## Seminar information archive

Seminar information archive ～07/24｜Today's seminar 07/25 | Future seminars 07/26～

#### FMSP Lectures

13:15-14:15 Room #128 (Graduate School of Math. Sci. Bldg.)

The BV space in variational and evolution problems (2) (ENGLISH)

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

**Piotr Rybka**(the University of Warsaw)The BV space in variational and evolution problems (2) (ENGLISH)

[ Abstract ]

https://www.ms.u-tokyo.ac.jp/kyoumu/docs/20160907.pdf を参照

[ Reference URL ]https://www.ms.u-tokyo.ac.jp/kyoumu/docs/20160907.pdf を参照

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

#### Seminar on Probability and Statistics

10:40-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Wavelet-based methods for high-frequency lead-lag analysis

**Yuta Koike**(Tokyo Metropolitan University, JST CREST)Wavelet-based methods for high-frequency lead-lag analysis

[ Abstract ]

We propose a novel framework to investigate the lead-lag effect between two financial assets. Our framework bridges a gap between continuous-time modeling based on Brownian motion and the existing wavelet methods for lead-lag analysis based on discrete-time models and enables us to analyze the multi-scale structure of lead-lag effects. We also present a statistical methodology for the scale-by-scale analysis of lead-lag effects in the proposed framework and develop an asymptotic theory applicable to a situation including stochastic volatilities and irregular sampling. Finally, we report several numerical experiments to demonstrate how our framework works in practice. This talk is based on a joint work of Prof. Takaki Hayashi (Keio University).

We propose a novel framework to investigate the lead-lag effect between two financial assets. Our framework bridges a gap between continuous-time modeling based on Brownian motion and the existing wavelet methods for lead-lag analysis based on discrete-time models and enables us to analyze the multi-scale structure of lead-lag effects. We also present a statistical methodology for the scale-by-scale analysis of lead-lag effects in the proposed framework and develop an asymptotic theory applicable to a situation including stochastic volatilities and irregular sampling. Finally, we report several numerical experiments to demonstrate how our framework works in practice. This talk is based on a joint work of Prof. Takaki Hayashi (Keio University).

#### Seminar on Probability and Statistics

11:30-12:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Second order fluctuations for zeros of arithmetic random waves

**Giovanni Peccati**(University du Luxembourg)Second order fluctuations for zeros of arithmetic random waves

[ Abstract ]

Originally introduced by Rudnick and Wigman (2007), arithmetic random waves are Gaussian Laplace eigenfunctions on the two-dimensional torus. In this talk, I will describe the high-energy behaviour of the so-called « nodal length » (that, is the volume of the zero set) of such random objects, and show that (quite unexpectedly) it is non-central and non-universal. I will also discuss the connected problem of counting the number of intersections points of independent nodal sets (equivalent to « phase singularities » for complex waves) in the high-energy regime. Both issues are tightly connected to the arithmetic study of lattice points on circles. One key concept in our presentation is that of ‘Berry cancellation phenomenon’ (see M.V. Berry, 2002), for which an explanation in terms of chaos expansions and integration by parts (Green formula) will be provided. Based on joint works (GAFA 2016 & Preprint 2016) with D. Marinucci (Rome Tor Vergata), M. Rossi (Luxembourg) and I. Wigman (King’s College, London), and with F. Dalmao (University of Uruguay), I. Nourdin (Luxembourg) and M. Rossi (Luxembourg).

Originally introduced by Rudnick and Wigman (2007), arithmetic random waves are Gaussian Laplace eigenfunctions on the two-dimensional torus. In this talk, I will describe the high-energy behaviour of the so-called « nodal length » (that, is the volume of the zero set) of such random objects, and show that (quite unexpectedly) it is non-central and non-universal. I will also discuss the connected problem of counting the number of intersections points of independent nodal sets (equivalent to « phase singularities » for complex waves) in the high-energy regime. Both issues are tightly connected to the arithmetic study of lattice points on circles. One key concept in our presentation is that of ‘Berry cancellation phenomenon’ (see M.V. Berry, 2002), for which an explanation in terms of chaos expansions and integration by parts (Green formula) will be provided. Based on joint works (GAFA 2016 & Preprint 2016) with D. Marinucci (Rome Tor Vergata), M. Rossi (Luxembourg) and I. Wigman (King’s College, London), and with F. Dalmao (University of Uruguay), I. Nourdin (Luxembourg) and M. Rossi (Luxembourg).

### 2016/10/31

#### Seminar on Geometric Complex Analysis

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

(JAPANESE)

**Yutaka Ishii**(Kyushu University)(JAPANESE)

#### Numerical Analysis Seminar

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Numerical methods of a quantum hydrodynamic model for semiconductor devices (日本語)

**Shohiro Sho**(Osaka University)Numerical methods of a quantum hydrodynamic model for semiconductor devices (日本語)

#### Operator Algebra Seminars

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

Dual cocycles and equivariant deformation quantization (English)

**Sergey Neshveyev**(Univ. Oslo)Dual cocycles and equivariant deformation quantization (English)

#### Seminar on Probability and Statistics

10:40-11:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Martingale expansion revisited

**Nakahiro Yoshida**(University of Tokyo, Institute of Statistical Mathematics, JST CREST)Martingale expansion revisited

[ Abstract ]

The martingale expansion is revisited in this talk. The martingale expansion for a martingale with mixed normal limit evaluates the tangent of the quadratic variation of the martingale and the torsion of an exponential martingale under the measure transform caused by the random limit of the quadratic variation. The martingale expansion has been applied to the realized volatility, quadratic form of an Ito process, p-variation and the QLA estimators of a volatility parametric model. An interpolation in time was used in martingale expansion. We discuss relation between martingale expansion and recently developed asymptotic expansion of Skorohod integrals by interpolation of distributions (a joint work with D. Nualart).

The martingale expansion is revisited in this talk. The martingale expansion for a martingale with mixed normal limit evaluates the tangent of the quadratic variation of the martingale and the torsion of an exponential martingale under the measure transform caused by the random limit of the quadratic variation. The martingale expansion has been applied to the realized volatility, quadratic form of an Ito process, p-variation and the QLA estimators of a volatility parametric model. An interpolation in time was used in martingale expansion. We discuss relation between martingale expansion and recently developed asymptotic expansion of Skorohod integrals by interpolation of distributions (a joint work with D. Nualart).

#### Seminar on Probability and Statistics

11:30-12:20 Room #123 (Graduate School of Math. Sci. Bldg.)

Error analysis for approximations to one-dimensional SDEs via perturbation method

**Nobuaki Naganuma**(Osaka University)Error analysis for approximations to one-dimensional SDEs via perturbation method

[ Abstract ]

We consider one-dimensional stochastic differential equations driven by fractional Brownian motions and adopt the Euler scheme, the Milstein type scheme and the Crank-Nicholson scheme to approximate solutions to the equations. We introduce perturbation method in order to analyze errors of the schemes. By using this method, we can express the errors in terms of directional derivatives of the solutions explicitly. We obtain asymptotic error distributions of the three schemes by combining the expression of the errors and the fourth moment theorem. This talk is based on a joint work with Prof. Shigeki Aida (Tohoku University).

We consider one-dimensional stochastic differential equations driven by fractional Brownian motions and adopt the Euler scheme, the Milstein type scheme and the Crank-Nicholson scheme to approximate solutions to the equations. We introduce perturbation method in order to analyze errors of the schemes. By using this method, we can express the errors in terms of directional derivatives of the solutions explicitly. We obtain asymptotic error distributions of the three schemes by combining the expression of the errors and the fourth moment theorem. This talk is based on a joint work with Prof. Shigeki Aida (Tohoku University).

#### Seminar on Probability and Statistics

13:50-14:40 Room #123 (Graduate School of Math. Sci. Bldg.)

Characterization of the convergence in total variation by Stein's method and Malliavin calculus

**Seiichiro Kusuoka**(Okayama University)Characterization of the convergence in total variation by Stein's method and Malliavin calculus

[ Abstract ]

Recently, convergence in distributions and estimates of distances between distributions are studied by means of Stein's equation and Malliavin calculus. However, in known results, the target distributions of the convergence were some specific distributions. In this talk, we extend the target distributions to invariant probability measures of diffusion processes. Precisely speaking, we prepare Stein's equation with respect to invariant measures of diffusion processes and consider the characterization of the convergence to the invariant measure in total variation by applying Malliavin calculus. This is a joint work with Ciprian Tudor.

Recently, convergence in distributions and estimates of distances between distributions are studied by means of Stein's equation and Malliavin calculus. However, in known results, the target distributions of the convergence were some specific distributions. In this talk, we extend the target distributions to invariant probability measures of diffusion processes. Precisely speaking, we prepare Stein's equation with respect to invariant measures of diffusion processes and consider the characterization of the convergence to the invariant measure in total variation by applying Malliavin calculus. This is a joint work with Ciprian Tudor.

#### Seminar on Probability and Statistics

14:50-15:40 Room #123 (Graduate School of Math. Sci. Bldg.)

Parameter estimation for diffusion processes with high-frequency observations

**Teppei Ogihara**(The Institute of Statistical Mathematics, JST PRESTO, JST CREST)Parameter estimation for diffusion processes with high-frequency observations

[ Abstract ]

We study statistical inference for security prices modeled by diffusion processes with high-frequency observations. In particular, we focus on two specific problems on analysis of high-frequency data, that is, nonsynchronous observations and the presence of observation noise called market microstructure noise. We construct a maximum-likelihood-type estimator of parameters, and study their asymptotic mixed normality. We also discuss on asymptotic efficiency of estimators.

We study statistical inference for security prices modeled by diffusion processes with high-frequency observations. In particular, we focus on two specific problems on analysis of high-frequency data, that is, nonsynchronous observations and the presence of observation noise called market microstructure noise. We construct a maximum-likelihood-type estimator of parameters, and study their asymptotic mixed normality. We also discuss on asymptotic efficiency of estimators.

#### Seminar on Probability and Statistics

15:40-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Markov chain Monte Carlo for high-dimensional target distribution

**Kengo Kamatani**(Osaka University, JST CREST)Markov chain Monte Carlo for high-dimensional target distribution

[ Abstract ]

The Markov chain Monte Carlo (MCMC) algorithms are widely used to evaluate complicated integrals in Bayesian Statistics. Since the method is not free from the curse of dimensionality, it is important to quantify the effect of the dimensionality and establish an optimal MCMC strategy in high-dimension. In this talk, I will review some high-dimensional asymptotics of MCMC initiated by Roberts et. al. 97, and explain some asymptotic properties of the MpCN algorithm. I will also mention some connection to Stein-Malliavin techniques.

The Markov chain Monte Carlo (MCMC) algorithms are widely used to evaluate complicated integrals in Bayesian Statistics. Since the method is not free from the curse of dimensionality, it is important to quantify the effect of the dimensionality and establish an optimal MCMC strategy in high-dimension. In this talk, I will review some high-dimensional asymptotics of MCMC initiated by Roberts et. al. 97, and explain some asymptotic properties of the MpCN algorithm. I will also mention some connection to Stein-Malliavin techniques.

#### Seminar on Probability and Statistics

16:50-17:40 Room #123 (Graduate School of Math. Sci. Bldg.)

New Functionals inequalities via Stein's discrepancies

**Giovanni Peccati**(Universite du Luxembourg)New Functionals inequalities via Stein's discrepancies

#### Seminar on Probability and Statistics

17:40-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Stochastic geometry and Malliavin calculus on configuration spaces

**Giovanni Peccati**(Université du Luxembourg)Stochastic geometry and Malliavin calculus on configuration spaces

[ Abstract ]

I will present some recent advances in the domain of quantitative limit theorems for geometric Poisson functionals, associated e.g. with random geometric graphs and random tessellations, obtained by means of Malliavin calculus techniques. One of our main results consists in a general (optimal) Berry-Esseen bound for stabilizing functionals, based on Stein’s method, iterated Poincaré inequalities and a variant of Mehler’s formula. Based on several joint works with S. Bourguin, R. Lachièze-Rey, G. Last and M. Schulte, as well as on the recent monograph that I co-edited with M. Reitzner.

I will present some recent advances in the domain of quantitative limit theorems for geometric Poisson functionals, associated e.g. with random geometric graphs and random tessellations, obtained by means of Malliavin calculus techniques. One of our main results consists in a general (optimal) Berry-Esseen bound for stabilizing functionals, based on Stein’s method, iterated Poincaré inequalities and a variant of Mehler’s formula. Based on several joint works with S. Bourguin, R. Lachièze-Rey, G. Last and M. Schulte, as well as on the recent monograph that I co-edited with M. Reitzner.

### 2016/10/28

#### Mathematical Biology Seminar

13:30-14:30 Room #126 (Graduate School of Math. Sci. Bldg.)

When is the allergen immunotherapy effective? (JAPANESE)

**Akane Hara**(Graduate School of Systems Life Sciences, Kyushu University)When is the allergen immunotherapy effective? (JAPANESE)

[ Abstract ]

Allergen immunotherapy is a method to treat allergic symptoms, for example rhinitis and sneezing in Japanese cedar pollen allergy (JCPA). In the immunotherapy of JCPA, patients take in a small amount of pollen over several years, which suppress severe allergic symptoms when exposed to a large amount of environmental pollen after the therapy. We develop a simple mathematical model to identify the condition for successful therapy. We consider the dynamics of type 2 T helper cells (Th2) and regulatory T cells (Treg) and both of them are differentiated from naive T cells. We assume that Treg cells have a much longer lifespan than Th2 cells, which makes Treg cells accumulate over many years during the therapy.

We regard that the therapy is successful if (1) without therapy the patient develops allergic symptoms upon exposure to the environmental pollen, (2) the patient does not develop allergic symptoms caused by the therapy itself, and (3) with therapy the patient does not develop symptoms upon exposure. We find the conditions of each parameter for successful therapy. We also find that the therapy of linearly increasing dose is able to reduce the risk of allergic symptoms caused by the therapy itself, rather than constant dose. We would like to consider application of this model to other kind of allergy, such as food allergy.

Allergen immunotherapy is a method to treat allergic symptoms, for example rhinitis and sneezing in Japanese cedar pollen allergy (JCPA). In the immunotherapy of JCPA, patients take in a small amount of pollen over several years, which suppress severe allergic symptoms when exposed to a large amount of environmental pollen after the therapy. We develop a simple mathematical model to identify the condition for successful therapy. We consider the dynamics of type 2 T helper cells (Th2) and regulatory T cells (Treg) and both of them are differentiated from naive T cells. We assume that Treg cells have a much longer lifespan than Th2 cells, which makes Treg cells accumulate over many years during the therapy.

We regard that the therapy is successful if (1) without therapy the patient develops allergic symptoms upon exposure to the environmental pollen, (2) the patient does not develop allergic symptoms caused by the therapy itself, and (3) with therapy the patient does not develop symptoms upon exposure. We find the conditions of each parameter for successful therapy. We also find that the therapy of linearly increasing dose is able to reduce the risk of allergic symptoms caused by the therapy itself, rather than constant dose. We would like to consider application of this model to other kind of allergy, such as food allergy.

### 2016/10/27

#### Applied Analysis

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Sign-changing solutions of the nonlinear heat equation with positive initial value

(ENGLISH)

**Fred Weissler**(Universite Paris 13)Sign-changing solutions of the nonlinear heat equation with positive initial value

(ENGLISH)

[ Abstract ]

#### Infinite Analysis Seminar Tokyo

15:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Non-unitary highest-weight modules over the $N=2$ superconformal algebra (JAPANESE)

**Ryou Sato**(Graduate School of Mathematical Scineces, The University of Tokyo)Non-unitary highest-weight modules over the $N=2$ superconformal algebra (JAPANESE)

[ Abstract ]

The $N=2$ superconformal algebra is a generalization of the Virasoro algebra having the super symmetry.

The character formulas associated with the unitary highest weight representations

are expressed in terms of the classical theta functions, and have the remarkable

modular invariance. Based on the method of the $W$-algebras,

Kac and Wakimoto, on the other hand, showed that the

characters for a certain class of non-unitary highest weight representations

can be written in terms of the mock theta functions associated with the affine ${sl}_{2|1}$.

Then they found a way to identify these formulas with

real analytic modular forms by using the correction terms given by Zwegers.

In this seminar, we explain a way to construct the above mentioned

non-unitary representations from the representations of the algebra affine ${sl}_{2}$,

based on the Kazama-Suzuki coset construction, namely not from the $W$-algebra method.

We also investigate the relations between the mock theta functions and the ordinary

theta functions, appearing in this method.

The $N=2$ superconformal algebra is a generalization of the Virasoro algebra having the super symmetry.

The character formulas associated with the unitary highest weight representations

are expressed in terms of the classical theta functions, and have the remarkable

modular invariance. Based on the method of the $W$-algebras,

Kac and Wakimoto, on the other hand, showed that the

characters for a certain class of non-unitary highest weight representations

can be written in terms of the mock theta functions associated with the affine ${sl}_{2|1}$.

Then they found a way to identify these formulas with

real analytic modular forms by using the correction terms given by Zwegers.

In this seminar, we explain a way to construct the above mentioned

non-unitary representations from the representations of the algebra affine ${sl}_{2}$,

based on the Kazama-Suzuki coset construction, namely not from the $W$-algebra method.

We also investigate the relations between the mock theta functions and the ordinary

theta functions, appearing in this method.

### 2016/10/25

#### Algebraic Geometry Seminar

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

Q-Gorenstein deformation theory and it applications to algebraic surfaces (English)

**Yongnam Lee**(KAIST/RIMS)Q-Gorenstein deformation theory and it applications to algebraic surfaces (English)

[ Abstract ]

The notion of Q-Gorenstein variety is important for the minimal model theory and the compact moduli theory of algebraic varieties in characteristic 0. Also Q-Gorenstein deformation theory can be applied to construct (simply connected) surfaces of general type with geometric genus 0 over the field of any characteristic. In this talk, some applications of Q-Gorenstein deformation theory to algebraic surfaces and some interesting examples related to Q-Gorenstein morphisms will be presented.

The notion of Q-Gorenstein variety is important for the minimal model theory and the compact moduli theory of algebraic varieties in characteristic 0. Also Q-Gorenstein deformation theory can be applied to construct (simply connected) surfaces of general type with geometric genus 0 over the field of any characteristic. In this talk, some applications of Q-Gorenstein deformation theory to algebraic surfaces and some interesting examples related to Q-Gorenstein morphisms will be presented.

#### Tuesday Seminar of Analysis

16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)

Asymptotics for the integrable discrete nonlinear Schr\"odinger equation (JAPANESE)

**Hideshi YAMANE**(Department of Mathematical Sciences, School of Science, Kwansei Gakuin University)Asymptotics for the integrable discrete nonlinear Schr\"odinger equation (JAPANESE)

### 2016/10/24

#### Seminar on Geometric Complex Analysis

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

Structure and equivalence of a class of tube domains with solvable groups of automorphisms (JAPANESE)

**Satoru Shimizu**(Tohoku University)Structure and equivalence of a class of tube domains with solvable groups of automorphisms (JAPANESE)

[ Abstract ]

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

In the study of the holomorphic equivalence problem for tube domains, it is fundamental to investigate tube domains with polynomial infinitesimal automorphisms. To apply Lie group theory to the holomorphic equivalence problem for such tube domains $T_\Omega$, investigating certain solvable subalgebras of $\frak g(T_{\Omega})$ plays an important role, where $\frak g(T_{\Omega})$ is the Lie algebra of all complete polynomial vector fields on $T_\Omega$. Related to this theme, we discuss the structure and equivalence of a class of tube domains with solvable groups of automorphisms. Besides, we give a concrete example of a tube domain whose automorphism group is solvable and contains nonaffine automorphisms.

#### Operator Algebra Seminars

16:45-18:15 Room #117 (Graduate School of Math. Sci. Bldg.)

Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

**Yusuke Isono**(RIMS, Kyoto Univ.)Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors

### 2016/10/18

#### Tuesday Seminar on Topology

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

Conformal field theory for extended W-algebras (JAPANESE)

**Yoshitake Hashimoto**(Tokyo City University)Conformal field theory for extended W-algebras (JAPANESE)

[ Abstract ]

This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of

"infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:

1. "factorization" in conformal field theory,

2. tensor structure of the category of M-modules and "module-bimodule correspondence".

This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of

"infinitesimal deformation of parameter" and Jack symmetric polynomials.

In this talk I will discuss the following subjects:

1. "factorization" in conformal field theory,

2. tensor structure of the category of M-modules and "module-bimodule correspondence".

### 2016/10/17

#### Seminar on Geometric Complex Analysis

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

(JAPANESE)

**Takaaki Nomura**(Kyushu University)(JAPANESE)

#### Operator Algebra Seminars

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

Unitarizability, Maurey-Nikishin factorization and Polish groups of finite type (English)

**Hiroshi Ando**(Chiba Univ.)Unitarizability, Maurey-Nikishin factorization and Polish groups of finite type (English)

### 2016/10/12

#### Number Theory Seminar

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

Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao) (English)

**Uwe Jannsen**(Universität Regensburg, The University of Tokyo)Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao) (English)

[ Abstract ]

We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.

We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.

### 2016/10/11

#### PDE Real Analysis Seminar

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

Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)

**Nam Quang Le**(Indiana University)Global solutions to the second boundary value problem of the prescribed affine mean curvature and Abreu's equations (English)

[ Abstract ]

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-Weinkove and the author solved this global problem under some restrictions on the sign or integrability of the affine mean curvature. In this talk, we explain how to remove these restrictions and obtain global solutions under optimal integrability conditions on the affine mean curvature. Our analysis also covers the case of Abreu's equation arising in complex geometry.

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