## Seminar information archive

Seminar information archive ～11/05｜Today's seminar 11/06 | Future seminars 11/07～

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

### 2019/10/23

#### Lie Groups and Representation Theory

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Symmetry breaking and unitary branching laws for finite-multiplicity pairs of rank one (English)

**Clemens Weiske**(Aarhus University)Symmetry breaking and unitary branching laws for finite-multiplicity pairs of rank one (English)

[ Abstract ]

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.

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.

### 2019/10/21

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Canonical almost complex structures on ACH Einstein manifolds

**Yoshihiko Matsumoto**(Osaka Univ.)Canonical almost complex structures on ACH Einstein manifolds

[ Abstract ]

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.

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.

### 2019/10/17

#### FMSP Lectures

13:00-15:05 Room #002 (Graduate School of Math. Sci. Bldg.)

Topic on minimal submanifolds (4/6) (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

**Chung-jun Tsai**(National Taiwan University)Topic on minimal submanifolds (4/6) (ENGLISH)

[ Abstract ]

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.

[ Reference URL ]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.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

#### Information Mathematics Seminar

16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)

5G Data Center in CASE+AI Era (Japanese)

**Hiroshi Fujiwara**(BroadBand Tower, Inc.)5G Data Center in CASE+AI Era (Japanese)

[ Abstract ]

Explanation of the recent situation of Data Center, and of its relation to

Connected、Autonomous、Sharing & Services、Electrification.

Explanation of the recent situation of Data Center, and of its relation to

Connected、Autonomous、Sharing & Services、Electrification.

#### Mathematical Biology Seminar

14:00-16:00 Room #052 (Graduate School of Math. Sci. Bldg.)

Complex spatiotemporal dynamics in a simple predator-prey model (ENGLISH)

Functional response of competing populations to environmental variability (ENGLISH)

**Merlin C. Koehnke**(Institute of Environmental Systems Research, School of Mathematics/Computer Science, Osnabrueck University) 14:00-15:00Complex spatiotemporal dynamics in a simple predator-prey model (ENGLISH)

[ Abstract ]

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] Beninc`a, E. et al. (2008). Chaos in a long-term experiment with a plankton community. Nature,

451(7180), 822.

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] Beninc`a, E. et al. (2008). Chaos in a long-term experiment with a plankton community. Nature,

451(7180), 822.

**Horst Malchow**(Institute of Environmental Systems Research, School of Mathematics/Computer Science, Osnabrueck University) 15:00-16:00Functional response of competing populations to environmental variability (ENGLISH)

[ Abstract ]

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

identied as well as parameter ranges of bistability. The latter are used for the numerical

study of diusive invasion phenomena. The Fickian diusivities, 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 diusion. An interesting feature of Fokker-Planck diusion is that for spatially varying diusion coecients the stationary solution is not a homogeneous distribution.

Instead, the densities accumulate in regions of low diusivity and tend to lower levels for

areas of high diusivity. Thus, the stationary distribution of the Fokker-Planck diusion can

be interpreted as a re

ection of dierent 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] Kohnke, 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

diusion 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.

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

identied as well as parameter ranges of bistability. The latter are used for the numerical

study of diusive invasion phenomena. The Fickian diusivities, 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 diusion. An interesting feature of Fokker-Planck diusion is that for spatially varying diusion coecients the stationary solution is not a homogeneous distribution.

Instead, the densities accumulate in regions of low diusivity and tend to lower levels for

areas of high diusivity. Thus, the stationary distribution of the Fokker-Planck diusion can

be interpreted as a re

ection of dierent 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] Kohnke, 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

diusion 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.

### 2019/10/16

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Developments in conformal bootstrap analysis

**Shinobu Hikami**(OIST)Developments in conformal bootstrap analysis

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On slopes of modular forms (ENGLISH)

**Liang Xiao**(BICMR, Peking University)On slopes of modular forms (ENGLISH)

[ Abstract ]

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.

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.

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Multidimensional continued fraction for Gorenstein cyclic quotient singularity

**Yusuke Sato**(University of Tokyo/ IPMU)Multidimensional continued fraction for Gorenstein cyclic quotient singularity

[ Abstract ]

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.

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.

### 2019/10/15

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Generalized Dehn twists on surfaces and surgeries in 3-manifolds (ENGLISH)

**Gwénaël Massuyeau**(Université de Bourgogne)Generalized Dehn twists on surfaces and surgeries in 3-manifolds (ENGLISH)

[ Abstract ]

(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.

(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.

### 2019/10/10

#### Information Mathematics Seminar

16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)

Post-Quantum Cryptography from Isogenies (Japanese)

**Katsuyuki Takashima**(Mitsubishi Electric Co./Kyushu Univ.)Post-Quantum Cryptography from Isogenies (Japanese)

[ Abstract ]

Explanation of the isogeny-based post-quantum cryptography

Explanation of the isogeny-based post-quantum cryptography

#### FMSP Lectures

13:00-15:05 Room #002 (Graduate School of Math. Sci. Bldg.)

Topic on minimal submanifolds (3/6) (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

**Chung-jun Tsai**(National Taiwan University)Topic on minimal submanifolds (3/6) (ENGLISH)

[ Abstract ]

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.

[ Reference URL ]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.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

#### Discrete mathematical modelling seminar

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Universal parabolic regularization of gradient catastrophes for the Burgers-Hopf equation and singularities of the plane into plane mappings of parabolic type (English)

**Boris Konopelchenko**(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)

[ Abstract ]

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.

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.

### 2019/10/09

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Relative K-homology group of $C^*$-algebras and almost flat vector bundle on manifolds with boundary

**Yosuke Kubota**(Riken)Relative K-homology group of $C^*$-algebras and almost flat vector bundle on manifolds with boundary

#### Number Theory Seminar

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Twisted doubling integrals for classical groups (ENGLISH)

**Yuanqing Cai**(Kyoto University)Twisted doubling integrals for classical groups (ENGLISH)

[ Abstract ]

In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.

In this talk, we present a family of Rankin-Selberg integrals (the twisted doubling method, in joint work with Friedberg, Ginzburg, and Kaplan) for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. This can be viewed as a generalization of the doubling integrals of Piatetski-Shapiro and Rallis. Time permitting, we will discuss the twisted doubling integrals for Brylinski-Deligne covers of classical groups.

In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.

In this talk, we present a family of Rankin-Selberg integrals (the twisted doubling method, in joint work with Friedberg, Ginzburg, and Kaplan) for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. This can be viewed as a generalization of the doubling integrals of Piatetski-Shapiro and Rallis. Time permitting, we will discuss the twisted doubling integrals for Brylinski-Deligne covers of classical groups.

### 2019/10/08

#### Tuesday Seminar on Topology

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

How can we generalize hyperbolic dynamics to group actions? (JAPANESE)

**Masaki Tsukamoto**(Kyushu University)How can we generalize hyperbolic dynamics to group actions? (JAPANESE)

[ Abstract ]

Hyperbolicity is one of the most fundamental concepts in the study of dynamical systems. It provides rich (expansive and positive entropy) and yet controllable (stable and having some nice measures) dynamical systems. Then, can we generalize this to group actions?

A naive approach seems difficult. For example, suppose $Z^2$ smoothly acts on a finite dimensional compact manifold. Then it is easy to see that its entropy is zero. In other words, there is no rich $Z^2$-actions in the ordinary finite dimensional world. So we must go to infinite dimension. But what kind structure can we expect in the infinite dimensional world?

The purpose of this talk is to explain that mean dimension seems to play an important role in such a research direction. In particular, we explain the following principle :

If $Z^k$ acts on a space $X$ with some hyperbolicity, then we can control the mean dimension of the sub-action of any rank $(k-1)$ subgroup $G$ of $Z^k$.

This talk is based on the joint works with Tom Meyerovitch and Mao Shinoda.

Hyperbolicity is one of the most fundamental concepts in the study of dynamical systems. It provides rich (expansive and positive entropy) and yet controllable (stable and having some nice measures) dynamical systems. Then, can we generalize this to group actions?

A naive approach seems difficult. For example, suppose $Z^2$ smoothly acts on a finite dimensional compact manifold. Then it is easy to see that its entropy is zero. In other words, there is no rich $Z^2$-actions in the ordinary finite dimensional world. So we must go to infinite dimension. But what kind structure can we expect in the infinite dimensional world?

The purpose of this talk is to explain that mean dimension seems to play an important role in such a research direction. In particular, we explain the following principle :

If $Z^k$ acts on a space $X$ with some hyperbolicity, then we can control the mean dimension of the sub-action of any rank $(k-1)$ subgroup $G$ of $Z^k$.

This talk is based on the joint works with Tom Meyerovitch and Mao Shinoda.

### 2019/10/07

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Cohomology of vector bundles and non-pluriharmonic loci (Japanese)

**Yusaku Chiba**(Ochanomizu Univ.)Cohomology of vector bundles and non-pluriharmonic loci (Japanese)

[ Abstract ]

We study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem. We especially study the examples of non-pluriharmonic loci in smooth toric varieties. I would like to explain the relation of non-pluriharmonic loci and polytopes.

We study cohomology groups of vector bundles on neighborhoods of a non-pluriharmonic locus in Stein manifolds and in projective manifolds. By using our results, we show variants of the Lefschetz hyperplane theorem. We especially study the examples of non-pluriharmonic loci in smooth toric varieties. I would like to explain the relation of non-pluriharmonic loci and polytopes.

### 2019/10/04

#### Discrete mathematical modelling seminar

17:30-18:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations (English)

**Anton Dzhamay**(University of Northern Colorado)Recurrence coefficients for discrete orthogonal polynomials with hypergeometric weight and discrete Painlevé equations (English)

[ Abstract ]

Over the last decade it became clear that the role of discrete Painlevé equations in applications has been steadily growing. Thus, the question of recognizing a certain non-autonomous recurrence as a discrete Painlevé equation and understanding its position in Sakai’s classification scheme, recognizing whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such example, becomes one of the central ones. Fortunately, Sakai’s geometric theory provides an almost algorithmic procedure of answering this question.

In this work we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for orthogonal polynomials can be expressed in terms of solutions of some discrete Painlevé equation. In this work we study orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form.

This is joint work with Galina Filipuk (University of Warsaw, Poland) and Alexander Stokes (University College, London, UK)

Over the last decade it became clear that the role of discrete Painlevé equations in applications has been steadily growing. Thus, the question of recognizing a certain non-autonomous recurrence as a discrete Painlevé equation and understanding its position in Sakai’s classification scheme, recognizing whether it is equivalent to some known (model) example, and especially finding an explicit change of coordinates transforming it to such example, becomes one of the central ones. Fortunately, Sakai’s geometric theory provides an almost algorithmic procedure of answering this question.

In this work we illustrate this procedure by studying an example coming from the theory of discrete orthogonal polynomials. There are many connections between orthogonal polynomials and Painlevé equations, both differential and discrete. In particular, often the coefficients of three-term recurrence relations for orthogonal polynomials can be expressed in terms of solutions of some discrete Painlevé equation. In this work we study orthogonal polynomials with general hypergeometric weight and show that their recurrence coefficients satisfy, after some change of variables, the standard discrete Painlevé-V equation. We also provide an explicit change of variables transforming this equation to the standard form.

This is joint work with Galina Filipuk (University of Warsaw, Poland) and Alexander Stokes (University College, London, UK)

### 2019/10/03

#### FMSP Lectures

13:00-15:05 Room #002 (Graduate School of Math. Sci. Bldg.)

Topic on minimal submanifolds (2/6) (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

**Chung-jun Tsai**(National Taiwan University)Topic on minimal submanifolds (2/6) (ENGLISH)

[ Abstract ]

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.

[ Reference URL ]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.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

### 2019/10/02

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Subfactors, K-theory and Equivariant Higher Twists (English)

**David E. Evans**(Cardiff University)Subfactors, K-theory and Equivariant Higher Twists (English)

### 2019/10/01

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Quantized SL(2) representations of knot groups (JAPANESE)

**Jun Murakami**(Waseda University)Quantized SL(2) representations of knot groups (JAPANESE)

[ Abstract ]

Let K be a knot and G be a group. The representation space of K for the group G means the space of homomorphisms from the knot group to G and is defined by using the group ring C[G], where C[G] is the ring of functions on G and has a commutative Hopf algebra structure. This construction can be generalized to any commutative Hopf algebras.

In this talk, we extend this construction to any braided Hopf algebras with braided commutativity. A typical example is BSL(2), which is the braided SL(2) introduced by S. Majid. Applying the above construction to BSL(2), we get the space of BSL(2) representations, which provides a quantization of SL(2) representations of a knot. This is joint work with Roloand van der Veen.

Let K be a knot and G be a group. The representation space of K for the group G means the space of homomorphisms from the knot group to G and is defined by using the group ring C[G], where C[G] is the ring of functions on G and has a commutative Hopf algebra structure. This construction can be generalized to any commutative Hopf algebras.

In this talk, we extend this construction to any braided Hopf algebras with braided commutativity. A typical example is BSL(2), which is the braided SL(2) introduced by S. Majid. Applying the above construction to BSL(2), we get the space of BSL(2) representations, which provides a quantization of SL(2) representations of a knot. This is joint work with Roloand van der Veen.

### 2019/09/30

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Rigidity of the directional moduli on pseudoconvex domains fibered by open Riemann surfaces

**Sachiko Hamano**(Osaka City Univ.)Rigidity of the directional moduli on pseudoconvex domains fibered by open Riemann surfaces

[ Abstract ]

G. Schmieder-M. Shiba observed conformal embeddings of a fixed open Riemann surface of positive finite genus into closed Riemann surfaces of the same genus, and they showed the range of each diagonal element of the period matrices. Now we shall consider a smooth deformation of open Riemann surfaces with a complex parameter. In this talk, we show the rigidity of directional moduli induced by elements of the period matrices on pseudoconvex domains fibered by open Riemann surfaces of the same topological type.

G. Schmieder-M. Shiba observed conformal embeddings of a fixed open Riemann surface of positive finite genus into closed Riemann surfaces of the same genus, and they showed the range of each diagonal element of the period matrices. Now we shall consider a smooth deformation of open Riemann surfaces with a complex parameter. In this talk, we show the rigidity of directional moduli induced by elements of the period matrices on pseudoconvex domains fibered by open Riemann surfaces of the same topological type.

### 2019/09/26

#### FMSP Lectures

13:00-15:05 Room #002 (Graduate School of Math. Sci. Bldg.)

Topic on minimal submanifolds (1/6) (ENGLISH)

[ Abstract ]

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.
[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

**Chung-jun Tsai**(National Taiwan University)Topic on minimal submanifolds (1/6) (ENGLISH)

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.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Tsai.pdf

### 2019/09/25

#### Numerical Analysis Seminar

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Finite volume method for the Keller-Segel system (Japanese)

**Guanyu Zhou**(Tokyo University of Science)Finite volume method for the Keller-Segel system (Japanese)

### 2019/08/20

#### thesis presentations

13:45-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

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