Menu

	French-Korean IRL in Mathematics International Research Laboratory

Documents

The IRL intends to gather two teams, the PDEA team (Partial Differential Equation, and Application) and the AGNT team (Algebraic, Complex, Differential Geometry and Number Theory), across the two contries France and Korea. The PDE team is more oriented towards Kinetic Theory, Mean Field Equations, Quantum Synchronization, and their Applications to Biology, Fluid Mechanics, Ocean-Atmosphere interface, or medicine. The team in Geometry, either Algebraic, Complex or Differential in strong interaction with Number Theory is more oriented towards Diophantine Approximation, Moduli spaces, Hodge Theory, p-adic Galois Representations, Langlands program, Iwasawa Theory of automorphic forms, Einstein Riemannian Equations.

1. One of the central problems in modern number theory is to understand the conjectural Langlands reciprocity between p-adic representations of the Galois group of a number field F (i.e., of a finite extension F/Q), and automorphic forms over F. Automorphic forms are a generalization of classical modular forms; perhaps the most notable example of the above reciprocity is the Shimura–Taniyama conjecture, whose proof [Wil, TW, BCDT] began with the famous work of Wiles and Taylor–Wiles, and which says that each elliptic curve over Q corresponds to a modular form. Questions on automorphic forms.
   1. For noncommutative cohomology: The algebraic side. Manin has given a map from vectors of cusp forms to a noncommutative cohomology set by means of iterated integrals. It is further studied that there a version of the Eichler-Shimura isomorphism with a non-abelian H1 in group cohomology. When it is integral weight this map is more explicit. Linear combinations of multiple Hecke L-values can be expressed as linear combinations of products of the usual Hecke L-series evaluated at the critical points. We would like to attach those values to period polynomial which played a important role in elliptic modular form theory]. Also we would like to establish Hecke actions on this noncommutative cohomology. The analytic side. Non-commutative cohomology theory uses iterated integral so that one may consider iterated Mellin transform. We would like to extend this theory for general type which may have a connection with Lerch Zeta functional equation.
   2. For Maass Wave forms: Maass wave forms are still very mysterious object, contains many conjecture such as Selberg eigenvalue conjecture. Selberg Zeta functions are still not much understood. We plan to study Maass wave form via Eicher-Shimura typo cohomology theory started by Lewis-Zagier and Bruggeman-Lewis-Zagier.
   3. For special values of autormorphic L functions: The special values of automorphic L functions can be studied via parabolic cohomology theory. We would like to study the automorphic L-values of outside of critical strip. Also study vanishing behavior at the center of critical strip via analytic approach. Questions on p-adic Galois representations, integral p-adic Hodge theory, and Galois deformation theory.
      1. The weight part of Serre’s conjecture for wildly ramified representations. The weight part of Serre’s conjecture for tamely ramified representations are well understood: at least, we do have a precise statement by Herzig. But the case for the wildly ramified representations is quite mystery: we don’t even have a conjectural description of Serre weights in this case.
      2. Semi-stable deformation rings in the residually reducible case In the residually reducible construct semi-stable deformation rings in Hodge–Tate weights (0,1,2) of such reducible residual representations.
      3. Semi-stable deformation rings: construct semi-stable deformation rings in parallel various Hodge–Tate weights These results can also be used to get some new results on modularity lifting theorems.
2. The group of algebraic geometry at IMJ-PRG has broad research interests reflecting the diversity of modern algebraic geometry.
   1. Chiodo and Zvonkine work on enumerative geometry of moduli spaces or stacks. This field has greatly expanded since the 90's in the direction of symplectic topology under the impulse of mirror symmetry and Gromov-Witten invariants.
   2. Christian Peskine and Frédéric Han work in projective geometry, and to a certain extent, also Nicolas Perrin although his work is more devoted to the study of homogeneous varieties, hence builds more on representation theory and algebraic groups. Perrin also has interests in Gromov-Witten invariants and enumerative geometry.
   3. Olivier Debarre has various main thematics of research. The three sorts of geometry he investigated are Fano varieties, abelian varieties, and hyper-Kaehler manifolds. It turns out that by auxiliary constructions, these different sorts of geometry are very much related. For example, associated with a cubic hypersurface of dimension 3, there is an abelian variety defined by Griffiths, known as the intermediate Jacobian. Associated with a cubic hypersurface of dimension 4, there are several hyper-Kaehler manifolds (we know now that there are infinitely many of them), the simplest being the Fano variety of lines.
   4. Claire Voisin works on Hodge theory and algebraic cycles and more generally the topology of algebraic varieties taken in a broad sense. Hodge theory can be studied in the general compact Kähler setting, but in the projective setting, the Hodge structures become polarized and there is a motive in the Grothendieck sense. So the Hodge structures in algebraic geometry are an incarnation of phenomena related to algebraic cycles. Nothing like that is expected in compact Kähler geometry, even if Hodge theory is still useful to restrict the topology of compact Kaehler manifolds The recent research interests of Claire Voisin are birational invariants related to Chow groups generalized Bloch conjecture for certain types of varieties and hyper-Kaehler geometr with emphasis on the Chow rings.
   5. Junyan Cao is on the complex analytic side of algebraic geometry. He is interested in classification theory for compact Kaehler manifolds (eg under some positivity assumption), deformation theory, and analytic methods (like currents).
   6. Gerard Freixas works originally in Arakelov geometry but he has become more and more intereste recently into its complex analytic aspects.
3. The interplay between Diophantine approximation and dynamical systems is a modern subject in Number Theory. Classical questions include the determination of the Hausdorff dimension of sets of points dynamically defined and approximable at a given rate by, say, rational points. Of special interest is the study of sets arising in uniform Diophantine approximation.
4. Algebraic geometry, with connections to transcendence theory, arithmetic and diophantine approximation, is another subject that some members intend to persue. Among the main themes of research tare the study of rational, elliptic and entire curves on varieties of general type, their relationship with their arithmetic and the study of higher codimensional cycles on special varieties (for instance hyperkaehler).One should also mention Hodge theory and Chow groups of complex projective varieties, with an emphasis on surfaces, Calabi-Yau varieties and hyperkaehler varieties.
5. Arithmetic geometry in connection with p-adic Hodge theory, p-adic Langlands program, Iwasawa theory, the cohomology of p-adic symmetric spaces, and p-adic dynamical systems is the main interest of several members of the program. This includes in particular the study of mod p and p-adic representations of p-adic Lie groups such as GL_2(Q_p); the study of the structure of deformation spaces of p-adic Galois representations, both from a theoretical and an algorithmic point of view; the study of (phi,Gamma)-modules in various settings, and the interaction with the theory of p-adic dynamical systems; the cohomology of symmetric spaces such as Drinfeld's upper half plane; the study of special values of complex and p-adic L-functions. We are very interested in the interactions between these topics as well as potential applications and generalizations.
6. Differential geometry and PDE is known to be very efficient tools when they are used together. Among them the geometric flows have striking properties; indeed, they behave as non-linear heat equation and hence are regularizing. The goal in using them is to produce nice objects as limit of these flows. Among them one can find Einstein Riemannian manifolds which are metric defined by an elliptic equation when written in suitable coordinate charts. They are very difficult to produce and lots of problems are still open. One technique consists in solving the Einstein (or a similar) equation admitting singularities. This gives some flexibility in the constructions. We then hope to smooth these singularities out. For some class of manifolds this method raises questions in algebraic geometry. Further one could dream of p-adic extensions of these techniques.
7. Yohan Brunebarbe's and Vincent Koziarz's work belongs to the domain of complex geometry, especially in kählerian and/or projective complexe geometry with applications to differential geometry and algebraic geometry in general characteristic. They study more specifically the fundamental groups of kählerian manifolds, or the period maps of such manifolds. The geometry of quotients of bounded symmetric domains is also at the heart of their interest (for instance quotients of the ball, Shimura manifolds).

PDE theory is a large branch in mathematics that covers both theoretical and applied thematic areas. The team is mainly concerned with, kinetic models, hydrodynamics, collective dynamics, self-organization, diffusive population models, cell mobility, multi-agent systems (chemotaxis, swarming), coalescence. The mathematical techniques that the team intends to develop concern new paradigms in reaction-diffusion related to kinetic methods (cross diffusion, non-local effects), nonlinear integrodifferential equations, topological methods for the search of steady states, spectral methods and stability, time-dependent rescalings for the study of large time behavior, entropy methods, Dynamical Systems and Ergodic Theory tools.

1. The Boltzman model is a fundament tool based on a non linear and non local operator but complicated to handle. The BGK model is a simpler model less costly in numerical simulations where the principle is to replace the Boltzmann collision operator by a relaxation operator whose attractor is a Maxwellian distribution, at the expense that the nonlinearity becomes exponential instead of quadratic. Some members of the team intend to develop theoretical existence theorems and numerical studies in the BGK model: existence theorems for models with bounded domains and interfaces corresponding for instance to interactions of two gases situated between two infinite parallel plates.
2. Regular reaction-diffusion systems in which no maximum principle is available can be studied by methods partly derived from kinetic theory, This is the case of degenerate reaction-diffusion systems in which part of the diffusion rates are 0, as well as cross-diffusion systems coming out of the theory of gases and the population dynamics. Some members of the team are working on the existence and smoothness theorems as well as asymptotic properties for cross diffusion models related to problems naturally arising in the theory of chemotaxis of bacteria. The methods that they intend to use include duality theorems, entropy structure studies, and the analysis of Turing-like patterns.
3. Kinetic equations have found a fertile new ground of applications in biology to describe bacterial collective motion. Indeed, it is known that E. Coli, and many other cells, move by run and tumble with a tumbling rate driven by internal states of the cell. Many biophysical observations can only be explained taking into account the detailed way individual decisions to tumble are taken, as the famous Berg experiments exhibiting traveling pulses. This area leads to understand rescaling and asymptotic analysis (and for instance the role the newly developed theory of flux limited Keller-Segel model), existence of traveling pulses or waves (and the recent discovery of accelerating waves also), various types of instabilities. Emergent questions are to explain, based on now available biophysical observations, abnormal diffusions and boundary conditions.
4. Mean-field equations for neural networks have been derived by several groups of physicists and studied by mathematicians. The most standard model is the Leaky, Noisy Integrate-and-Fire Fokker-Planck equation which describes the probability to find a neuron at the potential v. A major recent discovery is blow-up which indicates the begining of network synchronization. Another class of models are time elapsed equations structured by the time after discharge where synchronized states are also observed, and of course these are closely related to similar phenomena in Kuramoto equations and other models arising in population flocking. For instance, the Lohe matrix model, a noncommutative coupled dynamical system analogous to the Kuramoto lattice, leads to questions analogous to the well-known Kuramoto case (mean-field limit, quantitative estimates of the convergence rate for the mean-field limit, fluctuations about the mean-field regime, onset of synchronization in either the original coupled system or its mean-field kinetic limit). The subject is flourishing presently both in Korea and France.
5. Coupled mean-field Schrodinger type equations for quantum networks have been proposed by several physicists. Among them, one of mathematically well studied one is the Schrodinger-Lohe system for quantum synchronization. Recently, several mathematical approaches for quantum synchronization have been proposed, e.g., Lyapunov functional approach, dimension reduction method based on two point correlation functions. The Schrödinger-Lohe model raises fascinating questions, for instance on the validity of the mean-field limit in an infinite dimensional setting, for which no systematic approach seem to exist at present. The complete synchronization for the Schrodinger-Lohe model is still open problem, not to mention the phase-transition analysis which is one of hot topics in nonequilibrium statistical physics.
6. Quasi-periodic systems for large systems can be studied using Dynamical Systems and Ergodic Theory. The Kolmogorov-Shannon complexity of those systems is low compared to the one of more disordered systems; algebraic tools are then needed to classify them. Gibbs formalism on Sturmian trees have been recently developed for low dimensional systems; the extension to large systems is expected to be very promising.
7. There is currently an increasing interest from the clinical community in using computational models to monitor the evolution of a tumor or its response to a treatment. Among the key parameters involved in such phenomena, those related to cell proliferation and tulor rheology are essential. The Inria team MONC, located at the Institut de Mathématiques de Bordeaux, has developed an expertise in the mathematical description of cell proliferation at the tissue scale. The team has developed coupled hyperbolic/elliptic system of PDEs to describe the tumor evolution as seen by oncologists on medical images. Recently, thanks to a tight collaboration with biologists who performed experiments on multi-cellular tumor spheroids (MCTS), the team has obtained significant results on the influence of the microenvironment on the cell proliferation on one side, and on the spheroid rheology, thanks to observation of the fusion of spheroids. Interestingly, thanks to the consideration of apparent surface tension, a new viscoelastic model of MCTS has been validated by experimental results. We aim to push forward the modeling by adding nonlinear biological process that lead to such apparent surface tension. Multiscal modelling, linking the biological cell scale interactions to the macroscale physical behaviours is a challenge addressed by the researchers of Bordeaux University.