## 22nd Andrzej Jankowski Memorial Lecture Mini Conference

2 Maj 2023

The mini conference will take place at the Faculty of Law and Administration of the University of Gdańsk, May 12-13, 2023.

## Speakers

- Aaron Naber (Northwestern University)
- Sławomir Kołodziej (Jagiellonian University)
- Sławomir Dinew (Jagiellonian University)
- Michał Miśkiewicz (University of Warsaw)
- Marta Lesniak (University of Gdańsk)
- Paweł Raźny (Jagiellonian University)

## Preliminary schedule

**Friday, 12 May**

Auditorium of the Faculty of Law and Administration

16:15 A. Naber, *Andrzej Jankowski Memorial Lecture*

**Saturday, 13 May**

Auditorium C of the Faculty of Law and Administration

09:00 – 09:30 P. Raźny, *A spectral sequence for isometric group actions*

09:35 – 10:20 S. Dinew, *The distance function – old and new*

coffee break

10:50 – 11:20 M. Leśniak, *Generating the mapping class group of a nonorientable surface by torsions*

11:25 – 12:25 S. Kołodziej, *Hölder continuous solutions to complex Monge-Ampère equations*

lunch break

14:00 – 14:45 M. Miśkiewicz, *Regularity of minimizing p-harmonic maps into spheres*

14:50 – 15:50 A. Naber, *Recent advances on the Structure of Spaces with Lower and Bounded Ricci Curvature*

## Abstracts

#### Aaron Naber, Ricci Curvature, Fundamental Groups and the Milnor Conjecture

Milnor conjectured in 1968 that if (M,g) is a Riemannian manifold with nonnegative Ricci curvature, then its fundamental group is finitely generated. This lecture will not require much background, and the first half will be dedicated toward understanding this conjecture and a discussion on the work accomplished toward it over the years. The second half of the lecture will describe recent work with Brue and Semola where we provide a counterexample. Specifically, we construct a manifold M^{7} with nonnegative Ricci whose fundamental group is ℚ/ℤ . The construction involves a new topological construction of such spaces, and on understanding the relationship between Ricci curvature and the mapping class group. The talk will mostly focus on explaining the broader ideas involved in the construction.

#### Aaron Naber, Recent advances on the Structure of Spaces with Lower and Bounded Ricci Curvature

The talk will introduce, hopefully at a basic level, the meaning and analysis of spaces with Ricci curvature bounds. We will discuss the process of limiting spaces with such bounds, and studying the singularities on these limits. The singularities come with a variety of natural structure which have been proven in the last few years, from dimension bounds to rectifiable structure, which is (measure-theoretically) a manifold structure on the singular set. If time permits we will discuss some recent work involving the topological structure of boundaries of such spaces. From a general perspective the analysis and ideas involved are applicable to a wide range of nonlinear pde’s with singularities, especially those which tend to arise in geometric analysis.

#### Sławomir Kołodziej, Hölder continuous solutions to complex Monge-Ampère equation

There are reasons to discuss Hölder continuous solutions:

- This family has some nice properties like validity of the subsolution theorem.
- Geometric applications, in particular to cone Kähler merics. Applied in afamous proof of Chern-Donaldson-Sun characterizing Fano manifolds admitting Kähler-Einstein metrics.

The complex Monge-Ampere equation will be considered in four different settings: in strictly pseudoconvex domains in ℂ^{n}, on compact Kähler manifolds, on compact Hermitian manifolds, and in open subsets of Hermitian manifolds.

#### Sławomir Dinew, The distance function – old and new

We recall the basic theory behind the distance function to a compact set in an Euclidean space. Then we apply these to provide a new proof of Oka’s lemma in the theory of holomorphic convexity.

#### Michał Miśkiewicz, Regularity of minimizing p-harmonic maps into spheres

Regularity of minimizing *p*-harmonic maps – i.e., minimizers of the Dirichlet *p*-energy among maps between two given manifolds – is known to depend on the topology of the target manifold. In particular, the case of maps into spheres has been studied intensively, but still some of the most basic questions concerning maps from *B ^{3}* into

*S*remain open. Minimizing maps in this context were shown to be regular when

^{3}*p=2*or

*p≥3*, and recently also when

*2<p<2.13*, leaving a peculiar gap in between. I will discuss known approaches to the problem and how these can be extended to cover

*2<p<2.36*and

*2.97<p<3*, thus shrinking the gap. This is joint work with Katarzyna Mazowiecka (University of Warsaw).

## Contact

Andrzej Szczepański: andrzej.szczepanski@ug.edu.pl