The SEMPARIS seminar webserver hosts annoucements of all seminars taking place in Paris area, in all topics of physics, mathematics and computer science. It allows registered users to receive a selection of announcements by email on a daily or weekly basis, and offers the possibility to archive PDF or Powerpoint files, making it available to the scientific community. [ More information ]
Upcoming Seminars  [Next 30 ]  
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Monday 11 December 2017, 11:00 at IPHT, Salle Claude Itzykson, Bât. 774  IPHTPHM (Séminaire de physique mathématique)  mathph 



Abstract:  I consider a simple outofequilibrium setup where a 1d quantum spin system on the infinite lattice is prepared in a domain wall product state, and then let evolve unitarily with the Hamiltonian of the XXZ spin chain. I explain how this ``quantum quench'' problem is related to the sixvertex model by analytic continuation, and how quantum inverse scattering methods on the six vertex side may be exploited to both derive exact results and gain some insights on such inhomogeneous quenches. I focus in this talk on the simplest quantity, the IzerginKorepin partition function of the classical model, which becomes a return probability (RP) after the quench, and may be expressed in the form of a Fredholm determinant. In the gapless regime, the decay of the RP at late times is found to be gaussian, with a rate that is, surprisingly, nowhere continuous as a function of anisotropy. Exactly at the Heisenberg point, the decay is exponential in $\sqrt{t}$, strongly suggesting diffusive transport. \par This particular quench also serves as an opportunity to discuss alternative approaches such as the generalized hydrodynamic treatment of integrable systems, or closely related problems such as arctic curves for dimer coverings, and conformal field theory description of 1d inhomogeneous systems. \\ \\ Reference: arXiv:1707.06625 
Monday 11 December 2017, 11:00 at LPTMC, tower 1312 room 523  SEMLPTMC (Séminaire du Laboratoire de Physique Théorique de la Matière Condensée)  condmat.meshall 



Abstract:  The paradigmatic condensedmatter models where zeroenergy localized Majorana fermions have been studied so far have the distinguishing feature of not conserving the number of fermions. Moreover, the accepted definition of "Majorana fermion" naturally belongs to this scenario. Is it possible to discuss Majorana fermions in "canonical" particleconserving settings? In this seminar I will present several exact and numerical results on Majorana fermions in particleconserving scenarios. I will start from the discussion of a model for bosons and fermions where, in a proper limit, the physics of the celebrated Kitaev's chain appears. I will continue by presenting exact results on Majorana fermions in ladder models where the two legs of the system can only exchange pair of particles. Finally, I will comment on the possibility of making experiments with Majorana fermions in particleconserving settings. References: Iemini, LM, Rossini, Fazio and Diehl, PRL *115*, 156402 (2015) Iemini, LM, Fallani, Zoller, Fazio, Dalmonte, PRL *118*, 200404 (2017) 
Monday 11 December 2017, 14:00 at LPTHE, Bibliothèque  SEMLPTHE (Séminaire du LPTHE)  condmat.statmech 



Abstract:  Entanglement and entropy are key concepts standing at the foundations of quantum and statistical mechanics, respectively. In the last decade the study of quantum quenches revealed that these two concepts are intricately intertwined. Although the unitary time evolution ensuing from a pure initial state maintains the system globally at zero entropy, at long time after the quench local properties are captured by an appropriate statistical ensemble with non zero thermodynamic entropy, which can be interpreted as the entanglement accumulated during the dynamics. Therefore, understanding the postquench entanglement evolution unveils how thermodynamics emerges in isolated quantum systems. An exact computation of the entanglement dynamics has been provided only for noninteracting systems, and it was believed to be unfeasible for genuinely interacting models. Conversely, here we show that the standard quasiparticle picture of the entanglement evolution, complemented with integrabilitybased knowledge of the asymptotic state, leads to a complete analytical understanding of the entanglement dynamics in the spacetime scaling limit. Our framework requires only knowledge about the steady state, and the velocities of the lowlying excitations around it. We provide a thorough check of our result focusing on the spin1/2 Heisenberg XXZ chain, and considering quenches from several initial states. 
Monday 11 December 2017, 15:00 at
LPT,
Room 114, Bldg 210, 1st floor, LPT Orsay ( SInJe : Séminaire Informel des Jeunes )  SEMINFOR (Séminaire informel)  hepph 



Abstract:  Heavy neutrinos are new neutral leptons that can be introduced to explain standard neutrino masses. They can be between eV ant GUT scale but some particularly interesting models with GeV neutrinos explain baryogenesis with minimal extension of Standard Model. As it will be presented, this mass region is within the reach of present (LHC, T2K...) and future experiments (SHiP, FCC, DUNE), looking for heavy neutrinos through their production and/or decay. 
Tuesday 12 December 2017, 11:30 at LPTHE, LPTHE library  STRLPTENSHE (Séminaire commun LPTENS/LPTHE)  hepth 



Abstract:  Supersymmetric localization is a powerful technique to evaluate a class of functional integrals in supersymmetric field theories. It reduces the functional integral over field space to ordinary integrals over the space of solutions of the offshell BPS equations. The application of this technique to supergravity suffers from some problems, both conceptual and practical. I will discuss one of the main conceptual problems, namely how to construct the fermionic symmetry with which to localize. I will show how a deformation of the BRST technique allows us to do this. This leads us to a formulation of twisted supergravity. I will then sketch a computation of the oneloop determinant of the graviton supermultiplet that enters the localization formula for BPS black hole entropy. 
Tuesday 12 December 2017, 14:00 at APC, 483 A  Malevitch  APCTH (Seminar of the theory group of APC)  hepth 



Abstract:  The distribution of galaxies provides a powerful way to probe the properties of our universe. In order to exploit this observable properly it is necessary to understand what we are really measuring when we look at the largescale structure. Since our universe is not completely homogeneous and isotropic, we only see a distorted picture of our sky. In this talk, I will discuss the various relativistic effects that distort our observations. I will show that even though these effects complicate the interpretation of galaxy surveys, they are very useful since they contain information about the theory of gravity and can therefore be regarded as a new opportunity for future surveys. 
Tuesday 12 December 2017, 16:00 at LPT, salle 114 LPT Orsay Bat 210 1er etage  SEMLPT (Séminaire de Physique des Particules du LPT)  hepph 



Abstract:  Flavour Changing Neutral Currents (FCNC) are an excellent probe for the search of New Physics. Therefore, LHCb has put a particular care in the study of B decays mediated by FCNC starting from Run I and with more data being presently acquired during Run II. Tensions between present data and Standard Model predictions have been found in some of these channels, hinting at a possible violation of Lepton Flavour Universality. I will review the status of these tensions, assessing with particular care the theoretical cleanness of the observables displaying such tensions. Then, I’ll discuss the possible explanations for such a pattern of anomalies both within and beyond the Standard Model. 
Tuesday 12 December 2017, 16:00 at IPHT, Salle Claude Itzykson, Bât. 774  IPHTHEP (Séminaire de physique des particules et de cosmologie)  hepph 



Abstract:  Over the years many different types of fits for the strong coupling constant have been performed. However one high precision result that currently significantly differs from the world average are results from event shapes at electron positron colliders. One possible source for the difference in these results could be the degeneracy between the fit of the strong coupling constant and nonperturbative parameter. In this talk I will explore the possibility to apply jet substructure techniques, specifically soft drop, in order to break the degeneracy between the nonperturbative parameter and the strong coupling constant. 
Tuesday 12 December 2017, 17:15 at DPTPHYSENS, Salle Jean Jaurès (29 rue d'Ulm)  SEMPHYSENS (Colloquium du Département de Physique de l'ENS)  quantph 



Abstract:  Recent de Broglie wave interference experiments with atoms have achieved wavepacket separations as large as 54 cm over time intervals of 2 sec [1, 2]. These experiments, and their impact on gravitational and quantum physics, will be discussed. [1] Kovachy, T. et al. Quantum superposition at the halfmetre scale. Nature 528, 530–533 (2015) [2] Asenbaum, P. et al. Phase Shift in an Atom Interferometer due to Spacetime Curvature across its Wave Function. Physical Review Letters 118, (2017) 
Wednesday 13 December 2017, 10:00 at
IHES,
Centre de conférences Marilyn et James Simons ( Séminaire de Géométrie Arithmétique ParisPékinTokyo )  MATHIHES (TBA)  hepth 



Abstract:  What motives are to algebraic varieties, exponential motives are to pairs (X, f) consisting of an algebraic variety over some field k and a regular function f on X. In characteristic zero, one is naturally led to define the de Rham and rapid decay cohomology of such pairs when dealing with numbers like the special values of the gamma function or the Euler constant gamma which are not expected to be periods in the usual sense. Over finite fields, the étale and rigid cohomology groups of (X, f) play a pivotal role in the study of exponential sums. Following ideas of Katz, Kontsevich, and Nori, we construct a Tannakian category of exponential motives when k is a subfield of the complex numbers. This allows one to attach to exponential periods a Galois group that conjecturally governs all algebraic relations among them. The category is equipped with a Hodge realisation functor with values in mixed Hodge modules over the affine line and, if k is a number field, with an étale realisation related to exponential sums. This is a joint work with Peter Jossen (ETH). 
Wednesday 13 December 2017, 11:30 at
IPN,
Bâtiment 100, Salle A015 ( $ $ )  IPNTHEO (Séminaire du groupe de physique théorique de l'IPN Orsay)  nuclth 



Abstract:  The reach of ab initio calculations has extended significantly over the past years. In particular systems revealing a closedshell character can be described efficiently up to the tin isotopic chain via controlled manybody expansions like coupledcluster the ory, the inmedium similarity renormalization group or selfconsistent Green func tion techniques. However, the transfer of such methods to openshell nuclei requires a highly nontrivial extension. In this talk we will discuss manybody perturbation theory (MBPT) as a lightweighted alternative and give a pedestrian introduction to MBPT based on a HartreeFock reference state. Loworder energy corrections for ground states of closedshell nuclei will be discussed using stateoftheart chiral interactions. The conceptual simplicity of MBPT allows for a direct generalization to openshell systems by using correlated reference states. First I will present a MBPT version us ing a multiconfigurational reference state arising from a prior nocore shellmodel calculation (NCSM). We present recent calculations of secondorder energy correc tions of spectra of even and odd carbon and oxygen isotopes and compare them to exact diagonalizations from largescale NCSM calculations. Additionally, we will provide first results on the dripline physics of the fluorine isotopic chain. In a complementary ansatz we use symmetrybroken reference states from a Hartree FockBogoliubov (HFB) calculation as starting point for the correlation expansion and reformulate MBPT in a quasiparticle setting yielding the socalled Bogoliubov MBPT (BMBPT). We present preliminary results for the oxygen chain and compare them to other stateoftheart manybody approaches and provide an outlook on future applications of symmetrybroken manybody approaches.  
Attachments: 
Wednesday 13 December 2017, 13:45 at LKB, Collège de France, salle 2  SEMPHYSENS (Colloquium du Département de Physique de l'ENS)  quantph 



Abstract:  Quantum measurement protocols based on dispersive cavityassisted interactions will be described. We will show how these protocols lead to performance improvements for precision atomic sensors and to new tests of quantum mechanics. We will also describe quantum imaging methods based on repeated coherent interactions in degenerate optical and electron cavities. 
Wednesday 13 December 2017, 14:00 at LPT, Amphi 1, bat 210, 2eme etage, LPT Orsay  SOUTENTH (Soutenance de thèse)  hepph 



Abstract:  Many particle systems may exhibit interesting properties depending on the interaction between their con stituents. Among them, it is possible to nd situations where highly ordered microscopic structures may emerge from these interactions. The central problem to identify the mechanisms which activate the ordered particle arrangements has been the subject matter of theoretical and experimental studies. In the past decades, it was rigorously proved that systems in two dimensions with suciently shortrange interactions and continuous degrees of freedom do not have longrange order. In contrast, numerical studies of systems featuring lack of positional order in two dimensions showed evidence of phase transitions. This apparent contradiction was explained by the KosterlitzThouless (KT)transition for the XY model showing that transitions may take place in positional isotropic bidimensional systems if they still have quasilong range (QLR) order. Such QLR order associated to the orientational order of the system, is lost when topological defects activated by thermal uctuations begin to unbind in pairs producing a transition. On the other hand, twodimensional systems with positional order at vanishing temperature may show a melting scenario including three phases solid/hexatic/ uid with transitions driven by a unbinding mechanism of topological defects according to the KosterlitzThoulessHalperinNelsonYoung (KTHNY)theory. This work is focused on the study of the two dimensional one component plasma 2dOCP a system of N identical punctual charges interacting with an electric potential in a twodimensional surface with neutralizing background. The system is a crystal at vanishing temperature and it melts at suciently high temperature. If the interaction potential is logarithmic, then the system on the at plane and the sphere is exactly solvable at a special temperature located at the uid phase. We use analytical approaches to compute exactly thermodynamic variables and structural properties which enables to study the crossover behaviour from a disordered phases to crystals for small systems nding interesting connections with the Ginibre Ensemble of the random matrix theory. We perform numerical Monte Carlo simulations of the 2dOCP with inverse power law interactions and periodic boundary conditions nding a hexatic phase for suciently large systems. It is found a weakly rst order transition for the hexatic/ uid transition by using nite size analysis and the multihistogram method. Finally, a statistical analysis of clusters of defects during melting conrms in a detailed way the predictions of the KTHNYtheory but also provides alternatives to detect transitions in twodimensional systems. 
Thursday 14 December 2017, 10:00 at IHP, 314  RENCTHEO (Rencontres Théoriciennes)  hepth 



Abstract:  I will first briefly review the mechanism of bubble nucleation as a possible instability channel for AdS vacua. Secondly I will discuss it in the specific context of effective supergravity models arising from string compactifications truncated to their closed string sector. By making use of positive energy theorems, I will show that such instabilities do not really occur within this setup. Finally, I will discuss the relation between the above analysis and the weak gravity conjecture (WGC) and will eventually argue that some expected universal instabilities for nonsupersymmetric AdS are crucially linked to the coupling between open and closed string sector. 
Thursday 14 December 2017, 11:00 at LPTHE, Bibliothèque  SEMLPTHE (Séminaire du LPTHE)  condmat.statmech 



Abstract:  Dynamical symmetry breaking (DSB) is a well known phenomenon in several areas of physics leading to a mass (gap) generation for fermions. The presence of an external magnetic field can be a strong catalyst for the DSB effect, leading to the generation of a dynamical mass for fermions even by very weak attractive interactions  the phenomenon called the magnetic catalysis. For example, in an external magnetic field, the critical value of the Nambu–JonaLasinio coupling for the dynamical fermion mass generation is reduced to zero. The effect is primarily due to the dimensional reduction in the dynamics of fermion pairing in a magnetic field. The lowest Landau level plays the crucial role in catalyzing spontaneous symmetry breaking, its role is similar to that of the Fermi surface in the BCS theory of superconductivity. We consider the phenomenon of magnetic catalysis in various models of quantum field theory with shortrange and longrange interactions. In application to graphene, magnetic catalysis can be used to explain the observation of quantum Hall plateaus in strong magnetic fields beyond the standard anomalous sequence of filling factors. 
Thursday 14 December 2017, 11:45 at IHP, 314  RENCTHEO (Rencontres Théoriciennes)  hepth 



Abstract:  Motivated by the physics of quarkgluon plasma, calculations of timedependent processes based on the AdS/CFT correspondence have lead to the notion that relativistic hydrodynamics may be useful for a much wider set of states than usually assumed. This is based on the observation of universal, attractorlike behaviour quite far from local equilibrium. I will describe recent attempts to understand such behaviour in the case of N=4 supersymmetric YangMills plasma. 
Thursday 14 December 2017, 14:00 at LPTM, 4.13 St Martin II  SEMLPTMUCP (Seminaires du LPTM , Universite de Cergy Pontoise)  mathph 



Abstract:  Mean field games were introduced in the mathematical and engineering community as a way to deal with optimization problems involving a large number of interacting agents. I will show there is a strong formal link between an important class of mean field games models and the nonlinear Schrödinger equation that allows for the implementation of effective approximation schemes to solve the mean field games equation. In particular, I will discuss the example of a simple (but nontrivial) model of population dynamics. 
Thursday 14 December 2017, 16:00 at LPT, 114  LPTPTH (Particle Theory Seminar of LPT Orsay)  hepph 



Abstract:  I present a Grand Unified Theory of Flavour, based on SO(10) together with a nonAbelian discrete group S4. The CSD2 flavon vacuum alignment is discussed. The Yukawa matrices are derived and neutrino masses emerge from the type I seesaw mechanism. A full numerical fit is performed with 15 input parameters generating 19 presently constrained observables. The model predicts a normal neutrino mass ordering with a CP oscillation phase of 260 degrees, an atmospheric angle in the first octant and neutrinoless double beta decay with m= 11 meV. I also discuss N2 leptogenesis, which fixes the second righthanded neutrino mass to be M2~ 2*10^11 GeV, in the natural range predicted by the model. 
Friday 15 December 2017, 10:00 at
IPHT,
Salle Claude Itzykson, Bât. 774 ( https://courses.ipht.cnrs.fr/?q=fr/node/194 )  COURS (Cours)  condmat 



Abstract:  The complex behavior of a large variety of systems can often be ascribed to the competition of many quasioptimal equilibria. In these cases metastability deeply affects both the structural and dynamical properties. Glasses are the prototype of such systems, with glassy behavior arising not only in condensed matter but also in a wide variety of fields ranging from optimization to computer science. \par In these lectures I will introduce different kinds of glassy systems and I will describe their fundamental similarities, which emerge from their statistical description. Then I will focus on a representative model, the random perceptron, and I will discuss the main techniques that can be used to solve it. \par These techniques provide a versatile theoretical toolbox that can be applied to several problems such as for example the physics of the jamming transition, its interpretation in the context of constraint satisfaction, as well as the design of new efficient algorithms to solve statistical problems. \\ 1.Introduction to glassy systems: structural glasses and constraint satisfaction problems. \\ 2.The random perceptron model. The replica approach. The replica symmetric solution and phase diagram. \\ 3.Replica symmetry breaking and marginal stability. The full RSB solution of the random perceptron. \\ 4.The jamming transition as a satisfiability threshold; critical exponents. \\ 5.Belief propagation approach and the algorithmic version of the ThoulessAndersonPalmer equations. \\ Supported by ``Investissements d'Avenir'' LabEx PALM (ANR10LABX0039PALM)  
Attachments: 
Monday 18 December 2017, 11:00 at IPHT, Salle Claude Itzykson, Bât. 774  IPHTPHM (Séminaire de physique mathématique)  mathph 



Abstract:  I will discuss the O(n) loop model on a random surface and its connection with Conformal Loop Ensemples (CLE) and Liouville Quantum Gravity (LQG). Last year Borot, Bouttier and Duplantier determined the statistics of nesting of loops in both models and found that their large deviation functions are related by the KnizhnikPolyakovZamolodchikov (KPZ) formula. In this talk I will give a probabilistic interpretation of this result by exhibiting a bijective relation between surfaces with loops and walks on the square lattice. The nesting statistics of the former are connected to the winding statistics of the latter. 
Monday 18 December 2017, 14:00 at IPHT, Salle Claude Itzykson, Bât. 774  IPHTSTA (Séminaire de Physique Statistique, CEA/Saclay)  condmat 


Tuesday 19 December 2017, 14:00 at LPTENS, LPTENS library  STRLPTENSHE (Séminaire commun LPTENS/LPTHE)  hepth 



Abstract:  TBA 
Tuesday 19 December 2017, 14:00 at APC, 483 A  Malevitch  APCTH (Seminar of the theory group of APC)  hepth 


Tuesday 19 December 2017, 14:30 at IHES, Amphithéâtre Léon Motchane  PTIHES (Séminaire de physique théorique de l'IHES)  hepth 



Abstract:  I will talk about loop infrared effects in de Sitter QFT. Namely about their types, physical meaning and origin and also about their resumation and physical consequences. The talk is based on arXiv:1701.07226. 
Wednesday 20 December 2017, 15:00 at IPHT, Salle Claude Itzykson, Bât. 774  IPHTSTA (Séminaire de Physique Statistique, CEA/Saclay)  condmat 



Abstract:  Exotic types of order that go beyond Landau's theory of spontaneous symmetry breaking continue to captivate the attention of condensed matter physicists. Using the language of spins often allows for construction of models that are analytically and/or numerically tractable, yet still highly relevant to experiments. In the recent years, much progress has been made in one and two dimensions, which are particularly wellsuited for a theoretical study. However, there are two indisputable reasons to go beyond 2D: firstly, most of the compounds surrounding us are, in fact, three dimensional. Second, but no less important, reason is that higher dimensions give rise to even richer and more exotic physical phenomena. One of the examples of exotic quantum order is the socalled topological phases. Originally proposed in two dimensions, topological phases have ground state degeneracies that depend solely on the topology that the system is embedded in. This topological degeneracy of the ground state is directly linked to the existence of gapped excitations, termed anyons, whose exchange statistics are strikingly different from the familiar bosons and fermions. Pointlike anyons can only exist in twodimensions, however the concept of topological order has been extended beyond 2D via two main avenues: (a) introducing extended (e.g., looplike) anyon excitations, and (b) fracton topological order. The latter phases host pointlike excitations that are immobile: strictly localized in space. In my talk, I will introduce a simple toy model with this type of order, starting from a layered construction of twodimensional toric codes. 
Thursday 21 December 2017, 11:00 at IPHT, Salle Claude Itzykson, Bât. 774  IPHTPHM (Séminaire de physique mathématique)  mathph 



Abstract:  I will present a new formalism, which takes as input a functor $E$ from a category of surfaces with their mapping classes as morphisms, to a category of topological vector spaces, together with glueing operations, as well as a small amount of initial data, and produces as output functorial assignments $S \mapsto \Omega_S$ in $E(S)$. This construction is done by summing over all excisions of homotopy class of pair of pants decompositions of $S$, and we call it ``geometric recursion''. The topological recursion of Eynard and Orantin appears as a projection of the geometric recursion when $E(S)$ is chosen to be the space of continuous functions over the Teichmuller space of $S$, valued in a Frobenius algebra  and the projection goes via integration over the moduli space. More generally, the geometric recursion aims at producing all kinds of mapping class group invariant quantities attached to surfaces. \\ \\ This is based on joint work with J.E. Andersen and N. Orantin. 
Thursday 4 January 2018, 16:00 at LPT, 114  LPTPTH (Particle Theory Seminar of LPT Orsay)  hepph 



Abstract:  A model suited for calculating correlation functions in QCD from the ultraviolet to the infrared is reviewed. The model consist in standard FaddeevPopov Lagrangian for Landau gauge with an extra mass term for gluons. It is shown that once this mass term is included, two and three point correlation functions can be calculated with good precision at oneloop order even at very low momenta in the quenched approximation. After that, the inclusion of quarks is analyzed. It is shown that in that case, the perturbative calculation only gives the qualitative behavior in some correlation functions. In particular, it is shown that the analysis of spontaneous chiral symmetry breaking requires to go beyond perturbation theory. A non perturbative scheme but controlled by two small parameters is discussed and the corresponding results are shown to agree with high precision to Monte Carlo numerical simulations also in the quark sector. 
Tuesday 9 January 2018, 14:00 at APC, 483 A  Malevitch  APCTH (Seminar of the theory group of APC)  hepth 



Abstract:  What is the holographic dual of an ordinary solid? Using insight from effective field theory (EFT), I will argue that an answer is provided by an SO(d) magnetic monopole in (d+1)dimensional AdS space. We call such field configuration “solidon”. The lowenergy spectrum of the boundary theory can be derived analytically from the gravity dual, and the result confirms that the effective theory consists of a set of phonons having dispersion relations that match those expected from EFT. Next we obtain some numerical solutions including backreaction and calculate the free energy, finding evidence that the solidon “melts” into a black hole as the temperature is raised, which we interpret as a solidtoliquid phase transition on the field theory side. 
Thursday 11 January 2018, 11:45 at IHP, 314  RENCTHEO (Rencontres Théoriciennes)  hepth 



Abstract:  We put forward the Axionic String Ansatz (ASA), which provides a unified description for the worldsheet dynamics of confining strings in pure YangMills theory both in $D=3$ and $D=4$ spacetime dimensions. The ASA is motivated by the excitation spectrum of long confining strings, as measured on a lattice, and by recently constructed integrable axionic noncritical string models. According to the ASA, pure gluodynamics in 3D is described by a noncritical bosonic string theory without any extra local worldsheet degrees of freedom. We argue that this assumption fixes the set of quantum numbers (spins, $P$ and $C$parities) of almost all glueball states. We confront the resulting predictions with the properties of approximately $1^2+2^2+3^2+5^2=39$ lightest glueball states measured on a lattice and find a good agreement. 
Friday 12 January 2018, 10:00 at
IPHT,
Salle Claude Itzykson, Bât. 774 ( https://courses.ipht.cnrs.fr/?q=fr/node/195 )  COURS (Cours)  math.AG 



Abstract:  Algebraic equations are widespread in mathematics and physics, and the geometry of their spaces of solutions can be complicated. In the case of an equation of two complex variables, the space of solutions is a Riemann surface. \par We will provide basic tools (going back to Riemann) for studying algebraic equations and describing the geometry of compact Riemann surfaces. \par We will consider a Riemann surface defined from the solution locus of a polynomial equation $P(x,y)=0$ in $\mathbb{C} \times \mathbb{C}$. We will study its topology and geometry, and learn how to integrate differential forms along closed contours. Then we will describe the moduli space of Riemann surfaces with a given topology: its dimension, topology, etc. \par We will introduce some of the many tools that have been invented since the time of Riemann for studying these objects. We will partly follow the Mumford Tata lectures, the Fay lectures, and the FarkasKra book. \\ \\ The plan is: \\  Compact Riemann surfaces, charts, atlas, toplogy. Meromorphic functions and oneforms. Theorems on poles and residues. Newton's polygon. \\  Integrals, periods, Abel map, Jacobian, divisors. Theta functions, prime form, fundamental form. Basis of cycles, homology and cohomology. \\  Moduli spaces of Riemann surfaces. DeligneMumford compactification, Chern classes, tautological ring. Kontsevich integral and KdV hierarchy. \\  If times permits: fiber bundles, Hitchin systems, link to integrable systems.  
Attachments: 

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