TITLE AND ABSTRACT OF TALKS AT INDO-FRENCH 2016

JAN 11 - JAN 24, 201611 Jan 2016
1) Kumar Murty
Title : The self power map and splitting of primes .
Abstract In recent years, there has been some interest in understanding the image of the self power map n 7→ nn
modulo a prime. This map arises in some digital signature schemes. It is difficult to analyze partly because it is notgeometric. We will discuss some problems related to this map and its connection to the splitting of primes in certain
number fields.
2) Jean-Pierre Demailly
Title: Recent progress towards the Green-Griffiths-Lang and Kobayashi conjectures
Abstract The study of entire holomorphic curves contained in projective algebraic varieties is intimately related
to fascinating questions of geometry and number theory. The aim of the lectures is to present recent progress on thegeometric side of the problem.
The Green-Griffiths-Lang conjecture stipulates that for every projective variety X of general type over C, there
exists a proper algebraic subvariety of X containing all non constant entire curves f : C → X. Using the formalismof directed varieties, we will show that this assertion holds true in case X satisfies a strong general type condition
that is related to a certain jet-semistability property of the tangent bundle TX . It is then possible to exploit thisresult to investigate the long-standing conjecture of Kobayashi (1970), according to which every general algebraic
hypersurface of dimension n and degree at least 2n + 1 in the complex projective space Pn+1 is hyperbolic.
3) Kausal Verma.
Title: The boundaries of quadrature domains
Abstract Quadrature domains in the plane are those on which a given test class of holomorphic functions satisfy ageneralized mean value property. It is well known that planar quadrature domains have algebraic boundaries (modulo
possibly finitely many points). The purpose of this talk will be to outline an alternate way to see this.
4) Benoit Grbert
Title: On reducibility of quantum harmonic oscillator on Rd with quasi periodic in time potential
Abstract We prove that a linear d-dimensional Schrdinger equation on Rd with harmonic potential x2 and small
t-quasiperiodic potential,i∂t u = −∂x2 u + |x|2 u + V (tω, x)u,x ∈ Rd ,reduces to an autonomous system for most values of the frequency vector ω. As a consequence any solution of sucha linear PDE remains bounded in all Sobolev norms.
5) Parimala, R.
Title: Obstructions to embedding maximal tori in classical groups
Abstract Let k be a number field. We describe an explicit Brauer-Manin obstruction to embedding tori inclassical groups. This extends the results of Prasad-Rapinchuk to a more general setting and explains the known
counterexamples to the Hasse principle.
(joint work with Eva Bayer-Fluckiger and Tingyu Lee.)1

12 Jan 2016
1) R. Munshi.
Title: A new form of circle method
Abstract Using the Petersson formula we formulate a device to analytically detect the equation n=m. We compare
this with the standard delta method, and give an application in the context of L-functions.
2) Eknath Ghate
Title: Reductions of Galois representations of small slopes
Abstract The reduction of the local Galois representation attached to an ordinary (slope 0) form at a prime paway from the level is well known to be reducible. In this talk we shall survey what is known about the reduction
when the slope is small but positive.
We concentrate on the case of slope 1, which was treated essentially completely in recent joint work with S.
Bhattacharya and S. Rozensztajn. We show that the local reduction is mostly reducible, with its exact shapedepends on the congruence class of the weight mod p − 1. Moreover, we show that in each such class there is a further
congruence class of weights mod p where the local reduction is irreducible. We also distinguish between the so called
peu and trs ramifie cases in the relevant non-semisimple reducible cases.
The proof uses the compatibility between the p-adic and mod p Local Langlands Correspondences with respect
to the process of reduction. This reduces the computation to the automorphic side. We will try and explain most of
the key ingredients used in the proof in a self-contained way.
3) Philip Boalch
Title: Non-perturbative symplectic geometry
Abstract Just as a Lie group is a multiplicative version of a Lie algebra, there is a class of holomorphic symplectic
varieties which are precise multiplicative versions of simpler (”additive”) symplectic varieties. In this theory the roleof the exponential map is played by the Riemann-Hilbert map, and its extension to the irregular case. I’ll describe
some examples, and show how a new theory of multiplicative quiver varieties emerges.
4) Manoj Kummini
Title: On the cohomology of normalized blow-ups
Abstract Let (R, m) be a Noetherian three-dimensional Cohen-Macaulay analytically unramified ring, e.g., the
local ring at a Cohen-Macaulay point on a three-dimensional variety. Write X = Proj ⊕n∈N I n tn , where (·) denotes
integral closure. (If R is normal, then X is the normalization of the blow-up of Spec(R) along the (not necessarily reduced) subscheme Spec(R/I).) We prove some consequences of the vanishing of H2 (X, OX ), with regard toconjectures of Itoh (J. Algebra, 1992) and of Lipman (Math. Res. Lett., 1994).
−1
1Firstly, Xis Cohen-Macaulay. Secondly, if R is equicharacteristic (true in the geometric situation) then lengthR (H (X, IOX ))−lengthR

I2
II

is zero (if the extended Rees ring A := ⊕n∈Z I n tn is Cohen-Macaulay) or at least three (if A is not

Cohen-Macaulay). Thirdly, H2E (X, I m OX ) = 0 for all integers m, where E is the exceptional divisor in X. Finally,n , n ≥ 1 satisfy I
n = I I]n−1 for everyf
if additionally R is regular and X is pseudo-rational, then the adjoint ideals If
n ≥ 3.
5) Gautam Bharali
Title: The dynamics of holomorphic correspondences on compact Riemann surfaces
Abstract Owing to the theorem of M. de Franchis, a compact hyperbolic Riemann surface does not admit any
interesting iterative holomorphic dynamics. However, these surfaces are rich in holomorphic correspondences. The
iteration of such a correspondence gives rise to many questions – which have begun to be investigated fairly recently– that are analogous to those in the theory of iterations of rational maps on the 2-sphere. A part of this talk will be
devoted to defining the analogue of the Fatou set for correspondences.
Let F be a holomorphic correspondence on a compact Riemann surface X. Under certain conditions, F admits
a canonical measure, µF , that is invariant under F in a specific sense and has good ergodic properties. The work ofDinh and Sibony gives several ways of describing µF , in view of which the support of µF seems to be a good analogueof the Julia set in the case of correspondences. To this end, we show that the support of µF is disjoint from the2(analogue of) the Fatou set of F . The proof of this also reveals a dynamically meaningful condition under which theanalogues of the Fatou and the Julia set partition the underlying Riemann surface (which is simply definitional in
the case of rational maps on the 2-sphere).

313 Jan 2016

1) V. Lakshmibai.
Title: Free resolutions of some schubert singularies
Abstract We construct free resolutions of certain closed subvarieties of affine spaces (the socalled opposite big
cells of Grassmannians). Our class covers the determinantal varieties, whose resolutions were first constructed by
A. Lascoux (Adv. Math., 1978). Our approach uses the geometry of Schubert varieties. An interesting aspect of
our work is its connection to the computation of the cohomology of homogeneous bundles (that are not necessarily
completely reducible) on partial flag varieties. This work is jointly with M. Kummini, P.Sastry and C.S. Seshadri. We
shall also present results for certain subvarieties of opposite big cells of the Lagrangian Grassmannias (work jointlywith R. Hodges).
2) Marc Chaperon
Title: Invariant manifolds and semi-conjugacy.
Abstract: A simple and efficient approach to invariant manifolds of dynamical systems is presented, based upongenerating maps and avoiding any functional analysis. This yields (semi-)conjugacy theorems as well. In the end, ageometric view of KAM theory (in function space!) is suggested, to which the previous approach might apply.
3) S. Kesavan
Title: On a degenerate algebraic Riccati equation.
Abstract The existence of solutions to a degenerate algebraic Riccati equation associated to an optimal control
problem with infinite time horizon is studied. Under some assumptions on the control system, it is shown that it ispossible to select a solution to this equation which provides a feedback control law which will stabilize the system.
4) Pathasarathy Chakraborty
Title: An invariant for ergodic actions of compact groups
Abstract An invariant for ergodic actions of compact groups
Abstract: We will introduce a numerical invariant for an ergodic action of a compact (quantum) group on a unitalC ∗ algebra. We will discuss its computation and end with an open question for compact Lie groups. This is joint
work with Arup Pal of ISI, Delhi.
5) Parameswaran, A. J.
Title: On a Theorem of Deligne and the Slice Theorem
Abstract The aim of this talk is to prove a structure theorem for doubly saturated affine schemes of finite type in
GL(V ) over algebraically closed fields. This is built on recent results of Deligne and the results of Serre. We obtain
Lunas tale slice theorem in positive characteristics as an application. This is a Joint work with V. Balaji.
414 Jan 20161)Raphael Krikorian
Title: On almost reducibility in quasi-periodic dynamics.
Abstract I will discuss the following question: Is any smooth or analytic orientation preserving diffeomorphismof the circle f with an irrational rotation number α almost reducible in the sense that there exists a sequence ofsmooth or analytic conjugations gn such that gn−1 ◦ f ◦ gn converges in the smooth topology to x 7→ x + α? I will also
discuss the similar question for pseudo-rotations of the disk (orientation and area preserving diffeomorphisms of the
2-disk fixing the origin and the boundary without fixed points).
2) Riddhi Shah.
Title: The embedding problem of infinitely divisible probability measures on Lie groups
Abstract We discuss infinitely divisible measure on Lie groups. We discuss the conditions under which an infinitely
divisible measure can be embedded in a continuous one-parameter convolution semigroup of probability measures.
We survey known results and discuss the latest results obtained (jointly with Dani and Guivarc’h). We also mention
some techniques developed by Dani and McCrudden which gives the embeddability in some new cases.
3) Mathieu Lewin.
Title: Bose-Einstein condensation: the physical phenomenon and its mathematical proof
Abstract: In 1924, the Indian physicist Satyendranath Bose discovered that, at very low temperature, somequantum particles can suddenly all behave in the same way. After the idea was developped by Albert Einstein, thisnew state of matter was called a Bose-Einstein condensate. This theoretical prediction was confirmed by experiments
many years later in 1995.
In this talk I will present the mathematical formalism of Bose-Einstein condensation, and then review recent works
obtained in collaboration with Phan Thanh Nam (Vienna) and Nicolas Rougerie (Grenoble). Of importance is an
abstract theorem about the structure of the convex set of bosonic quantum states, which says that infinitely manyexchangeable particles have to be somewhat independent.
4)B V Rajarama Bhat
Title: Tensor product systems of Hilbert spaces
Abstract One parameter semigroups of *- endomorphisms of the algebra of all bounded operators on a Hilbert
space are known as E-semigroups. They are classified up to cocycle conjugacy by tensor product systems of Hilbert
spaces. We give a brief account of this theory.
5) Jeremy Faupin
Title: On spectral and scattering theories of non-relativistic QED
Abstract In this talk I will present a mathematical model commonly employed to describe an atom interactingwith photons, sometimes called the standard model of non-relativistic QED. This model goes back to the early days of
Quantum Mechanics and was elaborated in details by Pauli and Fierz in 1938. The energy of the total physical system(atom + photons) is associated with a self-adjoint operator an Hamiltonian acting on a suitable Hilbert space. The
Hamiltonian generates the dynamics of the states of the system. To understand phenomena such as relaxation of theatom to its ground state by emission of photons, one is led to analyse the spectrum of the Hamiltonian and study
scattering theory. I will review some of the main results of the theory and discuss a few open problems.
5

15 Jan 20161) T. N. Venkataramana
Title: Monodromy and Arithmetic Groups
Abstract A quick overview of the Deligne-Mostow theory will be presented. This will lead to questions onarithmeticity of monodromy groups associated to families of cyclic coverings of the projective line. We discuss a
result to the effect that the monodromy group is arithmetic if the number of branch points of the cyclic cover isgreater than twice the degree of the cover.
2) Michel Waldschmidt
Title: Two invariants related to two conjectures due to Nagata.
Abstract: Seshadri’s constant is related to a conjecture due to Nagata. Another conjecture, also due to Nagata
and solved by Bombieri in 1970, is related with algebraic values of meromorphic functions. The main part of Bombieri’s
proof leads to a Schwarz Lemma in several variables, the proof of which gives rise to another invariant associated withsymbolic powers of the ideal of functions vanishing on a finite set of points. This invariant is an asymptotic measureof the least degree of a polynomial in several variables with given order of vanishing on a finite set of points. Recent
works on the resurgence of ideals of points and the containment problem compare powers and symbolic powers of
ideals.
3) J. Sengupta
Title: Determination of modular forms from twisted central L Values.
Abstract:This will be a survey style talk on the topic mentioned in the title.
4) Anne-Marie Aubert
Title: The Harish-Chandra philosophy of cusp forms adapted to enhanced Langlands parameters
Abstract It is joint work with Ahmed Moussaoui and Maarten Solleveld. Let G be the group of F -rational pointsof a connected reductive algebraic group over a local non-archimedean field F . Let WF be the Weil group of F , letG∨ be the complex dual group of G, and let Z(G∨ )WF denote the fixed points by WF of the center of G∨ . Let G∨
sc∨∨
and G∨ad be the simply connected cover of the derived group of G , and the adjoint group of G , respectively. We
denote by ZG∨ (ϕ) the centralizer in G∨ of ϕ(WF × SL2 (C)), and by Z1G∨(ϕ) the inverse image of ZG∨ (ϕ)/Z(G∨ )WF
sc∨∨(ϕ) = ZG∨
(ϕ).
under the isogeny Gsc → Gad . If G is an inner form of a split group, then we have Z1G∨
sc
sc
Enhanced Langlands parameters for G are pairs (ϕ, ρ) formed by a relevant Langlands parameter ϕ : WF ×
SL2 (C) → G∨ o WF enhanced by an irreducible complex representation ρ of the group of components of Z1G∨(ϕ).
sc
We will introduce a notion of cuspidality for enhanced L-parameters for G, which conjecturally puts the supercuspidal irreducible representations of G in bijection with such enhanced L-parameters. We will also define a cuspidal
support map for enhanced L-parameters, and a decomposition of the set of them into Galois analogues of the Bernstein
components of the set Irr(G) of irreducible smooth representations of G.
It will reveal a new structure in the space of enhanced L-parameters for G, that of a disjoint union of twisted
extended quotients. This is an analogue of the ABPS conjecture (about Irr(G)) on the Galois side of the local
Langlands correspondence. Only, on the Galois side it is no longer conjectural. These results will be useful to reduce
the problem of finding a local Langlands correspondence for G-representations to the corresponding problem forsupercuspidal representations of Levi subgroups of G.
The main machinery behind this comes from perverse sheaves on algebraic groups. The extension of Lusztig’s
generalized Springer correspondence to disconnected complex reductive groups H provides a bijection between, on the
one hand, pairs consisting of a unipotent element u in H and an irreducible representation of the component groupof the centralizer of u in H, and, on the other hand, irreducible representations of a set of twisted group algebras ofcertain finite groups. Each of these twisted group algebras contains the group algebra of a Weyl group, which comesfrom the neutral component of H. The link with a given Langlands parameter ϕ is obtained by taking for H the
1 1centralizer in G∨
sc of ϕ(WF ), and for u the image by ϕ of (1, ( 0 1 )).
5) Shiva Shankar
Title: The Central Definition of Control Theory
6

Abstract I shall explain the notion of a ’controllable (dynamical) system’ due to R.E.Kalman, and its post-modern
generalisation due to J.C.Willems.

7

16 Jan 2016

1) Pierre Colmez
Title: The p-adic Langlands program
Abstract Following Wiles’s proof of Fermat’s last theorem, a p-adic version of the Langlands program has emerged.
I will give an introduction to this circle of ideas with an emphasis on the group GL2 .
2) Xavier Viennot.
Title: At the crossroad of algebra, combinatorics and physics: the 2-species PASEP
Abstract The PASEP (partially asymmetric exclusion model) is a model of particles hopping on a one-dimensional
strip of N sites with open boundaries. It has been intensively studied by physicists and appears as a toy model inthe physics of dynamical systems. The distribution of stationary probabilities has been computed using the socalled matrix ansatz (Derrida, Evans, Hakim, Pasquier). The solution involves a certain quadratic algebra and a
representation of this algebra with infinite tridiagonal matrices related to the Askey-Wilson polynomials. Thesepolynomials are a famous famous family of orthogonal polynomials which sit at the top of the hierarchy of classicalorthogonal polynomials.
In recent years intensive works has been done by combinatorists in order to give combinatorial interpretations ofthese stationary probabilities in terms of some weighted tableaux: permutations tableaux (Postnikkov, Steingrimsson,
Williams, Corteel), alternative tableaux (X.V.), tree-like tableaux (Aval, Boussicault, Nadeau), straircase tableaux(Corteel, Williams). The combinatorics of permutations, Catalan numbers and orthogonal polynomials plays a crucialrole.
The PASEP model has been extended in physics with two types of particles. In this talk we introduce rhombic
alternative tableaux which give combinatorial interpretation of the stationary probabilities. These tableaux definedfrom a tiling of some portions of the triangular lattice, are enumerated by the Lah numbers which are counting
certain assembles of permutations. Their weighted generating function give a combinatorial interpretation of the 2species PASEP (with 3 parameters). We describe a bijection between rhombic alternative tableaux and assembles of
permutations and get a bijective proof of the formula giving the partition function of the model. The general 2-species
model with 5 parameters is related to Koorwinder-Macdonald polynomials (Corteel, Mandelshtam, Williams). (Thisis a joint work with Olya Mandelshtam, Berkeley).
3) Arnaldo Nogueira.
Title: Piecewise contractions and its applications
Abstract Our talk is concerned with the study of the dynamical systems which are defined piece- wise contraction
maps (PC maps). Certain mathematical models, like contracting outer billiards and switched flow systems, aredescribed by PC maps. In the setup of the models usually the PC maps are defined on a convex set and it is expectedthat most orbits of a typical map be eventually periodic. Our aim is to present results about the asymptoticalbehavior of a typical map. Our approach is the following: We fix an Iterated Function System {ϕ1 , . . . , ϕn } where
the maps are Lipschitz contractions defined in a (convex) set D. Every partition of the set D into n convex subsets,{P1 , . . . , Pn }, defines a piecewise contraction map f : DD, in the natural way: f (x) = ϕi (x) for every x ∈ Pi . We willshow that in the one-dimensional case a typical PC map, in the metrical sense of the parameter set, is asymptoticallyperiodic and has at most n periodic orbits. A result for the multi-dimensional case will also be discussed. Joint workwith Benito Pires and Rafael Rosales.
818 Jan 2016

1) J.-L. Colliot-Thélène
Title: Integral points and Brauer-Manin obstruction : a survey
Abstract One extension of the Chinese remainder theorem is strong approximation for simply connected algebraic
groups. For groups and more generally varieties which are not geometrically simply connected, strong approximation
fails. The talk will describe classes of varieties for which this failure may be accounted for by the integral version of
the Brauer-Manin obstruction, a classical tool for the study of rational points.
2) V. S. Sunder
Title: From graphs to free products
Abstract We investigate a construction which associates a finite von Neumann algebra M (Γ, µ) to a finite weighted
graph (Γ, µ). The simplest examples - and in fact, the ‘building blocks’ for the most general case - yield natural
Fock-type models of operators with Wigner’s semi-circular distribution, circular operators and operators with free
Poisson distributions. Pleasantly, but not surprisingly, the von Neumann algebra associated to to a ‘flower with npetals’ is the von Neumann algebra LFn := λ(Fn )00 (generated by the left-regular representation) of the free groupon n generators.
In general, the algebra M (Γ, µ) is a free product, with amalgamation over a finite-dimensional abelian subalgebra
corresponding to the vertex set, over algebras associated to subgraphs ‘with one edge’ (actually a pair of dual edges).
3) Sujatha, R
Title: Derived birational invariants and unramified cohomology.
Abstract: We introduce some new birational invariants arising from unramified cohomology studied in the context
of Voevodsky’s theory in a derived category setting. This is joint work with Bruno Kahn.
4) Sukumar Adhikari
Title: A classical Ramsey-type result of Schur.
Abstract: The classical Ramsey-type result of Schur completes its hundred years of existence this year. It has
seen generalizations in several ways. We shall discuss some of these generalizations, related open questions and somerecent results.
5) Wieslawa Niziol
Title : Syntomic complexes and p-adic nearby cycles.
Abstract For a semistable scheme over a mixed characteristic local ring I will present a proof of a comparison
isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where
no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field.
I will also explain how to combine this local comparison isomorphism with the theory of finite dimensional BanachSpaces and finitness of étale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture
for formal schemes with semistable reduction. This is a joint work with Pierre Colmez.
919 Jan 2016

1) C.S. Rajan
Title: A Torelli type theorem involving base changes for elliptic surfaces
Abstract Given two elliptic surfaces over a base curve C, we show that any compatible family of effective isometriesof the Néron-Severi lattices of the base changed elliptic surfaces for all finite maps B → C arises from an isomorphismof the elliptic surfaces upto a finite base change. Without the effectivity hypothesis, we can conclude that the two
elliptic fibrations are isomorphic (joint work with S. Subramanian).
2) Michel Brion
Title: Commutative algebraic groups up to isogeny.
Abstract The commutative algebraic groups over a field k form an abelian category C, which has been studied
by Serre and Oort when k is algebraically closed. In particular, they showed that the homological dimension of C
is 1 in characteristic 0, and 2 in positive characteristics. In the talk, we will discuss the category of commutative
algebraic groups up to isogeny, which turns out to be much simpler. In particular, its homological dimension is 1
over an arbitrary field.
3) Amalendu Krishna
Title: Zero cycles on affine varieties.
Abstract The aim of this talk is to study the Affine Roitman torsion problem for 0-cycles on singular varieties.
We show that the affine Roitman torsion problem has positive solution for all affine varieties over an algebraicallyclosed field. We shall describe some applications of this solution. In particular, we shall show that an old open
question of Murthy has a positive solution.
4) Colin Vincent
Title: Towards higher-dimensional Heegaard-Floer homology.
Abstract: We associate to a Weinstein domain and to a symplectomorphism fixing pointwise its boundary a
Heegaard-Floer type homology. I will discuss invariance properties and relations with the contact homology of thecontact structure supported by the corresponding open book. This is joint work with Ko Honda.
5) M Krishna
Title: Eigenvalue statistcs for random operators
Abstract: We present the results on eigenvalue statistics for random Schrodinger operators and the Andersonmodel done by the group at IMSc.
10

20 Jan 2016

1) Bruno Khan.
Title: Motives with modulus
Abstract I will report on joint work with Shuji Saito and Takao Yamazaki, in which we construct and study atriangulated category of motives with modulus that extends Voevodsky’s category in such a way as to encompass
non-homotopy invariant phenomena.
2) V.R. S. Varadhan
Title: Large Deviations for Brownian local times. A second look.
Abstract Since Brownian motion is not positive recurrent, the large deviation behavior of its local time, or
occupation measure is not coercive, unless the space of probability measures on Rd is compactified. For many
problems this needs to be done in a translation invariant manner. This is an attempt to do that.
3) B. Ramakrishnan
Title: Theory of newforms of half-integral weight.
Abstract:In this lecture, we present a survey of our collaborative works on the theory of newforms of half-integral
weight and the extension of the Kohnen plus space to even levels. We also report on the extension of the theory of
newforms for the Eisenstein series space obtained by our collaborator.
11

21 Jan 2016

1) Mahan, MJ
Title: Limits of limit sets
Abstract Let Gn be a sequence of Kleinian groups converging to G. We deal with the dependence of the convergence of the corresponding Cannon-Thurston (CT) maps on the nature of convergence of the Kleinian groups. Moreprecisely, we shall deal with the following two questions:
1) If Gn converges to G strongly, do the corresponding CT maps converge uniformly?
2) If Gn converges to G algebraically, do the corresponding CT maps converge pointwise?
This is joint work with Caroline Series.
2) Oilvier Ramaré
Title: The Brun-Titchmarsh inequality
Abstract A given interval length N being fixed, the maximal number of primes ρ(N ) that are in an interval of thislength may contain is a major quantity already investigated by Hardy and Littlewood in 1922. The Brun-Titchmarsh
inequality in the form given by Montgomery and Vaughan in 1973 provides the upper bound 2N/log(N ) for ρ(N ); a first unsolved question is to decide whether this factor 2 is tight or not. A similar bound holds for primes inarithmetic progressions and breaking the factor 2 is known to be equivalent to showing that no Siegel zero exists, or
equivalently that enough small primes split in some quadratic extensions.
This talk will present a history of the problem and conclude with a new result. By using a functional inequality that mixes the Selberg sieve and the Large sieve, we prove, together with Soroosh Yazdani, that ρ(N ) ≤2N/(log(xN + 5.66 + o(1)). The method gives ground to the conjecture that the constant 5.66 can be replaced byan arbitrary large constant.
3) Uma, V.
Title: Equivariant K-theory of regular compactifications: further de- velopments.
Abstract: In this talk we shall describe the G̃ × G̃-equivariant K-ring of X, where G̃ is a factorial cover of aconnected complex reductive algebraic group G, and X is a regular compactification of G. Further- more, using thedescription of KG̃×G̃ (X), we describe the ordinary K- ring K(X) as a free module of rank the cardinality of the Weyl+group, over the K−ring of a toric bundle over G/B, with fibre the toric variety T , associated to a smooth subdivisionof the positive Weyl chamber. This generalizes our previous work on the wonderful compactification. Further, we givean explicit presentation of KG̃×G̃ (X) as well as K(X) as an algebra over the KG̃×G̃ (Gad ) and K(Gad ) respectively,
where Gad is the wonderful compactification of the adjoint semisimple group Gad . Moreover, in the case when X
is a regular compactification of Gad , we give a geometric interpretation of the above presentations in terms of theequivariant and ordinary Grothendieck ring of a canonical toric bundle over Gad ).
4) Laurent Manivel
Title: Hyperkahler varieties as parameter spaces
Abstract Hyperkahler varieties are an important class of complex varieties, with a rich and beautiful general
theory but only few examples, in particular compact ones. I will discuss the possibility to construct hyperkahler
varieties as parameter spaces for cycles on varieties with special Hodge structures. In particular I will explain such a
construction in connection with the exceptional complex Lie groups (joint work with Atanas Iliev).
5) C. R. E. Raja
Title: Groups with expansive automorphisms
Abstract Structure of totally disconnected locally compact groups that admit expansive automorphisms would
be presented. This is a joint work with H. Glockner of Universitat Paderborn.
12

22 Jan 201618) Gilles Pisier
Title: On the non-commutative Khintchine inequalities
Abstract This is joint work with Éric Ricard. We give a proof of the Khintchine inequalities in non-commutative
Lp -spaces for all 0 < p < 1. This case remained open since the first proof given by Françoise Lust-Piquard in 1986
for 1 < p < ∞. These inequalities are valid for the Rademacher functions or Gaussian random variables, but also for
more general sequences, e.g. for lacunary Fourier series or the analogues of Gaussian variables in free probability. In
the most classical setting, the non-commutative Khintchine inequalities deal with Rademacher series of the formXS=
rk (t)xk
kwhere (rk ) are the Rademacher functions on the Lebesgue interval where the coefficients xk are in the Schatten qclass or in a non-commutative Lq -space associated to a semifinite trace τ . Let us denote simply by k.kq the norm (orquasi-norm) in the latter Banach (or quasi-Banach) space, that we will denote by Lq (τ ). When τ is the usual traceon B(`2 ), we recover the Schatten q-class. By the non-commutative Khintchine inequalities, we mean the following:
For any 0 < q < ∞ there are positive constants αq , βq such that for any finite set (x1 , . . . , xn ) in Lq (τ ) we haveZ1/q(βq )−1 |||(xk )|||q ≤kS(t)kqq dt≤ αq |||(xk )|||q
where |||(xk )|||q is defined as follows:
If 2 ≤ q < ∞
(
(0.1)

def

|||(xk )|||q = max

Xx∗k xk

1
2

,

X

xk x∗k

)

1

2

q

q

and if 0 < q ≤ 2:
(
(0.2)

def

|||x|||q =inf

xk =ak +bk

Xa∗k ak

1

2

+

X

bk b∗k

)

1

q

2

.
q

Note that βq = 1 if q ≥ 2, while αq = 1 if q ≤ 2 and the corresponding one sided bounds are easy. The difficulty is
to verify the other side. We do this by “extrapolating” the case 1 ≤ q < 2.
2) Arul Shankar
Title: Polynomials with squarefree discriminant
Abstract A basic question of analytic number theory is that of understanding the density of squarefree values
taken by an integer polynomial. We will study this question for a special class of integer polynomials, namely,
discriminants of polynomials in one variable. This is joint work with Manjul Bhargava and Xiaoheng Wang.
3) Emmanuel Giroux.
Title: Existence of Lefschetz fibrations on Stein and Weinstein domains
Abstract In this talk, I will present our joint work with John Pardon showing that every Stein or Weinstein
domain may be realized (up to deformation) as a Lefschetz fibration over the disk. The proof is an application of
Donaldson’s quantitative transversality techniques.
4) Ravi Raghunathan
Title: Integral representations of L-functions and Hecke operators.
Abstract We investigate the holomorphy of automorphic L-functions by exploiting relations between period
functions given by Hecke operators.
5) K.M.Tamizhmani
Title: Singularities, Algebraic entropy and Integrability of discrete Systems
Abstract:In this talk, we discuss about two integrability detectors, namely, singularity confinement approach andalgebraic entropy method. for discrete systems. The power of these methods are demonstrated by deriving discrete
Painleve equations and linearizable systems.
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23 Jan 20161) S.G. Dani
Title: Surjectivity of the exponential map
Abstract The question of ”exponentiality” (viz. surjectivity of the exponential map) has attracted considerable
attention in literature. One of the pending issues is to relate the exponentiality of a Lie group with the exponentialityof its radical. In this talk I shall discuss the general question of exponentiality and describe certain conditions underwhich such an implication holds.
2) Sinnou David
Title: On Lehmer Problem
Abstract We shall discuss the various aspects of Lehmer’s problem from the classical statement by Lehmer tothe more recent group variety set up.

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Laurent Lafforgue
Title : Syntactic categories for Nori motives.
Abstract : The purpose of the talk will be to report about a preprint (http://arxiv.org/abs/1506.06113) signedwith Luca Barbieri-Viale and Olivia Caramello. All the results are due to Olivia Caramello, starting from a question
raised by Luca Barbieri-Viale after I had talked to him about the theory of ”classifying toposes” which I had learnt
from her. This preprint gives a completely new interpretation and generalisation of Madhav Nori’s constructionin the theory of motives. Let’s recall Nori’s construction had associated a universal Q-linear abelian category of”motives” to any homological or cohomological functor with values in finite dimensional Q-linear vector spaces. Nori
himself had applied his general categorical construction to the Betti homology functor to define an actual categoryof motives for schemes over a field of characteristic 0. Caramello’s construction shows Nori’s category is equivalent
to the so-called ”syntactic category” of a first-order ”regular” theory (in the sense of logic) associated to such ahomological or cohomological functor. It does’nt need anymore the finite-dimensions hypothesis so that it associatesa universal Q-linear abelian category of motives to any homological or cohomological functor with coefficients in a fieldof characteristic 0, such as l-adic or p-adic cohomology. In particular, it works for schemes over an arbitrary field ofarbitrary characteristic. Furthermore, the universal Q-linear abelian categories associated to different cohomological
functors are proved to be equivalent if and only if the different first-order ”regular” theories associated to thesefunctors are the same. As a consequence, the existence of a category of motives in the classical sense is equivalent to
the propery that the classical cohomological functors all define the same ”regular” theory. In other words, if motives
actually exist, they can be considered as objects in logic.

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