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Jaigyoung Choe (GAIA)

Title. Noncommutative Minimal Surfaces

Abstract. We define noncommutative minimal surfaces in an algebra with generators U,V satisfying [U ,V ]∼1 and give a method of construction by generalizing the well known Weierstrass representation formula. Then the noncommutative Enneper's surface, catenoid and helicoid will be constructed.

François Dahmani (I.F.)

Title. Isomorphisms problems for groups with hyperbolic geometries

Abstract. Already in the early study of combinatorial group theory, "explicit" computation problems in finitely presented groups were considered by Dehn, mainly as a test to know whether one understands a group by its presentation. As, as Hilbert's X-th problem on solving Diophantin equations, these problems turned into algorithmically decidability problems. Dehn considered the word problem the conjugacy problem (how to detect that two given elements are respectively equal, or conjugated) and,  more difficult, the isomorphism problem (how to detect that two given presentation define isomorphic groups). All these problems are algorithmically undecidable in the realm of finitely presented groups. From the perspective of geometric group theory, one can restrict to classes of groups with nice geometrical features, and then these problems become more tractable. In the case of Gromov hyperbolic groups, all three are solvable. In so-called relatively hyperbolic groups, in many cases, we can also solve these problems. Many ideas of geometry, dynamics, are involved in these developments.

J.P.Demailly (I.F.)

Title. Strong openness conjecture and generalized Nadel vanishing theorem (after Guan-Zhou, Hiep and Cao)

Abstract. This will be a survey talk covering recent results on the solution of the strong openness conjecture by Q.Guan-X.Zhou and Pham H.Hiep, and extensions of the Nadel vanishing theorem using techniques due to Ch.Mourougane, J.Cao and the lecturer. As a consequence, one gets optimal results of Nadel type for pseudoeffective line bundles over compact Kähler manifolds.

Stéphane Druel (I.F.)

Title. On Fano foliations

Abstract. In this talk I shall discuss codimension 1 Fano foliations on complex projective manifolds (these are foliations whose anti-canonical class is ample). I will concentrate on the special class of Mukai foliations. As I shall explain, such foliations have algebraic (and rationally connected leaves), except for some well understood foliations. This a joint work with Carolina Araujo.

Damien Gayet (I.F.)

Title.Universal components of random algebraic sets

Abstract. In this talk, I will explain that any compact algebraic hypersurface S of Rⁿ appears with a positive probability c_S as a component of a random algebraic hypersurface of high degree d, with c_S not depending on d. This is a joint work with Jean-Yves Welschinger.

Stéphane Guillermou (I.F.)

Title. Microlocal sheaf theory in symplectic geometry

Abstract. The microlocal sheaf theory of Kashiwara-Schapira associated with any sheaf on a manifold M its microsupport, which is a conic subset of the cotangent bundle T*M. This microsupport was known to be a coisotropic subset since the beginning of the theory. However it was only used recently by Tamarkin to study the symplectic geometry of the cotangent bundle. Building on Tamarkin's idea we can recover some well-known results of symplectic geometry using the microlocal sheaf theory (Arnold's non-displaceability and Gromov-Eliashberg theorem).

Kengo Hirachi (Univ. of Tokyo)

Title. Volume renormalization of complete Einstein-Kaehler manifolds and Burns-Epstein invariant

Abstract. Strictly pseudoconvex domain with complete Einstein-Kaehler metric has infinite volume.
We define its finite part (the renormalized volume) by considering the expansion of the volume of
subdomains which exhaust the domain.  For 2-dimensional domains, we show that the renormalized volume agrees with the Burns-Epstein invariant of the boundary.  For higher dimensions, we give
a variational formula of the renormalized volume under the deformation of domains in terms of the CR invariant of the boundary.

David Hyeon (GAIA)

Title. Generic state polytopes and stability

Abstract. Given an algebraic group G and a point v of its representation V, its state polytope with respect to a maximal torus T is the convex hull of the T-weights of v. State polytopes determine whether v is GIT semistable. We consider how the state polytopes change according to the choice of T, define the notion of generic state polytopes analogous to generic initial ideals, and show that when G is reductive, every point is semistable with respect to a general maximal torus T.

Dano Kim (GAIA)

Title. On the singularity of plurisubharmonic functions and their multiplier ideal sheaves

Abstract. A plurisubharmonic (psh) function plays an important role in the study of compact complex (projective) manifolds, appearing as a local weight function of a singular hermitian metric of a line bundle. In general, the singularity of a psh function f can be highly 'transcendental', but the multiplier ideal sheaves of positive multiples of f contain detailed algebraic information of the singularity, in particular the approximation of f by algebraic singularities.
 In the first part of the talk, we will discuss recent results concerning monotone decreasing subsequences of this approximation. In the second part, we will show that there can exist an infinite family of mutually inequivalent singular hermitian metrics of one line bundle on a smooth complex projective variety such that all the metrics have the same multiplier ideal sheaves for all their positive multiples.

Kang-Tae KIM (GAIA)

Title. Generalizations of Forelli's theorem

Abstract. The best known theorem in SCV concerning the   complex analyticity of a function must be Hartogs' analyticity theorem; Forelli's theorem may be the second.  The theorem says that any function that is indefinitely differentiable at a point is holomorphic in a neighborhood if the function is holomorphic along every complex line passing through the point mentioned above. This theorem, since it came out in 1977, was believed to be hard to be generalized (if not impossible) for a long time, until E. Chirka wrote a complex two dimensional generalization in Tr. Math. Inst. Steklova 2006.  Since then, together with Chirka's work, papers by Kim-Poletsky-Schmalz (JGA 2009), Joo-Kim-Schmalz (Math Ann 2013) showed that there are two directions for its generalizations. One remaining question in these line of research was whether the condition of analyticity along complex line can be replaced by the flow Riemann surfaces of a holomorphic vector field with resonance.  This has been resolved by Joo-Kim-Schmalz recently (2014) and I would like to report how it is done  in the presentation. The article is available in the ArXiv.1402.6390v1.

Greg McShane (I.F.)

Title. Geometric identities

Abstract. This talk is a survey of so-called geometric identities. In addition to the results mentioned for open surfaces below we will mention recent progress on work for closed surfaces by F. Luo and S.P. Tan.

Let S be an orientable surface of genus g with n boundary components. If 3g - 3 + n > 0 then S admits a Riemannian metric of constant curvature -1 a hyperbolic structure of finite area such that the boundary is totally geodesic. The space of all such hyperbolic structures is naturally a manifold of dimension 2 x (3g - 3 + n). For simplicity, we suppose that n=1 so that S has a single boundary component of length L. A geometric identity is a relation between the lengths of the closed simple geodesics on the surface S which holds for any choice of hyperbolic structure on S. The known geometric identities fall into 3 groups:

• Basmajian Identities

• McShane Identities

• Bridgeman Identitieq
  

We will explain briefly how each of these groups of identities is proven. The proofs follow from the existence of a decomposition of some geometric object related to the surface into two partsone of which is neglible and the other which further decomposes into pieces which can be classified and their ``size" computed. We will discuss applications of these identities in particular the symplectic volume of moduli space.We will discuss applications of these identities in particular the symplectic volume of moduli space.

Hervé Pajot (I.F.)

Title. Curvature estimates and the geometric traveling salesman problem in the Heisenberg group

Abstract. The geometric traveling salesman problem could be stated as follows: If (X,d) is a metric space, under which (quantitative) conditions is a compact subset E of X contained in a rectifiable curve ? In this talk, I will present some recent results related to this question in the case of the Heisenberg group and I will discuss some open questions. Most of the results are based on Alexandrov-type curvature estimates in the Heisenberg group.

Jinsung Park (GAIA)

Title. PSL(2,C)-Chern-Simons invariant for hyperbolic manifolds

Abstract. The PSL(2,C)-Chern-Simons invariant is a complex-valued invariant for Riemannian manifolds. In particular, this invariant is an important invariant for hyperbolic manifolds.  In this talk, I will talk on the PSL(2,C)-Chern-Simons invariant for some hyperbolic manifolds of infinite volume, whose conformal boundary is a Riemann surface of higher genus.  I will explain an equality of this invariant with  tau function and some related questions.

Pierre Will (I.F.)

Title. On discrete groups in complex hyperbolic geometry

Abstract. The complex hyperbolic plane is the simplest generalisation of the Poincaré disc in higner (complex) dimension. The goal of this talk is to present results concerning generalisations of Fuchsian and Kleinian groups in that frame. In particular, I will discuss discrete representations of free groups in PU(2,1), which is the isometry group of the complex hyperbolic plane.