## Future seminars

Seminar information archive ～10/13｜Today's seminar 10/14 | Future seminars 10/15～

### 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.

### 2019/10/16

#### 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.

### 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.

### 2019/10/21

#### 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.

### 2019/10/24

#### FMSP Lectures

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

Topic on minimal submanifolds (5/6) (ENGLISH)

**Chung-jun Tsai**(National Taiwan University)Topic on minimal submanifolds (5/6) (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.

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.

#### Logic

13:30-15:00 Room #156 (Graduate School of Math. Sci. Bldg.)

Generalizations of Bennet's results about partially conservative sentences (JAPANESE)

**Yuya Okawa**(Chiba University)Generalizations of Bennet's results about partially conservative sentences (JAPANESE)

### 2019/10/25

#### Colloquium

15:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Arithmeticity of discrete subgroups (英語)

**Yves Benoist**( CNRS, Paris-Sud)Arithmeticity of discrete subgroups (英語)

[ Abstract ]

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.

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.

### 2019/10/28

#### Seminar on Geometric Complex Analysis

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

**Junjiro Noguchi**(Univ. of Tokyo)### 2019/10/29

#### Tuesday Seminar on Topology

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

Strong stability of minimal submanifolds (ENGLISH)

**Chung-Jun Tsai**(National Taiwan University)Strong stability of minimal submanifolds (ENGLISH)

[ Abstract ]

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.

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.

### 2019/10/31

#### Applied Analysis

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

Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)

**Marius Ghergu**(University College Dublin)Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)

[ Abstract ]

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).

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 Lectures

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

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

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

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

### 2019/11/05

#### Tuesday Seminar on Topology

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

Magnitude homology of geodesic space (JAPANESE)

**Kiyonori Gomi**(Tokyo Institute of Technology)Magnitude homology of geodesic space (JAPANESE)

[ Abstract ]

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.

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.

### 2019/11/08

#### Colloquium

15:30-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

### 2019/11/18

#### Seminar on Geometric Complex Analysis

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

**Ken-ichi Yoshikawa**(Kyoto Univ.)### 2019/11/27

#### Operator Algebra Seminars

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

The homotopy groups of the automorphism groups of Cuntz-Toeplitz algebras

**Taro Sogabe**(Kyoto University)The homotopy groups of the automorphism groups of Cuntz-Toeplitz algebras

### 2019/12/02

#### Seminar on Geometric Complex Analysis

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

**Nobuhiro Honda**(Tokyo Tech.)