## 過去の記録

### 2019年11月05日(火)

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00

Magnitude homology of geodesic space (JAPANESE)
[ 講演概要 ]
Magnitude is an invariant which counts effective number of points' on a metric space. Its categorification is magnitude homology. This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton, and then for any metric spaces by Leinster and Shulman. The definition of the magnitude homology is easy, but its calculation is rather difficult. For example, the magnitude homology of the circle with geodesic metric was known partially. In my talk, I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption. In this result, the magnitude homology is described in terms of geodesics. Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.

#### 解析学火曜セミナー

16:50-18:20   数理科学研究科棟(駒場) 128号室
Ngô Quốc Anh 氏 (ベトナム国家大学ハノイ校 / 東京大学)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
[ 講演概要 ]
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity $\Delta^m u = \pm u^\alpha$ in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.

### 2019年10月31日(木)

#### 応用解析セミナー

16:00-17:30   数理科学研究科棟(駒場) 128 (TBD)号室
Marius Ghergu 氏 (University College Dublin)
Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)
[ 講演概要 ]
We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).

#### FMSPレクチャーズ

13:00-15:05   数理科学研究科棟(駒場) 002号室

Chung-jun Tsai 氏 (National Taiwan University)
Topic on minimal submanifolds (6/6) (ENGLISH)
[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

#### 情報数学セミナー

16:50-18:35   数理科学研究科棟(駒場) 122号室

[ 講演概要 ]

4回構成の本講義ではまず1回目に量子計算の基礎的な枠組みを学んだ後、2,3回目でNISQアルゴリズムとして有望視される量子化学計算と機械学習への量子計算機の応用についてそれぞれ解説し、4回目に量子誤り訂正について解説を行う。

### 2019年10月30日(水)

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室
Andrew Macpherson 氏 (IPMU)
A Tannakian perspective on rigid analytic geometry (English)
[ 講演概要 ]
Raynaud's conception of analytic geometry contends that the category of analytic spaces over a non-Archimedean field is a (suitably "geometric") localisation of the category of formal schemes over the ring of integers at a class of modifications "along the central fibre". Unfortunately, as with all existing presentations of non-Archimedean geometry, this viewpoint is confounded by a proliferation of technical difficulties if one does not impose absolute finiteness conditions on the formal schemes under consideration.

I will argue that by combining Raynaud's idea with a Tannakian perspective which prioritises the module category, we can obtain a reasonable framework for rigid analytic geometry with no absolute finiteness hypotheses whatsoever, but which has descent for finitely presented modules.

#### Lie群論・表現論セミナー

16:30-18:00   数理科学研究科棟(駒場) 128号室
Quentin Labriet 氏 (Reims University)
On holographic transform (English)
[ 講演概要 ]
In representation theory, decomposing the restriction of a given representation $¥pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context,one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $¥pi¥vert_{G'}$ and its irreducible components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform.

I will illustrate this construction by two examples :

- the diagonal case where one considers the restriction problem for $¥pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)¥times SL(2,R)$ and $G'=SL(2,R)$.

- the conformal case for the restriction of a scalar valued holomorphic discrete series representation $¥pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$.

I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.

### 2019年10月29日(火)

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Chung-Jun Tsai 氏 (National Taiwan University)
Strong stability of minimal submanifolds (ENGLISH)
[ 講演概要 ]
It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario. Besides a strong local uniqueness property, a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow. We will also discuss some famous local (complete, non-compact) models. This is based on a joint work with Mu-Tao Wang.

### 2019年10月28日(月)

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室

On Kiyoshi Oka's unpublished papers 1943 (Japanese)
[ 講演概要 ]
いわゆる岡の解決した不分岐リーマン領域に対する3大問題(Oka IX, 1953)は、実はこの未発表論文(VII～XI)で終わっている。　Oka　VII、VIIIで示された連接性、不定域イデアルの理論はこれ等を、分岐リーマン領域へ確立しようとする試みより生まれたことが、この未発表論文から明らかになる。この講演では、この未発表論文で擬凸問題がどのように解決されたかを紹介する。 分岐リーマン領域の場合の擬凸問題は、Fornaessによる反例が与えられたとはいえ、情況は不明で未解決問題として今も残っている（岡の夢）ことにも言及したい。

### 2019年10月25日(金)

#### 談話会・数理科学講演会

15:30-16:30   数理科学研究科棟(駒場) 123号室
Yves Benoist 氏 ( CNRS, Paris-Sud)
Arithmeticity of discrete subgroups (英語)
[ 講演概要 ]
By a theorem of Borel and Harish-Chandra,
an arithmetic group in a semisimple Lie group is a lattice.
Conversely, by a celebrated theorem of Margulis,
in a higher rank semisimple Lie group G
any irreducible lattice is an arithmetic group.

The aim of this lecture is to survey an
arithmeticity criterium for discrete subgroups
which are not assumed to be lattices.
This criterium, obtained with Miquel,
generalizes works of Selberg and Hee Oh
and solves a conjecture of Margulis. It says:
a discrete irreducible Zariski-dense subgroup
of G that intersects cocompactly at least one
horospherical subgroup of G is an arithmetic group.

### 2019年10月24日(木)

#### FMSPレクチャーズ

13:00-15:05   数理科学研究科棟(駒場) 002号室

Chung-jun Tsai 氏 (National Taiwan University)
Topic on minimal submanifolds (5/6) (ENGLISH)
[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

#### 基礎論セミナー

13:30-15:00   数理科学研究科棟(駒場) 156号室

[ 講演概要 ]

$T + \varphi \vdash \psi$ ならば $T \vdash \psi$ が成立することをいう.
1979 年 Guaspari は複数の理論に対して,

その一般的な状況を解明するという問いを残していた.
この問いに対し, 1986年 Bennet は特に2つの理論に対する分析を行い,

この Bennet の結果は任意有限個の理論に拡張可能であることが判明した.

#### 応用解析セミナー

16:00-17:30   数理科学研究科棟(駒場) 128号室
オム　ジュンヨン 氏 (東京大学)

[ 講演概要 ]

### 2019年10月23日(水)

#### Lie群論・表現論セミナー

16:30-18:00   数理科学研究科棟(駒場) 128号室
Clemens Weiske 氏 (Aarhus University)
Symmetry breaking and unitary branching laws for finite-multiplicity pairs of rank one (English)
[ 講演概要 ]
Let (G,G’) be a real reductive finite multiplicity pair of rank one, i.e. a rank one real reductive group G with reductive subgroup G’, such that the space of symmetry breaking operators (SBOs) between all (smooth admissible) irreducible representations is finite dimensional.

We give a classification of SBOs between spherical principal series representations of G and G’, essentially generalizing the results on (O(1,n+1),O(1,n)) of Kobayashi—Speh (2015). Moreover we show how to decompose unitary representations occurring in (not necessarily) spherical principal series representations of G in terms of unitary G’ representations, by making use of the knowledge gathered in the classification of the SBOs and the structure of the open P’orbit in G/P as a homogenous G’-space, where P’ is a minimal parabolic in G’ and P is a minimal parabolic in G. This includes the construction of discrete spectra in the restriction of complementary series representations and unitarizable composition factors.

### 2019年10月21日(月)

#### 複素解析幾何セミナー

10:30-12:00   数理科学研究科棟(駒場) 128号室

Canonical almost complex structures on ACH Einstein manifolds
[ 講演概要 ]
Einstein ACH (asymptotically complex hyperbolic) manifolds are seen as a device that establishes a correspondence between CR geometry on the boundary and Riemannian geometry in “the bulk.” This talk concerns an idea of enriching the geometric structure of the bulk by adding some almost complex structure compatible with the metric. I will introduce an energy functional of almost complex structures and discuss an existence result of critical points when the given ACH Einstein metric is a small perturbation of the Cheng-Yau complete K?hler-Einstein metric on a bounded strictly pseudoconvex domain. The renormalized Chern-Gauss-Bonnet formula is also planned to be discussed.

### 2019年10月17日(木)

#### FMSPレクチャーズ

13:00-15:05   数理科学研究科棟(駒場) 002号室

Chung-jun Tsai 氏 (National Taiwan University)
Topic on minimal submanifolds (4/6) (ENGLISH)
[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

#### 情報数学セミナー

16:50-18:35   数理科学研究科棟(駒場) 122号室
AIと量子計算（暗号理論を含む）を主題とする

CASE+AI時代の５Gデータセンター (Japanese)
[ 講演概要 ]

イバーセキュリティの重要性」の３つがあげられる。本連携講座では、このトレ
ンドに沿って、「AIと量子計算」を主テーマとし暗号理論の最前線にも触れなが
ら議論を深めていきたいと考えている。今回は、その第1回として、情報社会に
おいて情報収集拠点と共に情報発信拠点であるデータセンターの最新事情につい
て述べる。ここで起きている産業のデジタル化の中で、自動車産業が「CASE」
（Connected、Autonomous、Sharing & Services、Electrification）へと大転換
していることを取り上げデータセンターとの関係について述べることとする。

#### 数理人口学・数理生物学セミナー

14:00-16:00   数理科学研究科棟(駒場) 052号室
Merlin C. Koehnke 氏 (Institute of Environmental Systems Research, School of Mathematics/Computer Science, Osnabrueck University) 14:00-15:00
Complex spatiotemporal dynamics in a simple predator-prey model (ENGLISH)
[ 講演概要 ]
A simple reaction-diffusion predator-prey model with Holling type IV functional response
and logistic growth in the prey is considered. The functional response can be interpreted as
a group defense mechanism, i.e., the predation rate decreases with resource density when the
prey density is high enough [1]. Such a mechanism has been described in diverse biological
interactions [2,3]. For instance, high densities of filamentous algae can decrease filtering
rates of filter feeders [4].
The model will be described and linked to plankton dynamics. Nonspatial considerations reveal that the zooplankton may go extinct or coexistence (stationary or oscillatory) between
zoo- and phytoplankton may emerge depending on the choice of parameters. However,
including space, the dynamics are more complex. In particular, spatiotemporal irregular
oscillations can rescue the predator from extinction. These oscillations can be characterized
as spatiotemporal chaos. The results provide a simple mechanism not only for the emergence
of inhomogeneous plankton distributions [5] but also for the occurrence of chaos in plankton communities [6]. Possible underlying mechanisms for this phenomenon will be discussed.
References
[1] Freedman, H. I., Wolkowicz, G. S. (1986). Predator-prey systems with group defence: the
paradox of enrichment revisited. Bulletin of Mathematical Biology, 48(5-6), 493–508.
[2] Tener, J. S.. Muskoxen in Canada: a biological and taxonomic review. Vol. 2. Dept. of Northern
Affairs and National Resources, Canadian Wildlife Service, 1965.
[3] Holmes, J. C. (1972). Modification of intermediate host behaviour by parasites. Behavioural
aspects of parasite transmission.
[4] Davidowicz, P., Gliwicz, Z. M., Gulati, R. D. (1988). Can Daphnia prevent a blue-green algal
bloom in hypertrophic lakes? A laboratory test. Limnologica. Jena, 19(1), 21–26.
[5] Abbott, M., 1993. Phytoplankton patchiness: ecological implicationsand observation methods.
In: Levin, S.A., Powell, T.M., Steele, J.H.(Eds.), Patch Dynamics. Lecture Notes in Biomathematics, vol. 96. Springer-Verlag, Berlin, pp. 37–49.
[6] Beninca, E. et al. (2008). Chaos in a long-term experiment with a plankton community. Nature,
451(7180), 822.
Horst Malchow 氏 (Institute of Environmental Systems Research, School of Mathematics/Computer Science, Osnabrueck University) 15:00-16:00
Functional response of competing populations to environmental variability (ENGLISH)
[ 講演概要 ]
The possible control of competitive invasion by infection of the invader and multiplicative
noise is studied. The basic model is the Lotka-Volterra competition system with emergent
carrying capacities. Several stationary solutions of the non-infected and infected system are
identi ed as well as parameter ranges of bistability. The latter are used for the numerical
study of di usive invasion phenomena. The Fickian di usivities, the infection but in particular the white and colored multiplicative noise are the control parameters. It is shown
that not only competition, possible infection and mobilities are important drivers of the
invasive dynamics but also the noise and especially its color and the functional response of
populations to the emergence of noise.
The variability of the environment can additionally be modelled by applying Fokker-Planck
instead of Fickian di usion. An interesting feature of Fokker-Planck di usion is that for spatially varying di usion coecients the stationary solution is not a homogeneous distribution.
Instead, the densities accumulate in regions of low di usivity and tend to lower levels for
areas of high di usivity. Thus, the stationary distribution of the Fokker-Planck di usion can
be interpreted as a re
ection of di erent levels of habitat quality [1-5]. The latter recalls the
seminal papers on environmental density, cf. [6-7]. Appropriate examples will be presented.
References
[1] Bengfort, M., Malchow, H., Hilker, F.M. (2016). The Fokker-Planck law of diffusion and
pattern formation in heterogeneous media. Journal of Mathematical Biology 73(3), 683-704.
[2] Siekmann, I., Malchow, H. (2016). Fighting enemies and noise: Competition of residents
and invaders in a stochastically fluctuating environment. Mathematical Modelling of Natural
Phenomena 11(5), 120-140.
[3] Siekmann, I., Bengfort, M., Malchow, H. (2017). Coexistence of competitors mediated by
nonlinear noise. European Physical Journal Special Topics 226(9), 2157-2170.
[4] Kohnke, M.C., Malchow, H. (2017). Impact of parameter variability and environmental noise
on the Klausmeier model of vegetation pattern formation. Mathematics 5, 69 (19 pages).
[5] Bengfort, M., Siekmann, I., Malchow, H. (2018). Invasive competition with Fokker-Planck
di usion and noise. Ecological Complexity 34, 134-13.
[6] Morisita, M. (1971). Measuring of habitat value by the \environmental density" method. In:
Spatial patterns and statistical distributions (Patil, C.D., Pielou, E.C., Waters, W.E., eds.),
Statistical Ecology, vol. 1, pp. 379-401. Pennsylvania State University Press, University Park.
[7] N. Shigesada, N., Kawasaki, K., Teramoto, E. (1979). Spatial segregation of interacting species.
Journal of Theoretical Biology 79, 83-99.

### 2019年10月16日(水)

#### 作用素環セミナー

16:45-18:15   数理科学研究科棟(駒場) 126号室

Developments in conformal bootstrap analysis

#### 代数学コロキウム

17:30-18:30   数理科学研究科棟(駒場) 056号室
Liang Xiao 氏 (BICMR, Peking University)
On slopes of modular forms (ENGLISH)
[ 講演概要 ]
In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin's crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.

（本講演は「東京北京パリ数論幾何セミナー」として，インターネットによる東大数理，Morningside Center of Mathematics と IHES の双方向同時中継で行います．今回は北京からの中継です．）

#### 代数幾何学セミナー

15:30-17:00   数理科学研究科棟(駒場) 122号室

Multidimensional continued fraction for Gorenstein cyclic quotient singularity
[ 講演概要 ]
Let G be a finite cyclic subgroup of GL(n,C). Then Cn/G is a cyclic quotient singularity. In the case n = 2, Cn/G possess the unique minimal resolution, and it is obtained by Hirzubruch-Jung continued fraction. In this talk, we show a sufficient condition of existence of crepant desingularization for Gorenstein abelian quotient singularities in all dimensions by using Ashikaga’s continuous fractions. Moreover, as a corollary, we prove that all three dimensional Gorenstein abelian quotient singularities possess a crepant desingularization.

### 2019年10月15日(火)

#### トポロジー火曜セミナー

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Gwénaël Massuyeau 氏 (Université de Bourgogne)
Generalized Dehn twists on surfaces and surgeries in 3-manifolds (ENGLISH)
[ 講演概要 ]
(Joint work with Yusuke Kuno.) Given an oriented surface S and a simple closed curve C in S, the "Dehn twist" along C is the homeomorphism of S defined by "twisting" S around C by a full twist. If the curve C is not simple, this transformation of S does not make sense anymore, but one can consider two possible generalizations: one possibility is to use the homotopy intersection form of S to "simulate" the action of a Dehn twist on the (Malcev completion of) the fundamental group of S; another possibility is to view C as a curve on the top boundary of the cylinder S×[0,1], to push it arbitrarily into the interior so as to obtain, by surgery along the resulting knot, a new 3-manifold. In this talk, we will relate two those possible generalizations of a Dehn twist and we will give explicit formulas using a "symplectic expansion" of the fundamental group of S.

### 2019年10月10日(木)

#### 情報数学セミナー

16:50-18:35   数理科学研究科棟(駒場) 122号室
AIと量子計算（暗号理論を含む）を主題とする

[ 講演概要 ]

#### FMSPレクチャーズ

13:00-15:05   数理科学研究科棟(駒場) 002号室

Chung-jun Tsai 氏 (National Taiwan University)
Topic on minimal submanifolds (3/6) (ENGLISH)
[ 講演概要 ]
The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.
[ 参考URL ]
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

#### 離散数理モデリングセミナー

17:00-18:00   数理科学研究科棟(駒場) 056号室
Boris Konopelchenko 氏 (INFN, sezione di Lecce, Lecce, Italy)
Universal parabolic regularization of gradient catastrophes for the Burgers-Hopf equation and singularities of the plane into plane mappings of parabolic type (English)
[ 講演概要 ]
Two intimately connected topics, namely, regularization of gradient catastrophes of all orders for the Burgers-Hopf equation via the Jordan chain and the singularities of the plane into plane mappings
associated with two-component hydrodynamic type systems of parabolic type are discussed.
It is shown that the regularization of all gradient catastrophes (generic and higher orders) for the Burgers-Hopf equation is achieved by the step by step embedding of the Burgers-Hopf equation into multi-component parabolic systems of quasilinear PDEs with the most degenerate Jordan blocks. Infinite parabolic Jordan chain provides us with the complete regularization. This chain contains Burgers and KdV equations as particular reductions.
It is demonstrated that the singularities of the plane into planes mappings associated with the two-component system of quasilinear PDEs of parabolic type are quite different from those in hyperbolic and elliptic cases. Impediments arising in the application of the original Whitney's approach to such case are discussed. It is shown that flex is the lowest singularity while higher singularities are given by ( k+1,k+2) curves which are of cusp type for k=2n+1, n=1,2,...,. Regularization of these singularities is discussed.

Presentation is based on two publications:

1. B. Konopelchenko and G. Ortenzi, Parabolic regularization of the gradient catastrophes for the Burgers-Hopf equation and Jordan chain, J. Phys. A: Math. Theor., 51 (2018) 275201.

2. B.G. Konopelchenko and G. Ortenzi, On the plane into plane mappings of hydrodynamic type. Parabolic case. Rev. Math. Phys.,32 (2020) 2020006. Online access. arXiv:1904.00901.