# Adam Chudecki

**Education.**

- Habilitation, 2019 - University of Warsaw, dissertation
*Congruences of null strings and their relation with the symmetries in weak and strong hyperheavenly spaces.* - Ph.D., 2009 - Lodz University of Technology, dissertation
*Plebański - Robinson*-*Finley hyperheavenly equations. Analysis and applications in theory of relativity*(thesis supervisor: prof. dr hab. Maciej Przanowski). - M.Sc., 2002 - Lodz University of Technology, dissertation
*Phase space of quantum systems*(thesis supervisor: dr hab. Jaromir Tosiek).

**Scientific interests.**

I am working on the problems related to general theory of relativity and differential geometry. I am fascinated by structures called the congruences of null strings. These structures are complex counterparts of the congruences of null geodesics. Their influence on the geometry of 4-dimensional complex spaces is remarkable. In particular, I am investigating hyperheavenly spaces and their generalizations, so-called weak hyperheavenly spaces. I am interesting at the symmetries and at real slices of such spaces (especially real neutral slices which lead to the spaces equipped with the metric of the neutral signature $(++--)$). Nowadays I am using the hyperheavenly spaces formalism for an analysis of the para-Hermite and para-Kähler spaces.

**Other interests.**

In my free time I ride a bike, occasionally climb the rocks of Jura Krakowsko-Częstochowska and swim. I love mountains, especially Tatra Mountains.

A list below contains the problems which I am dealing with nowadays.

**Exact and general solutions of para-Kähler spaces**. This issue deals with spaces of the types $[\textrm{any}] \otimes [\textrm{D}]^{nn}$. Especially interesting are Einstein spaces of the types $[\textrm{I}] \otimes [\textrm{D}]^{nn}$ because no explicit examples of such spaces are known nowadays. Also, new examples of the heavenly spaces of the type $[\textrm{I}] \otimes [\textrm{O}]$ could be found during the analysis of this problem.**Para-Kähler Einstein geometry of the neutral spaces of the types $[\textrm{D}]\otimes [\textrm{N}]$.**Cosmological constant $\Lambda$ is necessarily nonzero in this case. There are two subtypes: $\{ [\textrm{D}]^{nn} \otimes [\textrm{N}]^{e},[--,++] \}$ and $\{ [\textrm{D}]^{nn} \otimes [\textrm{N}]^{e},[++,++] \}$. It is supposed that the first of these types could be solved with all the generality. Interesting task is to find examples of the second type.**Para-Hermite Einstein geometry of the neutral spaces of the types****$[\textrm{D}]\otimes [\textrm{N}]$.**There are two subtypes: $[\textrm{D}]^{ee} \otimes [\textrm{N}]^{n}$ and $[\textrm{D}]^{ee} \otimes [\textrm{N}]^{e}$ with different properties of the congruences of null geodesics. Explicit examples are unknown.**Spaces equipped with three or four distinct congruences of null strings of the same duality.**The metric of a space equipped with three congruences of null strings is known - this is a space of the type $[\textrm{I,II}]^{eee} \otimes [\textrm{any}]$ (with a special subcase $[\textrm{II}]^{nee} \otimes [\textrm{any}]$). However, the metric of a space of the type $[\textrm{I}]^{eeee} \otimes [\textrm{any}]$ remains unknown. Can the results be specialized to the types $[\textrm{I}]^{eeee} \otimes [\textrm{D}]^{nn}$ and $[\textrm{I}]^{eeee} \otimes [\textrm{O}]$ (which are para-Kähler)? What types of the traceless Ricci tensor are admitted by such spaces?**Spaces equipped with four distinct congruences of null geodesics.**The problem is to find the metric of a space which is equipped with four distinct shearfree congruences of null geodesics. It is not even known if such a structure exists. If "yes", it is admitted only by spaces of the type $[\textrm{I}]$ with matter. What types of mater are admitted by such spaces?**Killing vectors in Walker spaces in dimension 4.**The main aim is to analyze the Killing symmetries in spaces of the types $[\textrm{deg}]^{n} \otimes [\textrm{any}]$. Is there an equivalent of the master equation in such spaces?**Null Killing vectors in spaces with****$C_{ab} \ne 0$.**It is known (A.C., 2014) that the existence of the null Killing vectors in Einstein spaces implies the existence of the self-dual and anti-self-dual congruences of null strings. Can these results be generalized to the case of non-vacuum spaces.**Type-**It is known (Plebański, Przanowski, Formański, 1998) that a superposition of two heavenly spaces of the types**$[\textrm{N}] \otimes [\textrm{N}]$ problem - a superposition.****$[\textrm{N}] \otimes [\textrm{O}]$ and $[\textrm{O}] \otimes [\textrm{N}]$ lead to the space of the type $[\textrm{N}] \otimes [\textrm{N}]$ with matter. The idea is to consider superposition of two SD spaces of the types $[\textrm{N}] \otimes [\textrm{O}]$ and $[\textrm{O}] \otimes [\textrm{N}]$? The general metric of the SD space of the type $[\textrm{N}]^{n} \otimes [\textrm{O}]^{n}$ is known. It is desired to find a general form of the metric of the type $[\textrm{N}]^{e} \otimes [\textrm{O}]^{n}$ and then investigate a superposition.****Type-**. Is it possible to reduce the vacuum Einstein field equations for the type $[\textrm{N}]^{e} \otimes [\textrm{N}]^{e}$ with a twist equipped with a single Killing vector to a single equation? Can the counterpart of the Hauser solution be found?**$[\textrm{N}] \otimes [\textrm{N}]$**spaces with a single symmetry**Type-**It is known that the existence of the nonzero cosmological constant simplifies a little type-$[\textrm{N}]$ problem with twist. With $\Lambda \ne 0$ there are only three subtypes, namely $\{ [\textrm{N}]^{e} \otimes [\textrm{N}]^{e},[--] \}$, $\{ [\textrm{N}]^{e} \otimes [\textrm{N}]^{e},[+-] \}$ and $\{ [\textrm{N}]^{e} \otimes [\textrm{N}]^{e},[++] \}$. It is possible to reconstruct known solutions and to find new ones?**$[\textrm{N}] \otimes [\textrm{N}]$**spaces with**$\Lambda \ne 0$**.

Issues 1-4 and 6 deal with neutral geometry (in particular: para-Kähler and para-Hermitian geometry). Issues 8-10 are related to the almost 50-years old problem, namely: to find the second twisting, vacuum type $[\textrm{N}]$ solution. In all these problems a complex methods and spinorial formalism seem to be the best tools for analysis.

A title of the Ph.D. thesis which can be proposed is "Hyperheavenly spaces and their real slices". For realization of this dissertation a basic knowledge of differential geometry, general theory of relativity and partial differential equations is needed. The dissertation can be finalized in 4-5 years.

**Journal articles:**

*Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part II*, Journal of Geometry and Physics 188, 104826 (2023); arXiv, link*Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part I*, Journal of Geometry and Physics 179, 104591 (2022); arXiv, link*Two-sided conformally recurrent self-dual spaces*, Journal of Geometry and Physics 159, 103933 (2021); arXiv; link*Classification of complex and real vacuum spaces of the type $[\textrm{N}] \otimes [\textrm{N}]$*, Journal of Mathematical Physics 59, 062503 (2018); arXiv; link*On twisting type $[\textrm{N}] \otimes [\textrm{N}]$ Ricci flat complex spacetimes with two homothetic symmetries*, Journal of Mathematical Physics 59, 042504 (2018) (współautor: M. Przanowski); arXiv; link*On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces*, Journal of Mathematical Physics 58, 112502 (2017); arXiv; link*On some examples of para-Hermite and para-K*, Journal of Geometry and Physics 112, 175–196 (2017); arXiv; link*ä*hler Einstein spaces with $\Lambda \ne 0$*Classification of the traceless Ricci tensor in 4-dimensional pseudo-Riemannian spaces of neutral signature*, Acta Physica Polonica B, Vol. 48, No. 1, 53-74 (2017); arXiv; link*All complex and real ASD Einstein spaces with $\Lambda$ admitting nonnull Killing vector*, International Journal of Geometric Methods in Modern Physics, Vol. 13, No. 2, 1650011 (2016); arXiv; link*Proper conformal symmetries in self-dual Einstein spaces*, Journal of Mathematical Physics 55, 082502 (2014) (współautor: M. Dobrski); arXiv; link*Null Killing vectors and geometry of null strings in Einstein Spaces*, General Relativity and Gravitation 46, 1714 (2014); arXiv; link*Killing Symmetries in $\mathcal{H}$ spaces with $\Lambda$*, Journal of Mathematical Physics 54, 102503 (2013) (współautor: M. Przanowski); arXiv; link*Homothetic Killing vectors in expanding $\mathcal{HH}$-spaces with $\Lambda$*, International Journal of Geometric Methods in Modern Physics, Vol. 10, No. 1, 1250077 (2013); arXiv; link*Classification of the Killing vectors in nonexpanding $\mathcal{HH}$-spaces with $\Lambda$*, Classical and Quantum Gravity 29, 135010 (2012); arXiv; link*Notes on para-Hermite-Einstein spacetimes*, International Journal of Geometric Methods in Modern Physics, Vol. 9, No. 1, 1250008 (2012) (współautorzy: M. Przanowski i S. Formański); link*Conformal Killing vectors in nonexpanding $\mathcal{HH}$-spaces with $\Lambda$*, Classical and Quantum Gravity 27, 205004 (2010); link*From hyperheavenly spaces to Walker and Osserman spaces: II*, Classical and Quantum Gravity 25, 235019 (2008) (współautor: M. Przanowski); link*From hyperheavenly spaces to Walker and Osserman spaces: I*, Classical and Quantum Gravity 25, 145010 (2008) (współautor: M. Przanowski); link*A simple example of type-$[\textrm{N}]\otimes[\textrm{N}]$ $\mathcal{HH}$-spaces admitting twisting null geodesic congruence*, Classical and Quantum Gravity 25, 055010 (2008) (współautor: M. Przanowski); link

**Conference articles:**

*Complex and real para-Kähler Einstein spaces*, Acta Physica Polonica B Proceedings Supplement, Vol. 16, No. 6 (2023); link*Two-sided Walker and para-Kähler Spaces and Real Slices of Hyperheavenly Spaces*, Acta Physica Polonica B Proceedings Supplement, Vol. 15, No. 1 (2022); link*On Some Solutions of the Type [D] Self-dual Spaces*, Acta Physica Polonica B Proceedings Supplement, Vol. 13, No. 2 (2020); link*Congruences of null strings and their relations with Weyl tensor and traceless Ricci tensor*, Acta Physica Polonica B Proceedings Supplement, Vol. 10, No. 2 (2017); link

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