Schedule

September 9, Monday. Talks

Time	(Room)	Speaker
14:00 – 14:40	(12-13)	G.Ishikawa
14:50 – 15:30	(12-13)	M.Takahashi
15:50 – 16:30	(12-13)	A.Glutsyuk
16:45 – 17:25	(13-27)	S.Gusein-Zade

September 10, Tuesday. Problem sessions and discussions

Time	(Room)	Speaker
10:00 – 10:30	(474)	A.Remizov
11:00 – 11:30	(474)	Dm.Tunitsky

September 10, Tuesday. Talks

Time	(Room)	Speaker
14:00 – 14:40	(12-24)	Sh.Izumiya
14:50 – 15:30	(12-24)	I.Bogaevsky
15:50 – 16:30	(12-24)	T.Yamamoto
16:45 – 17:25	(14-02)	S.Voronin

September 11, Wednesday. Cultural program & conference dinner.

September 12, Thursday. Problem sessions and discussions

Time	(Room)	Speaker
10:00 – 10:30	(468)	Japanese 1
11:00 – 11:30	(468)	Japanese 2

September 12, Thursday. Talks

Time	(Room)	Speaker
15:00 – 15:40	(14-04)	Y.Kabata
15:50 – 16:30	(14-04)	A.Remizov
16:40 – 17:20	(14-04)	A.Davydov

September 13, Friday. Talks

Time	(Room)	Speaker
15:00 – 15:40	(16-08)	H.Teramoto
15:50 – 16:30	(16-08)	Dm.Tunitsky
16:40 – 17:20	(16-08)	E.Astashov

Titles and abstracts of the participants' talks are listed below alphabetically.

Evgeny Astashov
Title: On the classification of simple singularities in an equivariant context.
Abstract: Given a pair of real or complex algebraic varieties, it is of interest to obtain a local classification of real- or complex-analytic maps between them up to right equivalence (or up to certain other equivalence relations). When working on such a classification one usually imposes certain restrictions on the ``complexity’’ of the maps considered in terms of codimension, modality, etc.Moreover, when the source and target of the maps are endowed with actions of a group, it is natural to consider equivariant maps, i.e., maps that ``commute’’ with those group actions.
Certain well-known classification results for singular multivariate functions, including those in the equivariant context, belong to V.I. Arnold, D. Siersma, and V.V. Goryunov. In my talk I will mention some more recent results and open questions related to the topic, as well as possible connections with the study ofparameter-dependent matrices, vector fields, and differential equations.
Partially supported by RFBR and JSPS (research project 19-51-50005)

Ilya Bogaevsky
Title: Fronts of singular Legendrian submanifolds
Abstract: A general method reducing fronts of singular Legendrian submanifolds to
normal forms is presented. As an example, we show how to obtain the
known simple normal forms of fronts of cylinders over the semicubic
parabola.
Partially supported by RFBR and JSPS (research project 19-51-50005)

Alexey Davydov
Title: Structural stability of dynamic inequalities on sphere and related problems.
Abstract: The problem of structural stability of generic smooth dynamic inequality with bounded admissible velocities on two dimensional sphere and some related ones are analyzed. We show that the problem is equivalent to the one of structural stability of such an inequality on the plane, when near the innity the inequality either has small time local transitivity property or have no admissible velocities at all. In particular, that implies the structural stability of such simplest dynamic inequalities on two-dimensional sphere.
Partially supported by RFBR and JSPS (research project 19-51-50005)

Alexey Glutsyuk
Title: On integrable billiards, curves with Poritsky property and Liouville net.
Abstract: A caustic of a given convex planar billiard is a curve C such that each line tangent to C is reflected from the billiard boundary to a line tangent to C. The famous Birkhoff Conjecture (late 1920-ths) deals with integrable billiards, that is, billiards whose boundary has an inner neighborhood foliated by closed caustics. The Birkhoff Conjecture states that the only integrable billiards are ellipses.
In the talk we present a brief survey of Birkhoff Conjecture and related results, some of them being motivated by a work of H.Poritsky (1950). To each planar convex closed curve C the string construction associates a family of bigger closed curves G(t) whose billiards have C as a caustic. Each curve G(t) induces a dynamical system on C: given two tangent lines to C through the same point in G(t), the tangency point of the left tangent line is sent to the tangency point of the right tangent line. A curve C is said to have Poritsky property, if the above dynamical system has an invariant length element on C that is the same for all t. Curves with Poritsky property are closely related to integrable billiards. H.Poritsky had shown that the only planar curves with Poritsky property are conics. We extend his result to surfaces of constant curvature. We consider curves with Poritsky property on arbitrary Riemannian surface and obtain the following results: 1) a formula for the invariant length element in terms of the geodesic curvature and the natural length parameter; 2) each germ of C5-smooth curve with Poritsky property is completely determined by its 4-jet.
We also present the following very recent joint result with Sergei Tabachnikov and Ivan Izmestiev. Given a curve C bounding a topological disc on a Riemannian surface. Then the following statements are equivalent:
1) The curve C has Poritsky property.
2) The corresponding family of string constructions has Graves property: smaller curves are caustics for bigger curves.
3) The metric on the concave side of C is Liouville.
Partially supported by RFBR and JSPS (research project 19-51-50005)

Sabir Gusein-Zade
Title: On a version of the Berglund-Hübsch-Henningson duality with non-abelian
symmetry groups.
Abstract: P.Berglund, T.Hübsch and M.Henningson found a method to construct mirror
symmetric Calabi-Yau manifolds using so-called invertible polynomials: see details below. They considered pairs (f,G) consisting of an invertible polynomial f and a (finite abelian) group G of diagonal
symmetries of f. To a pair (f,G) one associates the Berglund-Hübsch-Henningson (BHH) dual pair (f^,G^). There were found some symmetries between dual invertible polynomials and dual pairs not
related directly with the mirror symmetry. (E.g., some of them hold when the corresponding manifolds are not Calabi-Yau. One of them was the so-called equivariant Saito duality as a duality between Burnside rings.
A.Takahashi suggested a conjectural method to find symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. The
equivariant Saito duality was generalized to a case of non-abelian groups. It turns out that the corresponding symmetry holds only under a special condition on the action of the subgroup of the permutation group called here PC (``parity condition''). An inspection of data on Calabi-Yau threefolds obtained from quotients by non-abelian groups (taken from tables computed by X.Yu) shows that the pairs found on the basis of the method of Takahashi have symmetric pairs of Hodge numbers (and thus are hopefully mirror symmetric) if and only if they satisfy PC.
The talk is based on a joint work with W.Ebeling.

Goo Ishikawa
Title: Several problems on singularities of implicit differential equations and related topics.
Abstract: In the joint paper by Davydov, Ishikawa, Izumiya, Sun "Generic singularities of implicit systems of first order differential equations on the plane", we classify generic singularities of family of phase curves of 1st order implicit differential equations up to smooth orbital equivalence.
Based on and derived from the paper, I would like to mention some problems on singularities of 1st, 2nd and 3rd order implicit ordinary differential equations, contact classification of cuspidal edges and swallowtails, singularities of tangent surfaces which are ruled by tangential geodesics to initial curves, singularities of projections of Lagrange and Legendre varieties, and so on.
Supported by JSPS and RFBR under the Japan - Russia Research Cooperative Program.

Shyuichi Izumiya
Title: Singularities of Cauchy horizons in globally hyperbolic space-times.
Abstract: can be found here
Supported by JSPS and RFBR under the Japan - Russia Research Cooperative Program.

Yutaro Kabata
Title: Projections and contact cylindrical surfaces of a surface around a parabolic point.
Abstract: The apparent contour of a smooth surface is considered as the set of
singularities of a projection mapping of the surface, and we
can investigate it in terms of singularity theory. What information of
a surface can we get from the type of the apparent contour or
projection? In this talk, we show some results to the question in terms
of contact cylindrical surfaces. Especially, we deal with surface germs
around parabolic points.
Supported by JSPS and RFBR under the Japan - Russia Research Cooperative Program.

Alexey Remizov
Title: Singularities of geodesic flows in signature varying metrics.
Abstract: I present a brief survey of recent results about singularities of geodesic flows in 2-dimensional signature varying metrics (such metrics are also called pseudo-Riemannian). Generically, such a metric generates on a regular curve. Degeneracy points are singular points of the corresponding geodesic flow, where the standard existence and uniqueness theorem fails. This lead to a curious phenomenon: geodesics cannot pass through a degenerate point in arbitrary tangential directions, but only in certain admissible directions. The talk is devoted to this phenomenon. From the technical viewpoint, the main tool is the method of local normal forms for vector fields with non-isolated singular points.
Partially supported by RFBR and JSPS (research project 19-51-50005)

Masatomo Takahashi
Title: Singularities of envelopes for first order differential equations of Clairaut type.
Abstract: A first order differential equations of Clairaut type has a family of classical solutions, and a singular solution when the contact singular set is not empty. The projection of a singular solution of Clairaut type is an envelope of a family of fronts (Legendre immersions). In this case, the envelope is always a front.
We investigate singular points of envelopes for first order ordinary differential equations, first order partial differential equations and systems of first order partial differential equations of Clairaut type, respectively.
This is a joint work with Saji Kentaro.
Supported by JSPS and RFBR under the Japan - Russia Research Cooperative Program.

Hiroshi Teramoto
Title: Automation algorithms of classification of singularities, current status and future perspectives.
Absract: We have developed automation algorithms of classification of singularities based on Comprehensive Groebner basis systems and Algebraic local cohomology. In this talk we will demonstrate our algorithms in various examples and present our future perspectives.
Supported by JSPS KAKENHI Grant Number JP19K03484 and JST PRESTO Grant
Number JPMJPR16E8.

Dmitry Tunitsky
Title: On contact classification of Monge-Ampère equations.
Abstract: The talk concerns some solved and unsolved problems of contact equivalence and classification of Monge-Ampère equations, that basically ascent to Sophus Lie.
Partially supported by RFBR and JSPS (research project 19-51-50005)

Sergey Voronin
Title: Functional invariants of pleated foliations.
Abstract: In the talk we will discuss existence of functional invariants of foliations near a cusp.
Partially supported by RFBR and JSPS (research project 19-51-50005)