lunch
1:45-3:15pm: J. A. Carrillo
Coffee break
3:30-5:00pm: J. A. Carrillo

Wednesday, February 17 (Winter School)
10:00-11:30am: B. Perthame
Coffee break
11:45am-1:15pm: B. Perthame
Free afternoon

Thursday, February 18 (Winter School)
9:00-10:30am: M. Thiriet
Coffee break
10:45-12:15am: M. Thiriet
Lunch
1:45-3:15pm: B. Perthame
Coffee break
3:30-5:00pm: B. Perthame

Friday, February 19 (Biology session)

9:00-9:15am: Welcome by the organizers
9:15-10:00am: T. Hillen
10:00-10:45am: P.-E. Jabin
Coffee break
11:15-12:00am: C. Schmeiser
12:00-12:45am: M. Rita D’Orsogna
Lunch

2:15-3:00pm: D. Ambrosi
3:00-3:45pm: L. Dumas
Coffee break
4:15-5:00pm: S. Martin

Monday, February 22 (Biology session and opening lecture)
9:00-9:45am: V. Calvez
9:45-10:30am: K. Fellner
Coffee break
11:00-11:45am: M. Doumic
11:45-12:30am: G. Raoul
Lunch
3:00-4:00pm: Opening Plenary Lecture, P.-L. Lions

Tuesday, February 23 (End Biology session and PDE sessions)

ICOR plenary speaker:
9:00-10:00am: B. Cornet

Coffee break
Biology session
10:30-11:10am: J. Mitchell
11:10-11:50am: A. Vidal
11:50-12:30am: N. Meunier

PDE sessions
10:30-11:10am: P. Markowich
11:10-11:50am: M. P. Gualdani
11:50-12:30am: A. Lachapelle
Lunch
2:00-3:20pm: D. Lannes (course 1)
3:40-5:00pm: F. Murat (course 1)

Evening: Reception and Jazz concert

Wednesday, February 24

ICOR plenary speaker:
9:00-10:00am: G. Still

Coffee break
PDE sessions
10:30-11:10am: M. Escobedo
11:10-11:50am: N. Ghoussoub
11:50-12:30: O. Kavian
Lunch
2:00-2:40pm: M. J. Caceres
2:40-3:20pm: J.A. Cañizo
in parallel with
2:00-3:20pm: D. Lannes (course 2)
3:40-5:00pm: F. Murat (course 2)

Thursday, February 25

ICOR plenary speakers:
9:00-10:00am: A. Xavier
2:30-3:30pm: S. Méléard

PDE sessions
10:30-11:10: T. Kano
11:10-11:50am: B. Valiño Alonso
11:50-12:30am: A. Nachbin
Lunch
2:00-2:40: C. Turner
2:40-3:20: A. Alvarez Cruz
in parallel with
2:00-3:20pm: D. Lannes (course 3)
3:40-5:00pm: F. Murat (course 3)

Evening: Social Diner

Friday, February 26

PDE sessions
ICOR plenary speaker:
9:00-10:00am: I. Gamba
Coffee Break
10:30-11:10am: S. Calogero
11:10-11:50am: R. Galeano Andrades

Location and Practical Information

2008-10-23T02:52:00.000-07:00

Conferences location

The Winter School (scheduled from February 15 to 18) as well as the first two days of the Session on "Mathematical Biology" (February 19 and February 22) will be held at the "Salón 250 Aniversario de la UH" in the Varona Building, located at one side of the Faculty of Mathematics and Computer Science inside the Campus of the La Havana University (Universidad de La Habana).

The university is located at the intersection of the San Lázaro St., the Neptuno St. and L St. in the Vedado district, Ciudad Habana. This is the main entrance with the stairs (escalinata) and the sculpture with the Alma Mater (see the picture on the right). Another entrance is located at the intersection of J St. and 27 de Noviembre St. in Vedado District. A map of the location can be found on Google Maps (http://maps.google.xx/): enter "23.134046, -82.382183".

Current information will be located in front of the building of the Faculty of Mathematics and Computer Science, inside the Campus.

From Tuesday 23 to Friday 26, every session will be held in the Hotel Parque Central, located in Neptuno street between the interceptions of Prado and Zulueta, in front of the Parque Central.

9th ICOR Registration

Monday 22 is the opening day of the 9th ICOR. Besides the Mathematical Biology session and the Opening Plenary Lecture, it is devoted to individual registration. People already attending the Winter School on Matematical Biology but also interested to attend the 9th ICOR, should be registrated.

Talks

Speakers will find slides projector and video projector connected to a computer which will accept your USB key (for a presentation with Acrobat or Power Point files) both in the University and in the Hotel Parque Central.

Accommodation/Hotel

According with your preferences, we recommend to be booked at a Hotel near the Campus or near the Capitolio. In addition, you can rent good rooms (can do so via internet) in hostals or family houses if they satisfy the official regulations (to our knowledge, can be booked good rooms near the Campus).

Near the Campus (Vedado area):

Medium standing: Hotel Vedado, Hotel Colina, St.Johns

High standing: Hotel Tryp Habana Libre

Very high standing: Hotel Nacional

Near the Capitolio (Habana Vieja area):

Medium standing: Hotel Caribbean, Hotel Deauville, Hotel Lido, Hotel Lincoln

High standing: Hotel Inglaterra, Hotel Plaza, Hotel Sevilla, Ambos Mundos

Very high standing: Hotel Telégrafo, Sta.Isabel, Los Frailes, Central Park

Visa

Each speaker will receive a "conference invitation" from the organizer committee. With that paper in hand one may ask to the cuban embassy a "scientific visa" (that is the only choice if you are coming from USA) or a "tourist card" (and, in that case, the conference invitation is obviously not necessary). The last procedure is quicker, cheaper and your travel agency should be able to do that formality for you. The only counterpart when you choose a "tourist card" is that you must be able to justify (hotel registration) that you have yet payed two nights in an hotel when you enter the cuban territory.

Money

There are two official currencies in Cuba. The "Peso Cubano" (CUP) (the usual money for cubans) and the "Peso Convertible " (1 CUC = 1.1 USD=24 CUP) with which foreigners must pay the Hotel, the Conference fees, and other services if required. Be careful, prices for both currency are ended by the $ sign! Of course in some places (Hotels, restaurants or in some Shopping Centers) the Visa card is also accepted. Traveller's checks and american express are refused. With Visa card one can get cash in a cash dispenser at the airport or in some hotels and places in some streets. It is also possible to bring cash and change it there (in CUC) when necessary. US dollar will be taxed on, 10% more than others currencies! Do not forget to change some money arriving at the airport (in order to pay the taxi to the hotel)

Airport

The arrival airport is the "José Marti's Aeropuerto" of La Habana. From the airport to the Habana center you must take a Taxi (no collective transport is available on that route). The journey from the airport to the town is between 10$ and 20$ (one must read $ = CUC here).

Invited speakers (to be confirmed)

2008-07-07T08:00:00.000-07:00

A. Alvarez Cruz (La Habana, Cuba)
D. Ambrosi (Milano, Italia)
M.J. Caceres (Granada, España)
S. Calogero (Granada, España)
V. Calvez (Lyon, France)
J.A. Cañizo (Barcelona, España)
I. Capuzzo Dolcetta (Roma, Italia)
J.A. Carrillo (Barcelona, España)
F. Dickstein (Rio de Janeiro, Brazil)
M.R. D'Orsogna (Los Angeles, USA)
M. Doumic-Jauffret (Paris, France)
L. Dumas (Paris, France)
M. Escobedo (Bilbao, España)
K. Fellner (Cambridge, UK)
R. Galeano Andrades (Cartagena, Colombia)
I. Gamba (Austin, USA)
N. Ghoussoub (Vancouver, Canada)
M.P. Gualdani (Austin, USA)
T. Hillen (Alberta, Canada)
P.-E. Jabin (Nice, France)
T. Kano (Kyoto, Japon)
O. Kavian (Versailles, France)
D. Lannes (Paris, France)
D. Levy (Maryland, USA)
P.-L. Lions (Paris, France)
P. Markowich (Cambridge, UK and Vienna, Austria)
S. Martin (Orsay, France)
N. Meunier (Paris, France)
J. Mitchell(Wisconsin-Madison, USA)
F. Murat (Paris, France)
A. Nachbin (Rio De Janeiro, Brazil)
B. Perthame (Paris, France)
G. Raoul (Cachan, France)
C. Schmeiser (Vienna, Austria)
M. Thiriet (Paris, France)
C. Turner (Cordoba, Argentina)
B. Valiño Alonso (La Habana, Cuba)
A. Vidal (Evry, France)

2008-02-03T13:29:00.000-08:00

Nassif Ghoussoub

(Banf, Canada)

Homogenization of maximal monotone vector fields via selfdual variational calculus

We use the theory of selfdual Lagrangians to give a variational approach to the homogenization of equations in divergence form, that are driven by a periodic family of maximal monotone vector fields. The approach has the advantage of using Gamma-convergence methods for corresponding functionals just as in the classical case of convex potentials, as opposed to the graph convergence methods used in the absence of potentials. A new variational formulation for the homogenized equation is also given.
This is joint work with Abbas Moameni and Ramon Zarate-Saiz

2008-02-02T00:51:00.000-08:00

Klemens Fellner

(University of Cambridge, UK)

Stability of stationary states of non-local evolution equations

(joint work with Gael Raoul, CMLA, ENS Cachan)

We study non-local evolution equations for a density of individuals, which interact through a given symmetric potential. Such models appear in many applications such as swarming and flocking, opinion formation, inelastic materials, .... In particular, we are interested in interaction potentials, which behave locally repulsive, but aggregating over large scales. A particular example for such potentials was recently given in models of the alignment of the directions of filaments in the cytoskeleton.

We present results on the structure and stability of steady states. We shall show that stable stationary states of regular interaction potentials generically consist of sums of Dirac masses. However the amount of Diracs depends delicately on the interplay between local repulsion and aggregation. In particular we shall see that singular repulsive interaction potentials introduce diffusive effects in the sense that stationary state are rendered continuously.

2008-02-01T13:56:00.000-08:00

Session B: Fluid mechanic and coastal dynamic

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A. C. Alvarez
(Habana, Cuba)

Solving Mild Slope equation in Matlab with

Domain decomposition method

(joint with M. Sarkis and D. Marchesin)

In this paper the mild slope equation is solved on Matlab environmental. The method take into account physical processes waves as diffraction, refraction reflection, shoaling, port resonance and breaking. Further the effects of wave dissipation by friction, nonlinear amplitude dispersion are include. The algorithm is based on minimization of nonlinear functional using damped Newton iteration with Armijo-Goldstein line search strategy. Each linear problem is solved with Finite Element Method. Domain decomposition method with Schur complement is used in the linear case.

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Tadayoshi Kano

(Kyoto, Japan)

Tunamis on a deep open sea and on a gentle sloping beach

Tunamis as shallow water waves:
(1) The shallow water wave equations show that tunamis propagate on a deep open sea by, at least, the sound speed. (2) On a gentle sloping beach, the shallow water wave equations show that the diminution of propagation speed caused by the diminution of the water-depth results the deformation and, eventually, the break down of tunamis. The shallow water wave equation have a corresponding conservation law on a sloping beach and shock formation corresponding tunamis breaking on the shore.
(3) A mathematical justification for tunamis as shallow water waves is given.

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David Lannes

(Paris, France)

A stability condition for two-fluid interfaces

The aim of this talk is to derive a nonlinear stability condition for the interface between two fluids. This stability condition generalized the nonlinear Taylor criterion for the one fluid case and the linear Kelvin condition for the two fluid case. We show that it can explain the persistance of various natural phenomena such as waves and internal waves.

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Andre Nachbin

(Rio de Janeiro, Brasil)

Asymptotic issues regarding nonlinear wave/topography interaction

The asymptotic issues of interest regard both the PDEs' (i.e. reduced models) as well as the solution's point of view. In particular for problems involving the interaction of coastal waves with quite arbitrary topography profiles. Phenomena studied are the apparent diffusion and waveform inversion of long waves in the presence of highly disordered topographies. I will report on recent results concerning the formulation of accurate reduced (PDE) models, where accuracy concerns asymptotic properties of solutions in respect to the full potential theory model.

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Cristina Turner

(Cordoba, Argentina)

Nonlinear stability of two-layer flows

We study the dynamics of two–layer, stratiﬁed shallow water ﬂows. This is a model in which two scenarios for eventual mixing of stratiﬁed ﬂows (shear-instability and internal breaking waves) are, in principle, possible. We ﬁnd that unforced ﬂows cannot reach the threshold of shear-instability, at least without breaking ﬁrst. This is a fully nonlinear stability result for a model of stratiﬁed, sheared ﬂow. Mathematically, for 2X2 autonomous systems of mixed type, a criterium is found deciding whether the elliptic domain is reachable –smoothly– from hyperbolic initial conditions. If the characteristic ﬁelds depend smoothly on the system’s Riemann invariants, then the elliptic domain is unattainable. Otherwise, there are hyperbolic initial conditions that will lead to incursions into the elliptic domain, and the development of the associated instability.

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Baldomero Valiño Alonso

(La Habana, Cuba)

Infinitely narrow soliton in shallow water equations
(joint with Amaury Álvarez Cruz, Instituto de Oceanología, Cuba and Alex Méril, Université des Antilles et Guyana, Guadeloupe, France)

V.P.Maslov have shown that infinitely narrow solitons appear in Burgers’s equation as limit cases of singular solutions of Korteveg-De Fries’s equation. In this work we seek infinitely narrow solitons for shallow water equations, in the case when the profile of the depth has an isolated singular point. Following V. P. Maslov, we look for that singular solution as an asymptotical development with smooth coefficients in an appropriate algebra of generalized functions. Singular solutions of hyperbolic systems of partial differential equations in general position may have only a finite number of structuraly stable and self similar singular solutions. When the time evolves, the singular solution preserves the shape and structure of the initial condition. According to Maslov, the coeffcients of the formal asymptotic expansion of the singular solution satisfy an infinite chain of ordinary differential equations (the Hugoniot-Maslov chain corresponding to the singular solution involved). The truncation of the chain makes possible its aproximate calculation. From the point of view of numerical calculation, this approach is advantageous because it reduces the problem of solving numerically the nonlinear system of partial di¤erential equations to solving a closed system of ordinary differential equations. We interpret "infinitely narros solitons" as examples of simplified generalized functions in the sense of J. F. Colombeau and we use the rules of Colombeau’s differential algebras to do the calculations.

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Session C: Mean field games

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Italo Capuzzo Dolcetta

(Roma, Italy)

A numerical approach to mean field games

Mean ﬁelds games models introduced by Lasry-Lions describe through new systems of pde’s the asymptotic behavior of differential games in which the number of players tends to +∞. In the talk we discuss some recent work in collaboration with Y. Achdou and F. Camilli about numerical methods for the approximation of stationary and nonstationary versions of such models. Particular attention will be given to the optimal planning problem, in which the positions of a very large number of identical rational agents, with common value function, evolve from an initial given spatial density to a desired target density at the ﬁnal horizon time.

Maria Pia Gualdani
(Austin, USA)

Global Existence of a Free Boundary Problem with Non-Standard Sources

We consider a nonlinear free boundary problem, proposed by J.M.Lasry and P.L. Lions in 2006 in the framework of mean field games. The model describes the evolution of a scalar price which is realized as a free boundary of a diffusion equation with evolving sources. The talk focuses on global existence, uniqueness and regularity of solutions. The proof uses tools from non-interacting stochastic particle systems and multiscale analysis. Joint work with L. Chayes, M.d.M. Gonzalez and I. Kim.

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Session D: Classical and quantum kinetic equations

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María José Cáceres Granados

A numerical solver for a nonlinear Fokker-Planck equation in Neuroscience

Due to the huge number of neurons in the visual cortex network a kinetic approach can be considered. In such a way that the dynamic of the system is analyzed by mean of a distribution function. Therefore, the system of ODEs, which describes the integrate-and-fire models, is replaced with a system of PDEs. The advantage of the kinetic models is the reduction of the number of equations and consequently also the time of computation is reduced. In this talk we show a numerical scheme for the Fokker-Planck system, which models the visual cortex network dynamics. The deterministic numerical solutions are compared with Monte Carlo simulations. Our numerical solver allows us to obtain the evolution on time of the distribution functions and therefore the evolution of the macroscopic quantities, since they are moments of the pdf. As a consequence, we validate models which are obtained via moment closure.

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Simone Calogero

(Granada, España)

Self-similar behavior in relativistic galactic dynamics

In this talk I will present a short proof of existence of self-similar solution to the Ultra-Relativistic Vlasov-Poisson system in 3 dimensions. The proof is considerably simpler than the one presented recently by Lemou, Méhats & Raphaël and, at the same time, our existence result is more general. Moreover, our method works also for the massless Einstein-Vlasov system, for which the existence of self-similar solutions was previously unknown. This talk is based on a joint work with Óscar Sánchez.

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José Alfredo Cañizo

(Barcelona, España)

Entropy inequalities and speed of convergence to equilibrium for the

growth-fragmentation equation

The growth-fragmentation equation models a group of cells which grow (or age) at a certain rate, and divide into two or more pieces. After a suitable rescaling, the shape of the distribution function of the population converges to a universal profile. By means of entropy-entropy dissipation inequalities for some unbounded fragmentation coefficients, we show that the speed of this convergence is exponential. This kind of inequalities may be used for other linear equations for which the General Relative Entropy principle is valid.

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Rafael Galeano Andrades

(Cartagena, Colombia)

A variational approach to the stationary Boltzmann equation

(joint with Pedro Ortega Palencia)

We define a functional whose critical points coincide with stationary solutions of Boltzmann equation and by classical method to find the minimum of functionals we find critical point.

2008-02-01T04:42:00.000-08:00

David Lannes (Paris, France)

A stability condition for two-fluid interfaces

2008-02-01T03:32:00.000-08:00

Pierre-Louis Lions

(Collège de France, Paris, France)

An Introduction To Mean Field Games

This talk is an introduction to a new class of models called Mean Field Games that arise as the limits of Nash equilibria in a game where the number of players goes to infinity. The rigorous derivation of these models requires some new developments in Analysis that have other applications (to Statistical Physics and Mechanics). And we discuss some examples of such models (distribution of wealth in economics, population dynamics...).

2008-01-25T14:08:00.000-08:00

José A. Carrillo de la Plata

(ICREA & Universitat Autònoma de Barcelona, Spain)

(The lecture of Prof. J. A. Carrillo will likely be in spanish,

depending on the audience)

Macroscopic and Kinetic Equations in Swarming and Chemotaxis

In these series of lectures I will review several models for the collective behavior of systems composed by many individuals. As a reference we will first discuss some swarming models proposed to describe some self-organized animal behaviors in fishes and birds for instance. Some typical principles of swarming will be described and minimal models derived. Continuum equations like kinetic or macroscopic equations appear when considering a large number of individuals and the corresponding mean-field limits and hydrodynamic equations will be treated. Some useful tools are distances between probability measures in which transport distances play an interesting role to connect to macroscopic models. At the macroscopic level, we will learn how to work with the transport distances in some typical examples. Classical chemotaxis models in which cells move due to a chemical response will be solved using this method and its use for some other aggregation equations in swarming will also be analysed.

Ecuaciones cinéticas y macroscópicas en "Swarming" y quimiotaxis

En esta serie de lecciones, haré un resumen del estudio de varios modelos del comportamiento colectivo de sistemas compuestos por muchos entes individuales. Como ejemplo de referencia, discutiremos algunos modelos del comportamiento de auto-organización en algunos animales como peces y pájaros, llamado "swarming". Los principios básicos del "swarming" darán lugar a modelos minimales para su descripción. Ecuaciones continuas de tipo cinética o macroscópicas se obtienen cuando el número de individuos es muy grande en lo que se conoce como límite de campo medio. Algunas herramientas matemáticas para estudiar dichos límites se analizarán como las distancias de tipo transporte entre medidas de probabilidad. Además, permiten conectar el estudio de dichas ecuaciones con ecuaciones macroscópicas. Modelos macroscópicos, entre ellos el de quimiotaxis en el que las células dirigen su movimiento hacia las regiones de mayor concentración de un cierto componente químico o algunos modelos de swarming, se pueden estudiar con dichas distancias.

2008-01-25T04:52:00.000-08:00

Aimé Lachapelle
(Paris, France)

Mean Field Games: a numerical methodology with application to a stylised model of technology switch

Mean Field Games (MFG) have been recently introduced by Lasry and Lions, and describe approximations of N-player games when N tends to infinity. After recalling briefly the mathematical framework, we will insist on the optimal control point of view. We will then present the monotonic algorithm we use for solving finite-horizon MFG problems. We will close with some simulations for a stylised model of technology switch involving non-uniqueness. Joint work with J. Salomon and G. Turinici.

2008-01-25T04:20:00.000-08:00

Otared Kavian

(Versaille, France)

Remarks on Inverse Problems arising from Electrical Impedance Tomography

In this talk we present a few inverse problems related to the technique known as Electrical Impedance Tomography, which has important applications in medical imaging. In particular we shall present some results recently published in joint paper with Y. Capdeboscq, J. Fehrenbach, F. de Gournay. In that paper, we discuss the reconstruction of the impedance from the local power density. This study is motivated by a new imaging principle which allows to recover interior measurements of the energy density by a non invasive method. We discuss the theoretical feasibility in two dimensions, and propose numerical algorithms to recover the conductivity in two and three dimension. The efficiency of this approach is documented by several numerical simulations.

2008-01-25T04:15:00.000-08:00

Irene Gamba

(Austin, Texas, USA)

Non-conservative models of multi-linear particle interactions

We look at extensions Kac N-particle model of pair interactions to an N-particle model which includes multi-particle interactions in Fourier/Laplace transformed space. Under molecular chaos assumption, one obtains a multi-linear integro-evolution equation that can be view as extensions of Boltzmann model dynamics of Maxwell type for conservative or dissipative interactions.

We shall study the evolution of the corresponding probability density solution. Under the assumption temporal invariance under scaling transformations of the phase space and contractive properties,
we obtain a full description of existence, uniqueness and long-time behavior to self-similar states, all from its spectral properties as well as the formation of power tails in their self similar state.

We also focus on a couple of new examples of multi-agent dynamics and information sharing and percolation models.

2008-01-25T04:11:00.000-08:00

Miguel Escobedo

(Bilbao, Spain)

Post Gel solutions for coagulation equations

We shall briefly present and motivate the Smoluchowski coagulation esuation. We will describe the phenomena of gelation and recall the conjecture about the behaviour of the solutions after gelation. We will finally present a recent result about the existence of solutions with such a behaviour.

2008-01-21T08:31:00.000-08:00

Baldomero Valiño Alonso

(La Habana, Cuba)

Infinitely narrow soliton in shallow water equations
(joint with Amaury Álvarez Cruz, Instituto de Oceanología, Cuba and Alex Méril, Université des Antilles et Guyana, Guadeloupe, France)

2008-01-21T08:28:00.000-08:00

Tadayoshi Kano

(Kyoto, Japan)

Tunamis on a deep open sea and on a gentle sloping beach

2008-01-21T08:24:00.000-08:00

Rafael Galeano Andrades

(Cartagena, Colombia)

A variational approach to the stationary Boltzmann equation

(joint with Pedro Ortega Palencia)

We define a functional whose critical points coincide with stationary solutions of Boltzmann equation and by classical method to find the minimum of functionals we find critical point.

2008-01-21T08:21:00.000-08:00

Italo Capuzzo Dolcetta

(Roma, Italy)

A numerical approach to mean field games

2008-01-21T08:18:00.000-08:00

Cristina Turner

(Cordoba, Argentina)

Nonlinear stability of two-layer flows

2008-01-21T08:16:00.000-08:00

Andre Nachbin

(Rio de Janeiro, Brasil)

Asymptotic issues regarding nonlinear wave/topography interaction

2008-01-21T05:36:00.000-08:00

Maria Pia Gualdani
(Austin, USA)

Global Existence of a Free Boundary Problem with Non-Standard Sources

2008-01-21T05:30:00.000-08:00

Simone Calogero

(Granada, España)

Self-similar behavior in relativistic galactic dynamics

2008-01-21T05:28:00.000-08:00

María José Cáceres Granados

A numerical solver for a nonlinear Fokker-Planck equation in Neuroscience

2008-01-21T05:24:00.000-08:00

A. C. Alvarez
(Habana, Cuba)

Solving Mild Slope equation in Matlab with Domain decomposition method
(joint with M. Sarkis and D. Marchesin)

2008-01-21T05:20:00.000-08:00

David Lannes

(Paris, France)

Stability/instability issues for two-fluid interfaces

This short course is devoted to simple considerations on the stability of the interface between two fluids. With simple tools, we will exhibit the main phenomena responsable for possible (Kelvin-Helmholtz) instabilities and explain qualitatively the main steps of the full nonlinear analysis.

2008-01-21T05:01:00.000-08:00

José Alfredo Cañizo

(Barcelona, España)

Entropy inequalities and speed of convergence to equilibrium for the

growth-fragmentation equation

2008-01-20T12:35:00.000-08:00

Vincent Calvez

(CNRS & ENS-Lyon, France)

The geometry of one-dimensional self-attracting particles

(joint work with José Antonio Carrillo, ICREA, Univiversidad Autònoma Barcelona)

We wish to present in this talk a brief overview of recent developments in the analysis of the generalized Keller-Segel system:
\[
\partial_t \rho(t,x) = \Delta\rho^m(t,x) + \chi \nabla\cdot \left(\rho(t,x) \nabla\left(\dfrac{|x|^{k}}{k}*\rho(t,x)\right)\right)\, , \quad t>0\, , \quad x\in \mathbb{R}^N\, .
\]
Here $\rho(t,x)$ denotes the density of diffusive cells which are self-attracted through a mean field potential. This model has raised a lot of interest in the field of mathematical biology since it generally exhibits a dichotomy between global existence (dispersion of the cells) and finite time blow-up (aggregation of the cells). We will explain how this system possesses the structure of a gradient flow for the Wasserstein metric on the space of probability densities. Interestingly enough the energy functional is the sum of two opposite contributions, being respectively convex and concave, and homogeneous. We will show some consequences of this gradient flow interpretation, concerning finite time blow-up and long time asymptotics.

2008-01-20T12:25:00.000-08:00

Thomas Hillen

(University of Alberta, Canada)

''Pointwise weak steady states of transport equations for cell movement in network tissue''

(joint work with K. Painter (Heriot-Watt), P. Hinow (Milwaukee) and Z.A. Wang (Vanderbilt))

Some metastatic cancer cells use an interesting mechanism to move through network tissue, such as collagen networks. I will present a transport equation model, which describes the movement of these cells along network fibers and the network degradation due to proteolytic enzymes which are produced by the cells. In numerical implementation we find that the resulting models shows generation of networks which resemble vasculature formation. We believe that pointwise weak steady states are the organizing centres behind these network patterns. I will introduce the notion of pointwise weak steady states and discuss their use to understand network formation.

2008-01-12T12:59:00.000-08:00

Nicolas Meunier

(Université Paris V, France)

Analysis of self-organization systems for cell polarization

(Joint work with V. Calvez and R. Voituriez)

In this work, we investigate the dynamics of a modified Keller-Segel type model. On the contrary to the classical configuration, the chemical production term is located on the boundary. In the one-dimensional case and in a particular case in dimension two, we prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass and they blow-up in finite time above the critical mass. Furthermore, in the one-dimensional case, using entropy techniques, we provide quantitative convergence results for the subcritical case. This work is completed with a more realistic model (still one-dimensional) for modeling purpose. In this new setting, the chemical is supplied by a quantity which evolves by exchanging particles at the boundary. Finally some results are given for the two-dimensional case and we also provide some links with cell polarisation that is an essential step for many biological processes and that is involved for instance in cell migration, division, or morphogenesis.

2008-01-12T12:25:00.000-08:00

Pierre-Emmanuel Jabin

(Université de Nice Sophia-Antipolis, France)

"Selection-mutation Dynamics for the Evolution of Traits in a Population"

This talk aims at introducing some models describing the evolution of a large population of individuals with different biological "traits". The reproduction rate of each individual depends on its trait and on the competition for ressources within the population. In addition mutations may be included, enabling new traits to develop. These models leads to difficult asymptotic problems as the time scale of the mutations is usually much larger than the time scale of births and death. One such asymptotic will in particular be investigated, corresponding to frequent small mutations.

2008-01-09T04:00:00.000-08:00

Session A: Mathematical Biology

________________________________________________________

Davide Ambrosi

(Politecnico di Milano, Italia)

The electromechanical coupling in cardiac mechanics

_________________________________________________________

Vincent Calvez

(CNRS & ENS-Lyon, France)

The geometry of one-dimensional self-attracting particles

(joint work with José Antonio Carrillo, ICREA, Univiversidad Autònoma Barcelona)

We wish to present in this talk a brief overview of recent developments in the analysis of the generalized Keller-Segel system:

\partial_t \rho(t,x) = \Delta\rho^m(t,x) + \chi \nabla\cdot \left(\rho(t,x) \nabla\left(\dfrac{|x|^{k}}{k}*\rho(t,x)\right)\right)\, , \quad t>0\, , \quad x\in \mathbb{R}^N\, .

Here $\rho(t,x)$ denotes the density of diffusive cells which are self-attracted through a mean field potential. This model has raised a lot of interest in the field of mathematical biology since it generally exhibits a dichotomy between global existence (dispersion of the cells) and finite time blow-up (aggregation of the cells). We will explain how this system possesses the structure of a gradient flow for the Wasserstein metric on the space of probability densities. Interestingly enough the energy functional is the sum of two opposite contributions, being respectively convex and concave, and homogeneous. We will show some consequences of this gradient flow interpretation, concerning finite time blow-up and long time asymptotics.

_________________________________________________________

Maria Rita D'Orsogna

(California State University at Northridge, Los Angeles, USA)

Modelling viral entry dynamics

Successful viral infection of a healthy cell requires complex host-pathogen interactions. In this talk we focus on the dynamics specific to the HIV virus entering a eucaryotic cell. We model viral entry as a stochastic engagement of receptors and coreceptors on the cell surface. We also consider the transport of virus material to the cell nucleus by coupling microtubular motion to the concurrent biochemical transformations that render the viral material competent for nuclear entry. We discuss both mathematical and biological consequences of our model, such as the formulation of an effective integrodifferential boundary condition embodying a memory kernel and optimal timing in maximizing viral probabilities.

_________________________________________________________

Marie Doumic-Jauffret

(INRIA, France)

Aggregation/fragmentation models for protein polymerization

Abnormal protein polymerization is at the origin of a number of neuro-degenerative diseases, e.g. Prion (Madcow), Huntington or Alzheimer diseases. This phenomenon can be described mathematically either in a discrete or in a continuous setting, by equations of fragmentation-coagulation type. We review here some results and some open problems on the study of prion proliferation equations.

_________________________________________________________

Laurent Dumas

(Université Paris VI, France)

A numerical simulation of the human arterial network based on non invasive measurements

The numerical simulation of the human arterial network of a given patient in order to find the main characteristics of its arteries and thus to prevent from cardiovascular diseases is a major subject of interest. A simple fluid-structure interaction model for the simulation of blood flow in arteries is studied here. Derived from the general 3D equations, it computes the section area of the artery and the volumetric flux of the flow for each longitudinal position and time. The inverse problem corresponding to the identification of the main parameters of this model is successfully solved by using a reduced number of non invasive measurements. A discussion on the amount of information needed is in particular studied. Some first results based on experimental measurements with a Doppler technique called echotracking are also presented.

__________________________________________________________

Klemens Fellner

(University of Cambridge, UK)

Stability of stationary states of non-local evolution equations

(joint work with Gael Raoul, CMLA, ENS Cachan)

__________________________________________________________

Thomas Hillen

(University of Alberta, Canada)

Pointwise weak steady states of transport equations for cell movement in network tissue

(joint work with K. Painter (Heriot-Watt), P. Hinow (Milwaukee) and Z.A. Wang (Vanderbilt))

__________________________________________________________

Pierre-Emmanuel Jabin

(Université de Nice Sophia-Antipolis, France)

Selection-mutation dynamics for the evolution of traits in a population

______________________________________________________

Doron Levy

(University of Maryland, USA)

Group Dynamics in Phototaxis

Microbes live in environments that are often limiting for growth. They have evolved sophisticated mechanisms to sense changes in environmental parameters such as light and nutrients, after which they swim or crawl into optimal conditions. This phenomenon is known as "chemotaxis" or "phototaxis." Using time-lapse video microscopy we have monitored the movement of phototactic bacteria, i.e., bacteria that move towards light. These movies suggest that single cells are able to move directionally but at the same time, the group dynamics is equally important. Following these observations, in this talk we will present a hierarchy of mathematical models for phototaxis: a stochastic model, an interacting particle system, and a system of PDEs. We will discuss the models, their simulations, and our theorems that show how the system of PDEs can be considered as the limit dynamics of the particle system. Time-permitting, we will overview our recent results on particle, kinetic, and fluid models for phototaxis. This is a joint work with Devaki Bhaya (Department of Plant Biology, Carnegie Institute), Tiago Requeijo (Math, Stanford), and Seung-Yeal Ha (Seoul, Korea).

_____________________________________________________

Sébastien Martin

(Université Paris-Sud 11, France)

Modelling of air flows and gas exchange in the human lung

This talk aims at describing some models of the respiratory system. The mechanical response of the lung can be described by either sophisticated models based on (Navier-)Stokes flows or simple mass-spring models, that are accurate enough to predict the behaviour of the system during a complex dynamical situation (the forced inspiratory-expiratory maneuver that is performed during spirometric evaluations). Oxygen absorption models are also described in order to observe the effects of changes in the parameters of the mechanical model in the efficiency of the lung. In particular, the models suggest that the airway smooth muscle (which is directly related to the bronchial rigidity) could have a structural function, to confer transport optimized mechanical properties to the bronchial tree.

_________________________________________________________

Nicolas Meunier

(Université Paris V, France)

Analysis of self-organization systems for cell polarization

(Joint work with V. Calvez and R. Voituriez)

In this work, we investigate the dynamics of a modified Keller-Segel type model. On the contrary to the classical configuration, the chemical production term is located on the boundary. In the one- dimensional case and in a particular case in dimension two, we prove, under suitable assumptions, the following dichotomy which is reminiscent of the two-dimensional Keller-Segel system. Solutions are global if the mass is below the critical mass and they blow-up in finite time above the critical mass. Furthermore, in the one-dimensional case, using entropy techniques, we provide quantitative convergence results for the subcritical case. This work is completed with a more realistic model (still one-dimensional) for modeling purpose. In this new setting, the chemical is supplied by a quantity which evolves by exchanging particles at the boundary. Finally some results are given for the two-dimensional case and we also provide some links with cell polarisation that is an essential step for many biological processes and that is involved for instance in cell migration, division, or morphogenesis.

______________________________________________________

Julie Mitchell

(University of Wisconsin - Madison, USA)

Using clustering and optimization in flexible protein docking

Modeling molecular flexibility continues to be a frontier in predicting how two proteins bind. Allowing full dynamic flexibility can be excessive, yet neither traditional rigid body docking nor small-scale rearrangements of amino acid side chains can capture many biologically relevant motions. We propose an intermediate approach, which is to combine rigid optimization with discrete deformations that represent a protein's ``natural'' motions. Through exhaustive low-resolution search, hierarchical clustering, and global optimization within each energy basin, we are able to achieve accurate docking predictions. Our optimization approach employs a Convex Global Underestimation method that has been adapted for energy functions defined on $R^3 \times SO(3)$.

_________________________________________________________

Gael Raoul

(ENS-Cachan, France)

Kimura's model: An integro-differential model to study evolution

Kimura's model is a simple intero-differential model used by theoretical biologists to study evolution. Simulations for this model can be run, and they show that that the population usually converges to a finite sum of Dirac Masses. This phenomenon is interpreted by biologists as a speciation process. We study the stability of such populations, and link our results to other existing theories used by biologists to study evolution.

__________________________________________________________

Christian Schmeiser

(University of Vienna, Austria)

Nonlinear friction in the cytoskeleton as a macroscopic limit of

the activity of cytoskeletal proteins

The mechanical stability of polymer networks in the cytoskeleton relies on the action of cross-linking proteins and transmembrane proteins creating adhesion to the substrate. It will be demonstrated that their averaged effect can be described as friction, when cross-linking and adhesion molecules are assumed as elastic and when their building and breaking are described as stochastic processes.

__________________________________________________________

Alexandre Vidal

(Université d'Evry Val d'Essonne, France)

A model of the Gonadotropin Releasing Hormone secretion by hypothalamic neuron clusters

The secretion of Gonadotropin Releasing Hormone (GnRH) by specific hypothalamic neurons plays a major role in the neuroendocrine control of the reproduction function in mammals. While the qualitative properties of the physiological secretion pattern is common to all females, the quantitative properties characterize the specificities of the ovulatory cycle in each species. The difficulty to obtain in vivo time series of GnRH release by the hypothalamus spurs us on to develop a controllable model to study this complex biological system.

I will describe the qualitative properties of the physiological GnRH secretion pattern and introduce the types of specifications that the biological knowledge can specify in various species. I will present a model of the GnRH neuron secreting activities based on reaction-diffusion coupling of fast-slow oscillators. Applying bifurcation theory and a fast-slow analysis to a reduced model, I will show the model ability to reproduce the qualitative and species-dependent quantitative properties of the GnRH secretion pattern. Finally, I will display and discuss the synchronization/desynchronization alternation of the cluster activities reproducing subtle observable features but biologically misunderstood.

________________________________________________________

2008-01-09T03:45:00.000-08:00

Marie Doumic-Jauffret

(INRIA, France)

Aggregation/fragmentation models for protein polymerization

2008-01-09T03:25:00.000-08:00

Laurent Dumas

(Université Paris VI, Frane)

A numerical simulation of the human arterial network based on non invasive measurements

2007-12-23T14:54:00.000-08:00

Christian Schmeiser

(University of Vienna, Austria)

Nonlinear friction in the cytoskeleton as a macroscopic limit of

the activity of cytoskeletal proteins

2007-12-19T15:08:00.000-08:00

Maria Rita D'Orsigna

(California State University at Northridge, Los Angeles, USA)

Modelling viral entry dynamics

2007-12-19T15:01:00.000-08:00

Doron Levy

(University of Maryland, USA)

Group Dynamics in Phototaxis

2007-12-19T14:46:00.000-08:00

Peter Markowich

(University of Cambridge, UK & University of Vienna, Austria)

On PDE Models for (some) Socio-Economic Problems:
Price Formation, Opinion Formation and Crowd Modeling

We present and analyse partial differential equation models in the socio-economic sciences:

1) a highly nonstandard free boundary problem in price formation

2) a Boltzmann equation for opinion formation in human societies with strong leaders

3) a continuity equation with an eikonal field equation coupling.

2007-12-15T13:54:00.000-08:00

Julie Mitchell

(University of Wisconsin - Madison, USA)

Using clustering and optimization in flexible protein docking

2007-02-01T00:54:00.000-08:00

Alexandre Vidal

(Université d'Evry Val d'Essonne, France)

A model of the Gonadotropin Releasing Hormone secretion by hypothalamic neuron clusters

2007-01-20T13:03:00.000-08:00

Marc Thiriet

(CNRS, Université P. et M. Curie and INRIA Rocquencourt, France)

Lecture 1 : Physiological flows -- a macroscopic point of view with application to blood circulation and body's ventilation.

- Basic equations: mass and momentum conservation (Navier-Stokes equations).

- Main features: unsteady, developing, three-dimensional flows of Newtonian or non-Newtonian fluid in deformable pipes (distensible and collapsible ducts).

Lecture 2 : Physiological flows -- a nano/microscopic point of view.

Biological conduit walls are living tissues that react to the stress field applied to them by flowing fluid (mechanotransduction) and adapt vessel lumen size accordingly. Additional behavioral equation must be coupled to correct the computational domain size predicted by the Navier-Stokes solution.

2007-01-17T05:34:00.000-08:00

Sébastien Martin

(Université Paris-Sud 11, France)

Modelling of air flows and gas exchange in the human lung

2006-01-13T12:34:00.000-08:00

Gael Raoul

(ENS-Cachan, France)

Kimura's model: An integro-differential model to study evolution

2006-01-13T12:17:00.000-08:00

Benoît Perthame

(Université P. et M. Curie, INRIA-Rocquencourt and Institut Universitaire de France, France)

Adaptive evolution: a population view

The two processes of mutations and selection, proposed by C. Darwin, can be written in mathematical words. In a very simple, general and idealized description, the environment can be considered as a nutrient shared by all the population. This alllows certain individuals, characterized by a 'phenotypical trait', to multiply faster because they are better adapted to use the environment. This leads to select the 'fittest trait' in the population. On the other hand, the new-born individuals undergo small variations of the trait under the effect of mutations. In these circumstances, is it possible to observ 'speciation' and to describe the dynamical evolution of the 'fittest' traits?

We will give a class of self-contained mathematical models of such population dynamics and show that an asymptotic view allows us to formalize precisely the concepts of monomorphic or polymorphic populations. We can describe the evolution of the 'fittest traits' and various forms of branching points. We will also show that numerical solutions are consistant with individual based stochastic simulations.

Eventhough the model is very simple, its analysis relates to remarkable recent progresses of nonlinear analysis.