## Seminar information archive

Seminar information archive ～12/08｜Today's seminar 12/09 | Future seminars 12/10～

### 2018/12/21

#### Algebraic Geometry Seminar

10:30-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Degenerations of p-adic volume forms (English)

**Mattias Jonsson**(Michigan)Degenerations of p-adic volume forms (English)

[ Abstract ]

Let X be an n-dimensional smooth projective variety over a non-Archimedean local field K. Also fix a regular n-form on X. This data induces a positive measure on the space of K'-rational points for any finite extension K' of K. We describe the asymptotics, as K' runs through towers of finite extensions of K, in terms of Berkovich analytic geometry. This is joint work with Johannes Nicaise.

Let X be an n-dimensional smooth projective variety over a non-Archimedean local field K. Also fix a regular n-form on X. This data induces a positive measure on the space of K'-rational points for any finite extension K' of K. We describe the asymptotics, as K' runs through towers of finite extensions of K, in terms of Berkovich analytic geometry. This is joint work with Johannes Nicaise.

### 2018/12/20

#### Tuesday Seminar on Topology

13:00-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Johnson-type homomorphisms and the LMO functor (ENGLISH)

**Anderson Vera**(Université de Strasbourg)Johnson-type homomorphisms and the LMO functor (ENGLISH)

[ Abstract ]

One of the main objects associated to a surface S is the mapping class group MCG(S). This group plays an important role in the study of 3-manifolds. Reciprocally, the topological invariants of 3-manifolds can be used to obtain interesting representations of MCG(S).

One possible approach to the study of MCG(S) is to consider its action on the fundamental group P of the surface or on some subgroups of P. This way, we can obtain some kind of filtrations of MCG(S) and homomorphisms, called Johnson type homomorphisms, which take values in certain spaces of diagrams. These spaces of diagrams are quotients of the target space of the LMO functor. Hence it is natural to ask what is the relation between the Johnson type homomorphisms and the LMO functor. The answer is well known in the case of the Torelli group and the usual Johnson homomorphisms. In this talk we consider two other different filtrations of MCG(S) introduced by Levine and Habiro-Massuyeau. We show that the respective Johnson homomorphisms can also be deduced from the LMO functor.

One of the main objects associated to a surface S is the mapping class group MCG(S). This group plays an important role in the study of 3-manifolds. Reciprocally, the topological invariants of 3-manifolds can be used to obtain interesting representations of MCG(S).

One possible approach to the study of MCG(S) is to consider its action on the fundamental group P of the surface or on some subgroups of P. This way, we can obtain some kind of filtrations of MCG(S) and homomorphisms, called Johnson type homomorphisms, which take values in certain spaces of diagrams. These spaces of diagrams are quotients of the target space of the LMO functor. Hence it is natural to ask what is the relation between the Johnson type homomorphisms and the LMO functor. The answer is well known in the case of the Torelli group and the usual Johnson homomorphisms. In this talk we consider two other different filtrations of MCG(S) introduced by Levine and Habiro-Massuyeau. We show that the respective Johnson homomorphisms can also be deduced from the LMO functor.

### 2018/12/19

#### Operator Algebra Seminars

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

Polynomial Time Algorithm for Computing N-th Moments of a Self-Adjoint Operator in Algebra Generated by Free Independent Semicircular Elements

**Rei Mizuta**(Univ. Tokyo)Polynomial Time Algorithm for Computing N-th Moments of a Self-Adjoint Operator in Algebra Generated by Free Independent Semicircular Elements

#### Number Theory Seminar

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

Cohomology vanishing for automorphic vector bundles (ENGLISH)

**Jean-Stefan Koskivirta**(University of Tokyo)Cohomology vanishing for automorphic vector bundles (ENGLISH)

[ Abstract ]

A Shimura variety carries naturally a family of vector bundles parametrized by the characters of a maximal torus in the attached group. We want to determine which of these vector bundles are ample, and also show cohomology vanishing results. For this we use generalized Hasse invariants on the stack of G-zips of Moonen-Pink-Wedhorn-Ziegler. It is a group-theoretical counterpart of the Shimura variety and carries a similar family of vector bundles. This is joint work with Y.Brunebarbe, W.Goldring and B.Stroh.

A Shimura variety carries naturally a family of vector bundles parametrized by the characters of a maximal torus in the attached group. We want to determine which of these vector bundles are ample, and also show cohomology vanishing results. For this we use generalized Hasse invariants on the stack of G-zips of Moonen-Pink-Wedhorn-Ziegler. It is a group-theoretical counterpart of the Shimura variety and carries a similar family of vector bundles. This is joint work with Y.Brunebarbe, W.Goldring and B.Stroh.

### 2018/12/18

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Dynamics of singular vortex patches (English)

**In-Jee Jeong**(Korea Institute for Advanced Study (KIAS))Dynamics of singular vortex patches (English)

[ Abstract ]

Vortex patches are solutions to the 2D Euler equations that are given by the characteristic function of a bounded domain that moves with time. It is well-known that if initially the boundary of the domain is smooth, the boundary remains smooth for all time. On the other hand, we consider patches with corner singularities. It turns out that, depending on whether the initial patch satisfies an appropriate rotational symmetry condition or not, the corner structure may propagate for all time or lost immediately. In the rotationally symmetric case, we are able to construct patches with interesting dynamical behavior as time goes to infinity. When the symmetry is absent, we present a simple yet formal evolution equation which describes the dynamics of the boundary. It suggests that the angle cusps instantaneously for $t > 0$.

This is joint work with Tarek Elgindi.

Vortex patches are solutions to the 2D Euler equations that are given by the characteristic function of a bounded domain that moves with time. It is well-known that if initially the boundary of the domain is smooth, the boundary remains smooth for all time. On the other hand, we consider patches with corner singularities. It turns out that, depending on whether the initial patch satisfies an appropriate rotational symmetry condition or not, the corner structure may propagate for all time or lost immediately. In the rotationally symmetric case, we are able to construct patches with interesting dynamical behavior as time goes to infinity. When the symmetry is absent, we present a simple yet formal evolution equation which describes the dynamics of the boundary. It suggests that the angle cusps instantaneously for $t > 0$.

This is joint work with Tarek Elgindi.

#### Tuesday Seminar on Topology

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

Discrete G-spectra and a model for the K(n)-local stable homotopy category (JAPANESE)

**Takeshi Torii**(Okayama University)Discrete G-spectra and a model for the K(n)-local stable homotopy category (JAPANESE)

[ Abstract ]

The K(n)-local stable homotopy categories are building blocks for the stable homotopy category of spectra. In this talk I will construct a model for the K(n)-local stable homotopy category, which explicitly shows the relationship with the Morava E-theory E_n and the stabilizer group G_n. We consider discrete symmetric G-spectra studied by Behrens-Davis for a profinite group G. I will show that the K(n)-local stable homotopy category is realized in the homotopy category of modules in discrete symmetric G_n-spectra over a discrete model of E_n.

The K(n)-local stable homotopy categories are building blocks for the stable homotopy category of spectra. In this talk I will construct a model for the K(n)-local stable homotopy category, which explicitly shows the relationship with the Morava E-theory E_n and the stabilizer group G_n. We consider discrete symmetric G-spectra studied by Behrens-Davis for a profinite group G. I will show that the K(n)-local stable homotopy category is realized in the homotopy category of modules in discrete symmetric G_n-spectra over a discrete model of E_n.

### 2018/12/17

#### Seminar on Geometric Complex Analysis

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

Newton polyhedra and order of contact on real hypersurfaces (JAPANESE)

**Joe Kamimoto**(Kyushu University)Newton polyhedra and order of contact on real hypersurfaces (JAPANESE)

[ Abstract ]

This talk will concern some issues on order of contact on real hypersurfaces, which was introduced by D'Angelo. To be more precise, a sufficient condition for the equality of regular type and singular type is given. This condition is written by using the Newton polyhedron of a defining function. Our result includes earlier known results concerning convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4. Furthermore, under the above condition, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.

The technique of using Newton polyhedra has many significant applications in singularity theory. In particular, this technique has been great success in the study of the Lojasiewicz exponent. Our study about the types is analogous to some works on the Lojasiewicz exponent.

This talk will concern some issues on order of contact on real hypersurfaces, which was introduced by D'Angelo. To be more precise, a sufficient condition for the equality of regular type and singular type is given. This condition is written by using the Newton polyhedron of a defining function. Our result includes earlier known results concerning convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4. Furthermore, under the above condition, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.

The technique of using Newton polyhedra has many significant applications in singularity theory. In particular, this technique has been great success in the study of the Lojasiewicz exponent. Our study about the types is analogous to some works on the Lojasiewicz exponent.

### 2018/12/14

#### Algebraic Geometry Seminar

10:30-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)

On the birationality of quint-canonical systems of irregular threefolds of general type (English)

**Zhi Jiang**(Fudan)On the birationality of quint-canonical systems of irregular threefolds of general type (English)

[ Abstract ]

It is well-known that the quint-canonical map of a surface of general type is birational.

We will show that the same result holds for irregular threefolds of general type. The proof is based on

a careful study of the positivity of the pushforwards of pluricanonical bundles on abelian varieties and Severi

type inequalities. This is a joint work with J.A. Chen, J.Chen, and M.Chen.

It is well-known that the quint-canonical map of a surface of general type is birational.

We will show that the same result holds for irregular threefolds of general type. The proof is based on

a careful study of the positivity of the pushforwards of pluricanonical bundles on abelian varieties and Severi

type inequalities. This is a joint work with J.A. Chen, J.Chen, and M.Chen.

### 2018/12/12

#### Number Theory Seminar

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A higher weight (and automorphic) generalization of the Hermite-Minkowski theorem (ENGLISH)

**Gaëtan Chenevier**(CNRS, Université Paris-Sud)A higher weight (and automorphic) generalization of the Hermite-Minkowski theorem (ENGLISH)

[ Abstract ]

I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose "weights" are in the interval {0,...,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.

I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose "weights" are in the interval {0,...,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.

### 2018/12/11

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Solutions with moving singularities for equations of porous medium type (English)

**Marek Fila**(Comenius University in Bratislava)Solutions with moving singularities for equations of porous medium type (English)

[ Abstract ]

We construct positive solutions of equations of porous medium type with a singularity which moves in time along a prescribed curve and keeps the spatial profile of singular stationary solutions. It turns out that there appears a critical exponent for the existence of such solutions. This is a joint work with Jin Takahashi and Eiji Yanagida.

We construct positive solutions of equations of porous medium type with a singularity which moves in time along a prescribed curve and keeps the spatial profile of singular stationary solutions. It turns out that there appears a critical exponent for the existence of such solutions. This is a joint work with Jin Takahashi and Eiji Yanagida.

#### Tuesday Seminar on Topology

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

On non-singular solutions to the normalized Ricci flow on four-manifolds (JAPANESE)

**Masashi Ishida**(Osaka University)On non-singular solutions to the normalized Ricci flow on four-manifolds (JAPANESE)

[ Abstract ]

A solution to the normalized Ricci flow is called non-singular if the solution exists for all time and the Riemannian curvature tensor is uniformly bounded. In 1999, Richard Hamilton introduced it as an important special class of solutions and proved that the underlying 3-manifold is geometrizable in the sense of Thurston. In this talk, we will discuss properties of 4-dimensional non-singular solutions from a gauge theoretical point of view. In particular, we would like to explain gauge theoretical invariants give rise to obstructions to the existence of 4-dimensional non-singular solutions.

A solution to the normalized Ricci flow is called non-singular if the solution exists for all time and the Riemannian curvature tensor is uniformly bounded. In 1999, Richard Hamilton introduced it as an important special class of solutions and proved that the underlying 3-manifold is geometrizable in the sense of Thurston. In this talk, we will discuss properties of 4-dimensional non-singular solutions from a gauge theoretical point of view. In particular, we would like to explain gauge theoretical invariants give rise to obstructions to the existence of 4-dimensional non-singular solutions.

#### Lie Groups and Representation Theory

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

A Pieri-type formula and a factorization formula for K-k-Schur functions

**Motoki Takigiku**(the University of Tokyo)A Pieri-type formula and a factorization formula for K-k-Schur functions

[ Abstract ]

We give a Pieri-type formula for the sum of K-k-Schur functions \sum_{\mu\le\lambda}g^{(k)}_{\mu} over a principal order ideal of the poset of k-bounded partitions under the strong Bruhat order, which sum we denote by \widetilde{g}^{(k)}_{\lambda}. As an application of this, we also give a k-rectangle factorization formula \widetilde{g}^{(k)}_{R_t\cup\lambda}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda}

where R_t=(t^{k+1-t}), analogous to that of k-Schur functions s^{(k)}_{R_t\cup \lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}.

We give a Pieri-type formula for the sum of K-k-Schur functions \sum_{\mu\le\lambda}g^{(k)}_{\mu} over a principal order ideal of the poset of k-bounded partitions under the strong Bruhat order, which sum we denote by \widetilde{g}^{(k)}_{\lambda}. As an application of this, we also give a k-rectangle factorization formula \widetilde{g}^{(k)}_{R_t\cup\lambda}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda}

where R_t=(t^{k+1-t}), analogous to that of k-Schur functions s^{(k)}_{R_t\cup \lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}.

### 2018/12/10

#### Tokyo Probability Seminar

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

Random polymer models and classical groups (ENGLISH)

https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/zygouras/

**Nikolaos Zygouras**(University of Warwick)Random polymer models and classical groups (ENGLISH)

[ Abstract ]

The relation between polymer models at zero temperature and characters of the general linear group GL_n(R) has been known since the first breakthroughs in the field around the KPZ universality through the works of Johansson, Baik, Rains, Okounkov and others. Later on, geometric liftings of the GL_n(R) characters appeared in the study of positive temperature polymer models in the form of GL_n(R)-Whittaker functions. In this talk I will describe joint works with E. Bisi where we have established that Whittaker functions associated to the orthogonal group SO_{2n+1}(R) can be used to describe laws of positive temperature polymers when their end point is free to lie on a line. Going back to zero temperature, we will also see that characters of other classical groups such as SO_{2n+1}(R); Sp_{2n}(R); SO_{2n}(R) do play a role in describing laws of polymers in various geometries. This occurence might be surprising given the length of time these models have been studied.

[ Reference URL ]The relation between polymer models at zero temperature and characters of the general linear group GL_n(R) has been known since the first breakthroughs in the field around the KPZ universality through the works of Johansson, Baik, Rains, Okounkov and others. Later on, geometric liftings of the GL_n(R) characters appeared in the study of positive temperature polymer models in the form of GL_n(R)-Whittaker functions. In this talk I will describe joint works with E. Bisi where we have established that Whittaker functions associated to the orthogonal group SO_{2n+1}(R) can be used to describe laws of positive temperature polymers when their end point is free to lie on a line. Going back to zero temperature, we will also see that characters of other classical groups such as SO_{2n+1}(R); Sp_{2n}(R); SO_{2n}(R) do play a role in describing laws of polymers in various geometries. This occurence might be surprising given the length of time these models have been studied.

https://warwick.ac.uk/fac/sci/statistics/staff/academic-research/zygouras/

### 2018/12/07

#### Operator Algebra Seminars

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)### 2018/12/06

#### Infinite Analysis Seminar Tokyo

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Integrability and TBA in non-equilibrium emergent hydrodynamics (ENGLISH)

**Francesco Ravanini**(University of Bologna)Integrability and TBA in non-equilibrium emergent hydrodynamics (ENGLISH)

[ Abstract ]

The paradigm of investigating non-equilibrium phenomena by considering stationary states of emergent hydrodynamics has attracted a lot of attention in the last years. Recent proposals of an exact approach in integrable cases, making use of TBA techniques, are presented and discussed.

The paradigm of investigating non-equilibrium phenomena by considering stationary states of emergent hydrodynamics has attracted a lot of attention in the last years. Recent proposals of an exact approach in integrable cases, making use of TBA techniques, are presented and discussed.

#### Operator Algebra Seminars

15:00-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)### 2018/12/05

#### Operator Algebra Seminars

17:15-18:45 Room #126 (Graduate School of Math. Sci. Bldg.)

The Gromov-Hausdorff Propinquity

**Frederic Latremoliere**(Univ. Denver)The Gromov-Hausdorff Propinquity

#### Operator Algebra Seminars

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)#### Seminar on Probability and Statistics

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

Lecture 2:Representation results for the Gaussian processes. Financial applications of fractional Brownian motion

**Yuliia Mishura**(The Taras Shevchenko National University of Kiev)Lecture 2:Representation results for the Gaussian processes. Financial applications of fractional Brownian motion

[ Abstract ]

Arbitrage with fBm: why it appears. How to present any contingent claim via self-financing strategy on the financial market involving fBm. Absence of arbitrage for the mixed models. Fractional -Uhlenbeck and fractional Cox-Ingersoll-Ross processes as the models for stochastic volatility.

Arbitrage with fBm: why it appears. How to present any contingent claim via self-financing strategy on the financial market involving fBm. Absence of arbitrage for the mixed models. Fractional -Uhlenbeck and fractional Cox-Ingersoll-Ross processes as the models for stochastic volatility.

#### Seminar on Probability and Statistics

15:00-17:00 Room #156 (Graduate School of Math. Sci. Bldg.)

Lecture 3:Statistical parameter estimation for the diffusion processes and in the models involving fBm

**Yuliia Mishura**(The Taras Shevchenko National University of Kiev)Lecture 3:Statistical parameter estimation for the diffusion processes and in the models involving fBm

[ Abstract ]

Drift parameter estimation in the standard diffusion model and its strong consistency. Hurst and drift parameter estimation in the models involving fBm and in the mixed models. Asymptotic properties. Estimation of the diffusion parameter.

Drift parameter estimation in the standard diffusion model and its strong consistency. Hurst and drift parameter estimation in the models involving fBm and in the mixed models. Asymptotic properties. Estimation of the diffusion parameter.

### 2018/12/04

#### Tuesday Seminar on Topology

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

Slopes and concordance of links (ENGLISH)

**Vincent Florens**(Université de Pau et des Pays de l'Adour)Slopes and concordance of links (ENGLISH)

[ Abstract ]

We define the slope of a link associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima-Yamasaki η-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. We present several examples and discuss the invariance by concordance. Joint with A. Degtyarev and A. Lecuona.

We define the slope of a link associated to admissible characters on the link group. Away from a certain singular locus, the slope is a rational function which can be regarded as a multivariate generalization of the Kojima-Yamasaki η-function. It is the ratio of two Conway potentials, provided that the latter makes sense; otherwise, it is a new invariant. We present several examples and discuss the invariance by concordance. Joint with A. Degtyarev and A. Lecuona.

#### Operator Algebra Seminars

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)#### Seminar on Probability and Statistics

15:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Lecture 1: Elements of fractional calculus

How to connect the fractional Brownian motion to the Wiener process. Stochastic integration w.r.t. fBm and stochastic differential equations involving fB

**Yuliia Mishura**(The Taras Shevchenko National University of Kiev)Lecture 1: Elements of fractional calculus

How to connect the fractional Brownian motion to the Wiener process. Stochastic integration w.r.t. fBm and stochastic differential equations involving fB

[ Abstract ]

Fractional integrals and fractional derivatives. Wiener and stochastic integration w.r.t. the fractional Brownian motion. Representations of fBm via a Wiener process and vice versa. Elements of the fractional stochastic calculus. Stochastic differential equations involving fBm: existence, uniqueness, properties of the solutions. Simplest models: fractional Ornstein-Uhlenbeck and fractional Cox-Ingersoll-Ross processes.

Fractional integrals and fractional derivatives. Wiener and stochastic integration w.r.t. the fractional Brownian motion. Representations of fBm via a Wiener process and vice versa. Elements of the fractional stochastic calculus. Stochastic differential equations involving fBm: existence, uniqueness, properties of the solutions. Simplest models: fractional Ornstein-Uhlenbeck and fractional Cox-Ingersoll-Ross processes.

### 2018/12/03

#### Operator Algebra Seminars

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

**Reiji Tomatsu**(Hokkaido Univ.)#### Seminar on Geometric Complex Analysis

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

Variational theories of complex analysis of several variables (JAPANESE)

**Genki Hosono**(University of Tokyo)Variational theories of complex analysis of several variables (JAPANESE)

[ Abstract ]

In complex analysis, there are some values and functions which are subharmonic under pseudoconvex variations.

For example, the variation of Robin constant (Yamaguchi) and of Bergman kernels (Maitani-Yamaguchi) were studied.

As a generalization, the curvature positivity of spaces of $L^2$ holomorphic functions is proved by Berndtsson.

These theories are known to have some relations with $L^2$ extension theorems.

In this talk, I will explain known results and discuss the variation problem of the Azukawa pseudometric, which is a generalization of the Robin constant.

In complex analysis, there are some values and functions which are subharmonic under pseudoconvex variations.

For example, the variation of Robin constant (Yamaguchi) and of Bergman kernels (Maitani-Yamaguchi) were studied.

As a generalization, the curvature positivity of spaces of $L^2$ holomorphic functions is proved by Berndtsson.

These theories are known to have some relations with $L^2$ extension theorems.

In this talk, I will explain known results and discuss the variation problem of the Azukawa pseudometric, which is a generalization of the Robin constant.

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