# Schedule for: 16w5080 - Complex Analysis and Complex Geometry

Arriving in Banff, Alberta on Sunday, May 1 and departing Friday May 6, 2016

Sunday, May 1 | |
---|---|

16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

20:00 - 22:00 | Informal gathering (Corbett Hall Lounge (CH 2110)) |

Monday, May 2 | |
---|---|

07:00 - 08:15 | Breakfast (Vistas Dining Room) |

08:00 - 08:15 | Introduction and Welcome by BIRS Station Manager (TCPL 201) |

08:15 - 09:00 |
Barbara Drinovec Drnovšek: Minimal hulls and minimally convex domains ↓ Minimal hulls and minimally convex domains were introduced in a series
of papers by Harvey and Lawson.
They are natural substitutes for polynomial hulls and strictly
pseudoconvex domains in the context of minimal surface theory.
We present a characterization of the minimal hull of a compact set $K$ in
$\mathbb R^n$ by sequences of conformal minimal discs
whose boundaries converge to $K$ in the measure theoretic sense. We also
study some properties of minimally convex domains.
This is a report on a joint work with Alarc\'on, Forstneri\v c and L\'opez. (TCPL 201) |

09:15 - 10:00 |
Richard Larkang: Chern classes of singular metrics on vector bundles ↓ For holomorphic line bundles, it has turned out to be useful to not just consider smooth metrics, but also
singular metrics which are not necessarily smooth, and which can degenerate. In relation to vanishing theorems and
other properties of the line bundle, one considers plurisubharmonicity properties of the possibly singular metric which
correspond to notions of positivity for the line bundle. In particular, having a positive singular metric means that the
first Chern form associated to the metric is a closed positive \((1,1)\)-current.
More recently, singular metrics on holomorphic vector bundles have been considered, Griffiths positivity of a singular
metric on a vector bundle is defined in terms of plurisubharmonicity. For a vector bundle with a Griffiths positive singular
metric, there is a naturally defined first Chern class which is a closed positive \((1,1)\)-current, but there are examples
where the full curvature matrix is not of order 0. I will discuss joint work with Hossein Raufi, Jean Ruppenthal and
Martin Sera, where we show that one can give a natural meaning to the k:th Chern form \(c_k(h)\) of a singular Griffiths
positive metric h as a closed \((k,k)\)-current of order 0, as long as h is non-degenerate outside a subvariety of codimension
at least \(k\). The proof builds on pluripotential theory, and in particular, one consider in the spirit of Bedford-Taylor products
like \((dd^c \varphi)^q \wedge T\), where \(\varphi\) is plurisubharmonic and \(T\) is a closed positive
\( (q,q) \)-current. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:15 |
Ahmed Zeriahi: Weak solutions to degenerate complex Monge-Ampere flows ↓ Studying the (long-term) behaviour of the K\"ahler-Ricci flow on mildly singular varieties, one is naturally lead to study
weak solutions of "degenerate parabolic complex Monge-Amp\`ere equations".
The purpose of this work, is to develop a viscosity theory for degenerate complex Monge-Amp\`ere flows on compact K\"ahler manifolds.
The main ingredient is the "parabolic comparison principle" which allows us to prove uniqueness of the solution to a general complex
Monge-Amp\`ere flow starting from a singular metric with bounded potentials in a given K\"ahler class.
Then using our previous results on the "degenerate elliptic side" of the complex Monge-Amp\`ere theory, we are able to construct barriers that allow us to prove the existence of a solution to the Cauchy problem by means of the classical method of Perron.
Our general theory allows in particular to define and study the behaviour of the (normalized) K\"ahler-Ricci flow on projective varieties with canonical singularities, generalizing results of Song and Tian. In the case when the variety is Calabi-Yau or of general type, we prove that the
K\"ahler-Ricci flow converges weakly in the sense of currents (strongly at the level of potentials) to the unique K\"ahler-Einstein metric on the the variety.
The case of intermediate Kodaira dimension is more tricky and will be briefly sketched if time permits. This is a joint work with P. Eyssidieux and V. Guedj which will appear in Advances in Math. (TCPL 201) |

11:30 - 12:15 |
Leandro Arosio: Models for holomorphic self-maps of the unit ball ↓ In order to study the forward or backward iteration of a holomorphic
self-map \(f\) of a complex manifold \(X\), it is natural
to search for a semi-conjugacy of \(f\) with some automorphism of a complex
manifold. Examples of this approach are given by the Schroeder, Valiron
and Abel equation in the unit disc \(D\).
Given a holomorphic self-map \(f\) of the ball \(B^q\), we show that it is
canonically semi-conjugate to an automorphism (called a canonical model)
of a possibly lower dimensional ball \(B^k\), and this semi-conjugacy
satisfies a universal property. This approach unifies in a common
framework recent works of Bracci, Gentili, Poggi-Corradini, Ostapyuk.
This is done performing a time-dependent conjugacy of the autonomous
dynamical system defined by \(f\), obtaining in this way a non-autonomous
dynamical system admitting a relatively compact forward (resp. backward)
orbit, and then proving the existence of a natural complex structure on a
suitable quotient of the direct limit (resp. subset of the inverse
limit). As a corollary we prove the existence of a holomorphic solution
with values in the upper half-plane of the Valiron equation for a
hyperbolic holomorphic self-map of \(B^q\). (TCPL 201) |

12:15 - 13:30 | Lunch (Vistas Dining Room) |

13:00 - 14:00 |
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) |

14:50 - 15:15 | Coffee Break (TCPL Foyer) |

15:15 - 16:00 |
Debraj Chakrabarti: L2-cohomology of annuli and Sobolev estimates for the ∂-bar problem ↓ We consider the question of $L^2$-estimates for the $\overline{\partial}$-problem on annuli,
a simple but interesting class of non-pseudoconvex domains. We relate this question with $W^1$-Sobolev
estimates on the "hole" of the annulus. We then consider special classes of non-smooth holes for which
the questions can be answered. This is joint work with Mei-Chi Shaw and Christine Laurent-Thiébaut. (TCPL 201) |

16:15 - 17:00 |
Turgay Bayraktar: Universality principles for random polynomials ↓ In this talk, I will present several universality principles concerned with zero distribution of random polynomials or more generally random holomorphic sections of high powers $L^{\otimes n}$ of positive line bundle $L\to X$ defined over a projective manifold endowed with a continuous metric. In one direction, universality phenomenon indicates that under natural assumptions, asymptotic distribution of (appropriately normalized) zeros of random polynomials is independent of the choice of probability law defined on random polynomials. Another form of universality is asymptotic normality of smooth linear statistics of zero currents. Finally, if time permits, I will also describe some recent results on universality of scaling limits of correlations between simultaneous zeros of random polynomials. (TCPL 201) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

Tuesday, May 3 | |
---|---|

07:00 - 08:15 | Breakfast (Vistas Dining Room) |

08:15 - 09:00 |
Andrew Zimmer: Characterizing domains by their automorphism group ↓ It is generally believed that (up to biholomorphism) very few domains have a large automorphism group and a nice boundary. For instance the Wong-Rosay Ball theorem says that a strongly pseudoconvex domain with non-compact automorphism group must be bi-holomorphic to the ball. Later, Bedford and Pinchuk proved that a convex domain of finite type and non-compact automorphism group must be bi-holomorphic to a domain defined by a polynomial. I will discuss a recent result which removes the finite type condition from the Bedford-Pinchuk result but at the cost of assuming that the automorphism group is slightly larger than non-compact. In particular, a smoothly bounded convex domain is biholomorphic to a domain defined by a polynomial if and only if an orbit of the automorphism group accumulates on at least two different complex faces of the set. The proof of this result combines rescaling arguments and ideas from the theory of metric spaces of non-positively curvature. (TCPL 201) |

09:15 - 10:00 |
Shulim Kaliman: Algebraic (volume) density property ↓ Let $X$ be a connected affine homogenous space of a linear algebraic group $G$ over $\mathbb C$.
(1) If $X$ is different from a line or a torus we show that the space of all algebraic vector fields on $X$ coincides
with the Lie algebra generated by complete algebraic vector fields on $X$. (2) Suppose that $X$ has a $G$-invariant
volume form $\omega$. We prove that the space of all divergence-free (with respect to $\omega$) algebraic vector
fields on $X$ coincides with the Lie algebra generated by divergence-free complete algebraic vector fields on
$X$ (including the cases when $X$ is a line or a torus). (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:15 |
Franc Forstneric: The parametric h-principle for minimal surfaces in R^n and null curves in C^n ↓ Let $M$ be an open Riemann surface. It was proved by Alarc{\'o}n and Forstneri{\v c}
that every conformal minimal immersion $M\to\mathbb R^3$ is isotopic to the real part of a holomorphic null curve
$M\to\mathbb C^3$. We prove the following substantially stronger result in this direction:
for any $n\ge 3$, the inclusion of the space of real parts of nonflat null holomorphic immersions $M\to\mathbb C^n$
into the space of nonflat conformal minimal immersions $M\to \mathbb R^n$ satisfies the parametric h-principle with approximation;
in particular, it is a weak homotopy equivalence. Analogous results hold for several other related maps.
For an open Riemann surface $M$ of finite topological type, we obtain optimal results by showing that the above
inclusion and several related maps are inclusions of strong deformation retracts; in particular, they are homotopy equivalences.
(Joint work with Finnur L{\'a}russon.) (TCPL 201) |

11:30 - 12:15 |
Alexander Tumanov: Symplectic non-squeezing for the discrete nonlinear Schrodinger equation ↓ The celebrated Gromov's non-squeezing theorem of 1985
says that the unit ball $B^n$ in $C^n$ can be symplectically embedded
in the "cylinder" $rB^1 \times C^{n-1}$ of radius r only if $r\ge 1$.
Hamiltonian differential equations provide examples of symplectic
transformations in infinite dimension. Known results on the non-squeezing
property in Hilbert spaces cover compact perturbations of linear
symplectic transformations and several specific non-linear PDEs, including
the periodic Korteweg - de Vries equation and the periodic cubic
Schr\"odinger equation. We prove a new version of the non-squeezing
theorem for Hilbert spaces. We apply the result to the discrete nonlinear
Schr\"odinger equation. This work is joint with Alexander Sukhov. (TCPL 201) |

12:15 - 13:30 | Lunch (Vistas Dining Room) |

14:45 - 15:15 | Coffee Break (TCPL Foyer) |

15:00 - 15:15 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL Foyer) |

15:15 - 16:00 |
Damir Kinzebulatov: Towards Oka-Cartan theory within some algebras of holomorphic functions ↓ We extend the basic sheaf-theoretic techniques of complex function theory
to work within some algebras of holomorphic functions (joint with Alex Brudnyi) (TCPL 201) |

16:15 - 17:00 |
Purvi Gupta: Rational density on compact real manifolds ↓ Motivated by the observation that every continuous complex-valued function on the unit circle can be approximated by rational combinations of a single function, we will discuss some conditions under which a manifold \(M\) admits \(N\) functions whose rational combinations are dense in the space of complex-valued \(C^k\)-functions on M. As a result, we will produce an optimal bound on \(N\) in terms of the dimension of M. This is joint work with R. Shafikov. (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Wednesday, May 4 | |
---|---|

07:00 - 08:15 | Breakfast (Vistas Dining Room) |

08:15 - 09:00 |
Jasna Prezelj: Positivity of metrics ↓ Let $p: Z \rightarrow Z$ be a submersion from a complex manifold $Z$ to a $1$-convex manifold $X$ with an exceptional set
$X.$ Let $E \rightarrow Z$ Let $a : X \rightarrow Z$ be a holomorphic section. Then there exist a conic neighbourhood $U$ of $a(X \setminus S)$
such that $U$ admits a K\"ahler metric and a metric on $E_U$ with positive Nakano curvature and with at most polynomial poles over
$p^{-1}(S).$ (TCPL 201) |

09:15 - 10:00 |
Alex Brudnyi: On the Sundberg approximation theorem ↓ Let $H^\infty$ be the Banach algebra of bounded holomorphic functions on the open
unit disk $D\subset \mathbb C$. We extend Sundberg’s theorem on uniform approximation of functions in
BMOA by $H^\infty$ functions to other classes of holomorphic functions on $D$. In our proofs we use a
new characterization of meromorphic functions on $D$ that extend to continuous maps of the maximal
ideal space of $H^\infty$ to the Riemann sphere. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:15 |
Francois Berteloot: Rescaling methods in complex analysis ↓ Rescaling methods are very efficient in complex analysis or geometry
because they can be combined with the theory of normal families. We will survey
some typical examples of such methods and in particular those
introduced by Sergey Pinchuk. (TCPL 201) |

11:30 - 12:15 | Eric Bedford (TCPL 201) |

12:15 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 17:30 | Free Afternoon (Banff National Park) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Thursday, May 5 | |
---|---|

07:00 - 08:15 | Breakfast (Vistas Dining Room) |

08:15 - 09:00 |
Evgeny Poletsky: Homotopic properties of holomorphic mappings ↓ Let $W$ be a domain in a complex manifold $M$. In 2008 B. J\"oricke found a way to extend holomorphic functions
from $W$ to another manifold and show that it is the envelope of holomorphy of $W$ and in 2013 F. L\'arusson and the
speaker used a similar approach to subextend plurisubharmonic functions from $W$ to a complex manifold. To define
these manifolds the authors considered the space $A(W,M)$ of analytic disks in $M$ whose boundaries lie in $M$. The
new manifolds were defined as the quotients of this space by equivalence relations, where equivalent analytic disks can
be connected by a continuous path or a homotopy in $A(W,M)$.
In 1983 L. Rudolph introduced quasipositive elements of braid groups that are fundamental groups of the complements
to some set W of planes in Cn. He proved that these elements are boundaries of analytic disks in $A(W,\mathbb C^n)$
and form a semi- group. The talk will be divided in two parts. In the first part we will discuss general constructions of
extensions of Riemann domains and subextensions of plurisubharmonic functions. In the second part we will address
the notion of quasipositive elements in general situation and explain why they form a semigroup.
An important question is whether this semigroup is embeddable into the fundamental group. That is equivalent of asking
whether two analytic disks are homotopic as analytic disks when their boundaries are equivalent in the fundamental group.
A similar problem was studied by M. Gromov and, recently, by F. Forstneri\v c and his colleagues for homotopies of
submanifolds in elliptic manifolds. In our case the ambient manifold is hyperbolic and the answer is not known. In the
special case is when $W$ is an analytic variety in $M$ we will show that this problem can be reduced to the problem
involving only real disks. (TCPL 201) |

09:15 - 10:00 |
Joel Merker: Ample Examples ↓ Tuan Huynh, Ph.D. student in Orsay, obtained (IMRN 2015) examples of
Kobayashi-hyperbolic hypersurfaces $\mathbb{ X}^n \subset
\mathbb{P}^{n+1}(\mathbb{ C})$ of low degree $2n+2$ for $n = 2, 3, 4,
5$, and of degree $\frac{ (n+3)^2}{4}$ for $n \geqslant 6$.
Song-Yan Xie, Ph.D. student in Orsay, established in 2015 the
ampleness of cotangent bundles (jets of order $1$) to generic complete
intersections $\mathbb{X}^n \subset \mathbb{P}^{n+c} (\mathbb{ C})$ of
codimension $c \geqslant n$ with degrees $d_1, \dots, d_c \geqslant
(n+c)^{(n+c)^2}$. This result answered fully a conjecture made by
Debarre in 2005.
The first part of the talk will present a variation of S.~Xie's proof,
based on multidimensional resultants, which conducts to an improvement
on the degree bound: $d_1, \dots, d_c \geqslant (n+c)^{n+c}$.
In order to reach an effective generic ampleness result about higher
order jet bundles, in link with Kobayashi's hyperbolicity conjecture,
taking inspiration from Masuda-Noguchi (1996), the second part of the
talk will focus on families of hypersurfaces $\mathbb{ X}^n \subset
\mathbb{ P}^{n+1} ( \mathbb{ C})$ having homogeneous polynomial
defining equations of the form:
\[
0
\,=\,
\sum_{\alpha_0+\alpha_1+\cdots+\alpha_{n+1}
\,=\,
{\sf fmn}}\,
A_{\alpha_0,\alpha_1,\dots,\alpha_{n+1}}(X)\,\,
\big((X_0)^d\big)^{\alpha_0}
\big((X_1)^d\big)^{\alpha_1}
\cdots
\big((X_{n+1})^d\big)^{\alpha_{n+1}},
\]
with {\sl Fermat-Masuda-Noguchi index} ${\sf fmn} \geqslant n^2 + n$,
and with polynomials $A_\bullet \big( X_0 \colon X_1 \colon
\cdots \colon X_{n+1}\big)$ homogeneous of relatively low degree ${\sf
deg}\, A_\bullet =: {\sf a} \geqslant n$, compared with the
dominant degree $d \geqslant n^{n^2}$.
Mainly, some appropriately truncated order-$n$ jet bundles will happen
to admit (a wealth of) global holomorphic sections, by means of a new
process of forming (huge) Macaulay-type matrices, thanks to an
application of Hartogs' theorem, a bit similarly as was performed by
Siu-Yeung (Invent. 1996) and by Siu (Invent. 2015).
The end of the talk will conclude by presenting a link between the
geometry of complex vector bundles and the first complete effective
computations of CR curvatures of CR manifolds up to dimension
$\leqslant 5$ performed by Samuel Pocchiola (ex-Ph.D. student in
Orsay) and Masoud Sabzevari (Shahrekord). (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:15 |
Herve Gaussier: Prime ends theory in higher dimension ↓ This is a joint work with Filippo Bracci. We try to extend the Carath\'eodory prime ends theory in higher dimension, defining the
"horosphere boundary" of complete hyperbolic (in the sense of Kobayashi) manifolds. We prove that a strongly pseudoconvex domain
together with its horosphere boundary, endowed with the horosphere topology, is homeomorphic to its Euclidean closure, whereas the
horosphere boundary of a polydisc is not even Hausdorff. As an application we study the boundary behaviour of univalent mappings. (TCPL 201) |

11:30 - 12:15 |
Nikolay Shcherbina: A domain with non-plurisubharmonic squeezing function ↓ We construct a strictly pseudoconvex domain with smooth boundary
whose squeezing function is not plurisubharmonic. This is a joint work
with J.E. Fornaess. (TCPL 201) |

12:15 - 13:30 | Lunch (Vistas Dining Room) |

14:45 - 15:15 | Coffee Break (TCPL Foyer) |

15:15 - 16:00 |
Ilya Kossovskiy: Borel theorem for CR-maps ↓ Following Henri Poincare, numerous results in Dynamics establish the curious phenomenon saying that two smooth objects
(e.g., vector fields), which can be transformed into each other by means of a formal power series transformation, can be also transformed
into each other by a smooth map. This is a kind of analogue of Borel Theorem on smooth realizations of formal power series. In CR-geometry, similar phenomena hold for real-analytic CR-manifolds, and the usual outcome is that two formally equivalent CR-manifolds are also equivalent holomorphically. However, in our recent work with Shafikov we proved that there exist real-analytic CR-manifolds, which are equivalent
formally, but still not holomorphically.
On the other hand, in our more recent work with Lamel and Stolovitch we prove that the following is true: if two 3-dimensional
real-analytic CR-manifolds are equivalent formally, then they are $C^\infty$ CR-equivalent. In this talk, I will outline the latter result. (TCPL 201) |

16:15 - 17:00 |
Rafael Andrist: The density property for Gizatullin surfaces of type [[0, 0, −r_2, −r_3]] ↓ I will give a brief introduction to the density property for Stein manifolds, which is a notion to express that a
manifold has ``many'' holomorphic automorphisms.
\smallskip
Although large classes of Stein manifolds with the density property are known, e.g.\ most of the homogeneous spaces of Stein Lie groups, they include only very few surfaces, namely $\mathbb{C}^2$, $\mathbb{C} \times \mathbb{C}^\ast$ and the smooth Danielewski surfaces. The lack of examples of such surfaces is due to the absence of the so-called ``compatible pairs'' of complete vector fields, which are usually the main tool for proving the density property.
\smallskip
Smooth Gizatullin surfaces provide a good class of candidates for surfaces with the density property, and they include the examples mentioned above. The next natural step is the investigation of Gizatullin surfaces of type $[[0,0,-r_2,-r_3]], \; r_2, r_3 \geq 2$, which can be described by the equations
\[
\left\{
\begin{array}{lcl}
y u &=& x P(x) \\
x v &=& u Q(u) \\
y v &=& P(x) Q(u)
\end{array}
\right.
\]
in $\mathbb{C}^4$ with coordinates $(x,y,u,v)$, where $P$ and $Q$ are polynomials of degree $r_2-1$ resp.\ $r_3-1$. We establish the density property for smooth Gizatullin surfaces of this type and describe a dense subgroup of the identity components of their holomorphic automorphism groups.
Joint work with Frank Kutzschebauch and Pierre-Marie Poloni. (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Friday, May 6 | |
---|---|

07:00 - 08:15 | Breakfast (Vistas Dining Room) |

08:15 - 09:00 |
Florian Bertrand: Riemann-Hilbert problems with singularities ↓ The study of analytic discs attached to a totally real submanifold M of $\mathbb C^n$ leads to the consideration
of a regular Riemann-Hilbert problem of a special form. Following this approach, Forstneric, and later on Globevnik,
characterized the existence and dimension of a family of deformations of a given analytic disc attached to M in terms of
certain indices. However, in case M admits some complex tangencies, the indices mentioned above are no longer
well-defined and the Forstneric-Globevnik method falls apart. In this talk, I will focus on a class of such singular
Riemann-Hilbert problems. We will see that they can be solved by a factorization technique that reduces them to regular
Riemann-Hilbert problems with geometric constraints. In particular, we will deduce the existence of stationary type discs
attached to finite type hypersurfaces. (TCPL 201) |

09:15 - 10:00 |
Elizabeth Wulcan: Direct images of semi-meromorphic currents ↓ I will discuss a joint work in progress with Mats Andersson, in which we study and develop a calculus for direct images of semi-meromorphic currents. In my talk I will focus on regularity properties of these and in particular show that the sheaf of such currents is stalkwise injective. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:15 |
Peter Ebenfelt: Stable umbilical points on perturbations of the sphere in C^2 ↓ The standard CR structure on the three dimensional sphere can be deformed in such a way that the deformed
structures have no (CR) umbilical points. A 1-parameter family of such deformations was essentially discovered by
E.~Cartan (and later studied by Cap, Isaev, Jacobowitz). The CR manifolds in this family, however, cannot be embedded in
$\mathbb C^2$. It is an open question whether the unit sphere can be perturbed in $\mathbb C^2$ such that no umbilical
points remain on the perturbed CR manifolds. In this talk, we shall discuss an approach to this problem, and describe some
recent results. One of the results that will be described guarantees stable (in a sense to be made precise in the talk)
umbilical points on generic perturbations of ”almost circular type”. This complements a previous result by the speaker
and Son Duong on existence of umbilical points on circular three-dimensional CR manifolds. (TCPL 201) |

11:30 - 12:00 |
Checkout by Noon ↓ 5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon. (Front Desk - Professional Development Centre) |

11:30 - 12:15 |
Zbigniew Blocki: Geodesics in the space of Kahler metrics and volume forms ↓ We discuss optimal regularity of geodesics in the
space of K\"ahler metrics of a compact K\"ahler manifold, as well
as the space of volume forms on a compact Riemannian manifold.
They are solutions of nonlinear degenerate elliptic equations:
homogeneous complex Monge-Amp\`ere equation and Nahm's equation
(introduced by Donaldson), respectively. The highest regularity
one can expect is $C^{1,1}$. (TCPL 201) |

12:15 - 13:30 | Lunch from 11:30 to 13:30 (Vistas Dining Room) |