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## Welcome on the site of the Maestro project

The research carried within this project is presented at the seminar

"Complex analysis and elliptic PDE's" (formerly "The Monge- Ampère equation"),

which is being held on Mondays from 14:15 to 15:45 in room 1016 of the Faculty

of Mathematics and Computer Science building, prof. Stanisława Łojasiewicza 6 str. in Krakow.

You are welcome to attend our meetings.

16.7 Pengfei Guan "Regularity estimates for scalar curvature equation " Abstract: Scalar curvature equation is a fully nonlinear PDE in differential geometry. It resembles some properties of Monge-Amp\`ere equation and exhibits other special features which are not true for higher dimensional Monge-Amp\`ere equation. The talk will concentrate on two problems: global regularity of immersed hypersurfaces in warped product spaces with nonnegative scalar curvature and interior $C^2$ estimate for immersed hypersurfaces in $R^{n+1}$ with positive scalar curvature (which is not true for $\sigma_k$ curvature equations when $k>2$ by examples of Pogorelov and Urbas). We will explain how special structures of the scalar curvature equation play the role in the estimates and discuss applications to the isometric embedding problem and notion of quasi local masses in general relativity.

11.6 Marcin Sroka "Quaternionic pluripotential theory"

28.5 Zbigniew Błocki "Monge-Ampère operator for plurisubharmonic functions with analytic singularities on Hermitian manifolds"

21.5 Constantin Schramov (Steklov Mathematical Institute and Higher School of Economics) "Fano threefolds of degree 22 with a Gm-action" ABSTRACT: It is known that the maximal possible anticanonical degree for smooth Fano threefolds of Picard rank 1 and index 1 is 22. The family of threefolds of this degree has many interesting features. In particular, it was extensively studied from the point of view of Tian's alpha-invariants and existence of Kaehler-Einstein metrics. It is known that some of them admit Kaehler-Einstein metrics, while some others do not. In some cases their existence can be deduced from a computation of alpha-invariants. In my talk I will tell about the geometry of Fano threefolds of degree 22 that admit the action of a one-dimensional torus Gm, and about their alpha-invariants. This is a joint work with Ivan Cheltsov.

14.5 Marcin Sroka "Quaternionic pluripotential theory"

7.5 Marcin Sroka "Quaternionic pluripotential theory"

23.4 DongSeon Hwang "On cascades of log del Pezzo surfaces" Abstract: There have been numerous attempts to classify log del Pezzo surfaces. In this talk, I will quickly summarize the known results to this goal and report my work on the classification of such surfaces. The result is obtained by generalizing the notion of ‘cascades’ of nonsingular del Pezzo surfaces to the singular case. This approach is turned out to be successful when the surface is toric or of Picard number one. At the end of the talk, I will also speak on the Singular Surface Database(SSDB) project which aims to realize this method.

16.4 Krzysztof Maciaszek "Cyclicity in the Dirichlet-type spaces"

9.4 Szymon Pliś "Bedford-Taylor type capacities on almost complex surfeces"

26.3 Szymon Pliś "Bedford-Taylor type capacities on almost complex surfeces"

19.3 Dominik Burek "McKay correspondence"

12.3 Sławomir Kołodziej "Hölder continuous subsolutions for the Monge-Ampère equation"

5.3 Giovanni Mongardi (Bologna) "Hilbert schemes of points and various equivalencies" Abstract: We will start with an introduction on irreducible symplectic (IHS) manifolds, with a particular focus on their birational geometry. Then, we analyze what happens when we take two K3 surfaces with the same derived category and their hilbert schemes of points. We will show how the birational geometry of IHS manifolds forces these schemes to be non birational in many cases. This is part of a joint work with C. Meachan and K. Yoshioka.

26.2 Żywomir Dinew "Gromov-Hausdorff limits of Kähler manifolds"

22.1 Żywomir Dinew "Gromov-Hausdorff limits of Kähler manifolds"

15.1 Marcin Sroka "Quaternionic Monge-Ampère equation IV: The Complex Attempt."

8.1 Marcin Sroka "Quaternionic Monge-Ampère equation III: Calabi Conjecture for HKT-manifolds."

18.12 Marcin Sroka "Quaternionic Monge-Ampère equation II: Quaternionic geometric structures"

11.12 Marcin Sroka "Quaternionic Monge-Ampere equation I: An introduction"

4.12 Grzegorz Kapustka "Compact Moduli spaces of del Pezzo surfaces and Kähler-Einstein metrics"

27.11 Sławomir Dinew "The $\partial\bar{\partial}$ lemma on Hermitian manifolds"

20.11 Tat Dat To "Geometric contraction of Chern-Ricci flow" Abstract: The Chern-Ricci flow is an evolution equation of Hermitian metrics by their Chern-Ricci form. It follows from previous works of Tosatti and Weinkove that the flow carries out blow-downs of exceptional curves on non-minimal surfaces. In this talk I provide more evidence that the Chern-Ricci flow is a natural evolution equation on complex manifolds and that its behavior reflects the underlying geometry.

13.11 Rafał Czyż "Extension and approximation of m-subharmonic functions"

6.11 Rafał Czyż "Extension and approximation of m-subharmonic functions"

23.10 Szymon Pliś "C^{1,1} regularity of degenerate Monge-Ampère equation"

16.10 Szymon Myga "Completion of the space of Kähler potentials"

9.10 Szymon Myga "Completion of the space of Kähler potentials"

2.10 Szymon Myga "Completion of the space of Kähler potentials"

6.9 Ngoc Cuong Nguyen "On the Hölder continuous subsolution theorem for the complex Monge-Ampère equation"

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12.6 David Witt Nyström "Complex homogeneous Monge-Ampère equations and the Hele-Shaw flow"

29.5 Szymon Myga "Minimum sets of solutions to real and complex Monge-Ampère equation".

22.5 Egmont Porten "Pseudoconcave submanifolds of complex space"

Abstract: Pseudoconcave CR manifolds attract particular interest since their structural properties resemble (and generalise) aspects known from complex manifolds, and also because of an abundant variety of examples originating from representation theory. In our talk, we discuss work in progress aiming at extending certain aspects of the theory, like the maximum principle and results on envelopes of holomorphy, to general real submanifolds of Cn (where CR dimension may vary with the point). A major part of this is due to joint work with Mauro Nacinovich (Rome 2).15.5 Sławomir Dinew "Compactness theorems for Kähler-Einstein manifolds in dimension 3 and up"

24.4 Xin Dong "Equality in Suita's conjecture"

10.4 Grzegorz Kapustka "Hyperkähler manifolds and incident planes."

3.4 Zbigniew Błocki "Monge-Ampère operator for plurisubharmonic functions with analytic singularities"

27.3 Xin Dong "Bergman kernel and its boundary asymptotics"

20.3 Sławomir Dinew "Compactness theorems for Kähler-Einstein manifolds in dimension 3 and up"

13.3 Sławomir Dinew "Compactness theorems for Kähler-Einstein manifolds in dimension 3 and up"

6.3 Żywomir Dinew "Further developments of the Wu-Yau theorem"

27.2 Żywomir Dinew "Further developments of the Wu-Yau theorem"

23.1 Nguyen Van Thien "Entire mappings and Monge-Ampere currents"

16.1 Krzysztof Maciaszek (excepionally in room 1094) "On the Maximal Rank Problem for the Complex Homogeneous Monge-Ampere Equation"

9.1 Szymon Pliś "Remarks on holomorphic discs"

12.12 Szymon Myga "Hermitian metrics that leave the total Monge-Ampere volume invariant"

5.12 Sławomir Dinew "Remarks on the obstacle problem"

28.11 Sławomir Dinew "From the Kähler-Ricci flow to moving free boundaries and shocks"

14.11 Rafał Czyż "Moser-Trudinger type inequality in \mathcal E_{p} classes"

7.11 Żywomir Dinew "Minimum sets and differential tests for plurisubharmonic functions"

24.10 Żywomir Dinew "Minimum sets and differential tests for plurisubharmonic functions"

17.10 Szymon Myga "Singular sets of solutions to real Monge-Ampere equation"

10.10 1)Nguyen Van Thien "An inequality for beta function with application to Pluripotential theory" 2)Szymon Myga "Singular sets of solutions to real Monge-Ampere equation"

3.10 Nguyen Van Thien "Convexity of Hessian integrals and Poincare type inequalities"

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30.5 Ngoc Cuong Nguyen "The Fu-Yau equation with negative slope parameter"

16.5 Szymon Myga "Harmonic discs of solutions to Homogeneous Monge-Ampere equation - negative result"

9.5 Dan Popovici "Hodge Theory for the Second Step of Certain Spectral Sequences"

25.4 Ngoc Cuong Nguyen "Log canonical thresholds and Monge-Ampère masses"

18.4 Sławomir Dinew "Sverak theorem"

11.4 Żywomir Dinew "On minimum sets of strictly plurisubharmonic functions"

4.4 Żywomir Dinew "On minimum sets of strictly plurisubharmonic functions"

21.3 Szymon Pliś "Pluripolar sets on almost complex surfaces"

14.3 Ibrahim Djire Karamoko "Pluripolar hulls of Poletsky"

29.2 Ngoc Cuong Nguyen "The Dirichlet problem for the complex Hessian equation on non-compact Hermitian manifolds"

25.1 Ngoc Cuong Nguyen "The Dirichlet problem for the complex Hessian equation on non-compact Hermitian manifolds"

11.1 Nguyen Van Thien "On delta m-subharmonic functions"

4.1 Nguyen Van Thien "On delta m-subharmonic functions"

21.12 Ngoc Cuong Nguyen "Gauduchon metrics with prescribed volume form"

14.12 Ngoc Cuong Nguyen "Gauduchon metrics with prescribed volume form"

7.12 Szymon Myga "Construction of non-polynomial solutions to \sigma_k equation"

30.11 Sławomir Dinew "Tangents to subsolutions (paper of Harvey and Lawson)"

23.11 Ngoc Cuong Nguyen "Fully non-linear elliptic equations on compact Hermitian manifolds"

16.11 Ngoc Cuong Nguyen "Fully non-linear elliptic equations on compact Hermitian manifolds"

9.11 Żywomir Dinew "Some conjectures in Kähler geometry''

26.10 Ngoc Cuong Nguyen "Fully non-linear elliptic equations on compact Hermitian manifolds"

19.10 Ngoc Cuong Nguyen "Fully non-linear elliptic equations on compact Hermitian manifolds"

12.10 Nguyen Van Thien "The weighted pluricomplex energy class $\mathcal E_{\chi}(\Omega)"

5.10 Nguyen Van Thien "The weighted pluricomplex energy class $\mathcal E_{\chi}(\Omega)"

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15.6 Szymon Myga "Monge-Kantorovich's optimal transportation problem"

8.6 Szymon Myga "Monge-Kantorovich's optimal transportation problem"

2.6 Workshop on geometric analysis, see the "workshop" tab

1.6 Workshop on geometric analysis, see the "workshop" tab

25.5 1) Maria Trybuła "Bergman distance on planar domains" 2)Sławomir Dinew "Remarks on Mukai threefolds with C^{*} action"

18.5 Maria Trybuła "Bergman distance on planar domains"

4.5 Duong Hong Phong "Estimates for Hessian and Monge-Ampère equations "

Abstract: There has been recently a lot of progress on techniques for a priori estimates for Hessian and Monge-Ampère equations on complex manifolds. We survey some of the more recent developments, with emphasis on open problems.

13.4 Rafał Czyż "On a problem of Błocki and Zwonek"

30.3 Sławomir Dinew "Gábor Székelyhidi's paper: Remark on the Calabi flow with bounded curvature"

9.3 Sławomir Dinew "Convexity of K- energy and uniqueness of constant scalar curvature metrics"

2.3 Ngoc Cuong Nguyen "Stability and regularity of solutions of the complex Monge-Ampère equation on Hermitian manifolds"

26.1 Ngoc Cuong Nguyen "Stability and regularity of solutions of the complex Monge-Ampère equation on Hermitian manifolds"

19.1 Sławomir Dinew "Convexity of K- energy and uniqueness of constant scalar curvature metrics"

12.1 Żywomir Dinew "Canonical measures and analytic Zariski decompositions"

15.12 Żywomir Dinew "Canonical measures and analytic Zariski decompositions"

8.12 Ngoc Cuong Nguyen "Monge-Ampère type equations on Hermitian manifolds"

1.12 Ngoc Cuong Nguyen "Monge-Ampère type equations on Hermitian manifolds"

24.11 Ngoc Cuong Nguyen "Monge-Ampère type equations on Hermitian manifolds"

17.11 Ngoc Cuong Nguyen "Monge-Ampère type equations on Hermitian manifolds"

3.11 Ngoc Cuong Nguyen "Monge-Ampère type equations on Hermitian manifolds"

27.10 Grzegorz Kapustka "Hyperkähler manifolds"

20.10 Szymon Pliś "On the viscosity solution of the Dirichlet problem"

13.10 Zbigniew Błocki "Isoperimetric inequalities and symmetrization"

6.10 Ngoc Cuong Nguyen "The Kähler rank of compact complex manifolds"

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9.9 during a joint meeting the following talks will be presented:

13:00 - 13:50 Evgeniya Dyachenko "An algebra of pseudodifferential operators with small parameters"

14:00 -14:50 Ahmed Zeriahi "Weak solutions to degenerate complex Monge-Ampère flows"

Evgeniya Dyachenko is a candidate for the postdoc position in IMUJ.

Both talks will be in room 1016

2.6 Sławomir Dinew "Minimum sets of plurisubharmonic functins"

5.5 Grzegorz Kapustka "Kähler-Einstein metric on Fano threefolds of genus 12"

28.4 Żywomir Dinew "Demailly's approximating sequence is not decreasing"

14.4 Rafał Czyż "Singular measures"

31.3 Sławomir Dinew "Regularity results for the complex Monge-Ampère equation"

17.3 Dongwei Gu "C^{1,1} regularity for ample line bundles"

10.3 Sławomir Kołodziej "Generalized Monge-Ampère capacities"

3.3 Sławomir Dinew "Monge-Ampère equations on quasi-projective varieties"

27.1 Ngoc Cuong Nguyen "An inequality between the m-capacity and the Euclidean volume form"

20.1 Dongwei Gu "Regularity of psh upper envelopes in big cohomology class"; Ngoc Cuong Nguyen "An inequality between the m-capacity and the Euclidean volume form"

13.1 Dongwei Gu "Regularity of psh upper envelopes in big cohomology class"

16.12 Szymon Pliś "The smoothing of m-subharmonic functions"

9.12 Grzegorz Kapustka "Kähler-Einstein metrics on Fano manifolds"

2.12 Sławomir Dinew "Strong openness conjecture"

18.11 Sławomir Dinew "Berndtsson's proof of the openness conjecture"

4.11 Sławomir Dinew "J-Flow"

21.10 Żywomir Dinew "The technique of Tian-Yau-Zelditch expansion"

14.10 Żywomir Dinew "Tsuji's L^2 approach"

7.10 Working meeting

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10.6 Grzegorz Kapustka "Kähler-Einstein metric on Fano manifolds"

3.6 Grzegorz Kapustka "Kähler-Einstein metric on Fano manifolds"

27.5 Żywomir Dinew "Kähler-Einstein submanifolds of the Fubini-Study projective space"

20.5 Żywomir Dinew "Kähler-Einstein submanifolds of the Fubini-Study projective space"

29.4 Żywomir Dinew "Kähler-Einstein submanifolds of the Fubini-Study projective space"

22.3 Dongwei Gu "Kähler-Einstein metrics on Fano manifolds"

15.3 Dongwei Gu "Kähler-Einstein metrics on Fano manifolds"

25.3 Dongwei Gu "Kähler-Einstein metrics on Fano manifolds"

18.3 Nguyen Ngoc Cuong "Hölder continous solutions to complex Hessian equations"

11.3 Frunza Ataev "Solutions to degenerate complex Hessian equations"

4.3 Frunza Ataev "Solutions to degenerate complex Hessian equations"

25.2 Frunza Ataev "Solutions to degenerate complex Hessian equations"

21.1 Frunza Ataev "Solutions to degenerate complex Hessian equations"

14.1 Frunza Ataev "Solutions to degenerate complex Hessian equations"

7.1 Frunza Ataev "Solutions to degenerate complex Hessian equations"

17.12 Frunza Ataev "Solutions to degenerate complex Hessian equations"

3.12 Szymon Pliś "Monge-Ampère operator for J-plurisubharmonic functions"

26.11 Szymon Pliś "Monge-Ampère operator for J-plurisubharmonic functions"

12.11 Ngoc Cuong Nguyen "The super-potential of closed positive currents and applications"

5.11 Ngoc Cuong Nguyen "The super-potential of closed positive currents and applications"

29.10 Ngoc Cuong Nguyen "The super-potential of closed positive currents and applications"

22.10 Ngoc Cuong Nguyen "The super-potential of closed positive currents and applications"

15.10 Ngoc Cuong Nguyen "The super-potential of closed positive currents and applications"

8.10 Dongwei Gu "Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds"

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11.6 Dongwei Gu "Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds"

4.6 Żywomir Dinew "Weak geodesics in the space of Kähler metrics"

28.5 Ngoc Cuong Nguyen "The J-flow in Kähler geometry"; Żywomir Dinew "Weak geodesics in the space of Kähler metrics"

21.5 Ngoc Cuong Nguyen "The J-flow in Kähler geometry"

7.5 Ngoc Cuong Nguyen "The J-flow in Kähler geometry"

23.4 Ngoc Cuong Nguyen "The J-flow in Kähler geometry"

16.4 Ngoc Cuong Nguyen "The J-flow in Kähler geometry"

2.4 Dongwei Gu "On the C^{2,alpha}-Regularity of the Complex Monge-Ampère Equation"

26.3 Frunza Ataev "Measures of finite pluricomplex energy"

19.3 Frunza Ataev "Measures of finite pluricomplex energy"

12.3 Frunza Ataev "Measures of finite pluricomplex energy"

5.3 Dongwei Gu "Ricci iterations on Fano manifolds"

27.2 Dongwei Gu "Ricci iterations on Fano manifolds"

26.1 Żywomir Dinew "The Ricci and scalar curvatures of the Bergman metric"

9.1 Ngoc Cuong Nguyen "Applications of the Moser-Trudinger type inequalities for complex Monge-Ampère operators"

7.1 Ngoc Cuong Nguyen "Applications of the Moser-Trudinger type inequalities for complex Monge-Ampère operators"

15.12 Ngoc Cuong Nguyen "Applications of the Moser-Trudinger type inequalities for complex Monge-Ampère operators"

12.12 Ngoc Cuong Nguyen "Applications of the Moser-Trudinger type inequalities for complex Monge-Ampère operators"

28.11 Ngoc Cuong Nguyen "Applications of the Moser-Trudinger type inequalities for complex Monge-Ampère operators"

24.11 Zbigniew Błocki "Geodesics in the space of Kähler metrics need not be C^3 (an example of Lempert and Vivas)"

17.11 Szymon Pliś "Dirichlet problem on almost complex manifolds"

7.11. Rafał Czyż "The Dirichlet problem and Dirichlet principle"; Szymon Pliś "Dirichlet problem on almost complex manifolds"

20.10 Rafał Czyż "The Dirichlet problem and Dirichlet principle"

13.10 Rafal Czyż "On Dirichlet's principle and problem"