## Seminar information archive

Seminar information archive ～01/18｜Today's seminar 01/19 | Future seminars 01/20～

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Age-structured epidemic model with infection during transportation (JAPANESE)

**Yukihiko Nakata**(Graduate School of Mathematical Sciences, University of Tokyo)Age-structured epidemic model with infection during transportation (JAPANESE)

### 2014/04/22

#### Tuesday Seminar of Analysis

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

Bounded small solutions to a chemotaxis system with

non-diffusive chemical (JAPANESE)

**Yohei Tsutsui**(The University of Tokyo)Bounded small solutions to a chemotaxis system with

non-diffusive chemical (JAPANESE)

[ Abstract ]

We consider a chemotaxis system with a logarithmic

sensitivity and a non-diffusive chemical substance. For some chemotactic

sensitivity constants, Ahn and Kang proved the existence of bounded

global solutions to the system. An entropy functional was used in their

argument to control the cell density by the density of the chemical

substance. Our purpose is to show the existence of bounded global

solutions for all the chemotactic sensitivity constants. Assuming the

smallness on the initial data in some sense, we can get uniform

estimates for time. These estimates are used to extend local solutions.

This talk is partially based on joint work with Yoshie Sugiyama (Kyusyu

Univ.) and Juan J. L. Vel\\'azquez (Univ. of Bonn).

We consider a chemotaxis system with a logarithmic

sensitivity and a non-diffusive chemical substance. For some chemotactic

sensitivity constants, Ahn and Kang proved the existence of bounded

global solutions to the system. An entropy functional was used in their

argument to control the cell density by the density of the chemical

substance. Our purpose is to show the existence of bounded global

solutions for all the chemotactic sensitivity constants. Assuming the

smallness on the initial data in some sense, we can get uniform

estimates for time. These estimates are used to extend local solutions.

This talk is partially based on joint work with Yoshie Sugiyama (Kyusyu

Univ.) and Juan J. L. Vel\\'azquez (Univ. of Bonn).

### 2014/04/21

#### Numerical Analysis Seminar

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

Shape optimization problems for time-periodic solutions of the Navier-Stokes equations (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

**Takashi Nakazawa**(Tohoku University)Shape optimization problems for time-periodic solutions of the Navier-Stokes equations (JAPANESE)

[ Reference URL ]

http://www.infsup.jp/utnas/

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Lagrangian mean curvature flows and some examples (JAPANESE)

**Hikaru Yamamoto**(The University of Tokyo)Lagrangian mean curvature flows and some examples (JAPANESE)

### 2014/04/19

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Strichartz estimates for incompressible rotating fluids (JAPANESE)

On the interpolation of functions for scattered data on random infinite points with a sharp error estimate (JAPANESE)

**Ryo Takada**(Tohoku University) 13:30-15:00Strichartz estimates for incompressible rotating fluids (JAPANESE)

**Masami Okada**(Tokyo Metropolitan Unversity) 15:30-16:30On the interpolation of functions for scattered data on random infinite points with a sharp error estimate (JAPANESE)

### 2014/04/16

#### Number Theory Seminar

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

On the cycle class map for zero-cycles over local fields (ENGLISH)

**Olivier Wittenberg**(ENS and CNRS)On the cycle class map for zero-cycles over local fields (ENGLISH)

[ Abstract ]

The Chow group of zero-cycles of a smooth and projective variety defined over a field k is an invariant of an arithmetic and geometric nature which is well understood only when k is a finite field (by higher-dimensional class field theory). In this talk, we will discuss the case of local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus over p-adic fields. The same statement holds for semistable K3 surfaces over C((t)), but does not hold in general for surfaces over C((t)) or over the maximal unramified extension of a p-adic field. This is a joint work with Hélène Esnault.

The Chow group of zero-cycles of a smooth and projective variety defined over a field k is an invariant of an arithmetic and geometric nature which is well understood only when k is a finite field (by higher-dimensional class field theory). In this talk, we will discuss the case of local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus over p-adic fields. The same statement holds for semistable K3 surfaces over C((t)), but does not hold in general for surfaces over C((t)) or over the maximal unramified extension of a p-adic field. This is a joint work with Hélène Esnault.

### 2014/04/15

#### Tuesday Seminar on Topology

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

On the rational string operations of classifying spaces and the

Hochschild cohomology (JAPANESE)

**Takahito Naito**(The University of Tokyo)On the rational string operations of classifying spaces and the

Hochschild cohomology (JAPANESE)

[ Abstract ]

Chataur and Menichi initiated the theory of string topology of

classifying spaces.

In particular, the cohomology of the free loop space of a classifying

space is endowed with a product

called the dual loop coproduct. In this talk, I will discuss the

algebraic structure and relate the rational dual loop coproduct to the

cup product on the Hochschild cohomology via the Van den Bergh isomorphism.

Chataur and Menichi initiated the theory of string topology of

classifying spaces.

In particular, the cohomology of the free loop space of a classifying

space is endowed with a product

called the dual loop coproduct. In this talk, I will discuss the

algebraic structure and relate the rational dual loop coproduct to the

cup product on the Hochschild cohomology via the Van den Bergh isomorphism.

#### PDE Real Analysis Seminar

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

An application of weighted Hardy spaces to the Navier-Stokes equations (JAPANESE)

**Yohei Tsutsui**(The University of Tokyo)An application of weighted Hardy spaces to the Navier-Stokes equations (JAPANESE)

[ Abstract ]

The purpose of this talk is to investigate decay orders of the L^2 energy of solutions to the incompressible homogeneous Navier-Stokes equations on the whole spaces by the aid of the theory of weighted Hardy spaces. The main estimates are two weighted inequalities for heat semigroup on weighted Hardy spaces and a weighted version of the div-curl lemma due to Coifman-Lions-Meyer-Semmes. It turns out that because of the use of weighted Hardy spaces, our decay orders of the energy can be close to the critical one of Wiegner.

The purpose of this talk is to investigate decay orders of the L^2 energy of solutions to the incompressible homogeneous Navier-Stokes equations on the whole spaces by the aid of the theory of weighted Hardy spaces. The main estimates are two weighted inequalities for heat semigroup on weighted Hardy spaces and a weighted version of the div-curl lemma due to Coifman-Lions-Meyer-Semmes. It turns out that because of the use of weighted Hardy spaces, our decay orders of the energy can be close to the critical one of Wiegner.

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Toward the graded Cartan invariants of the symmetric groups (JAPANESE)

**Shunsuke Tsuchioka**(the University of Tokyo)Toward the graded Cartan invariants of the symmetric groups (JAPANESE)

[ Abstract ]

We propose a graded analog of Hill's conjecture which is equivalent to K\\"ulshammer-Olsson-Robinson's conjecture on the generalized Cartan invariants of the symmetric groups.

We give justifications for it and discuss implications between the variants.

Some materials are based on the joint work with Anton Evseev.

We propose a graded analog of Hill's conjecture which is equivalent to K\\"ulshammer-Olsson-Robinson's conjecture on the generalized Cartan invariants of the symmetric groups.

We give justifications for it and discuss implications between the variants.

Some materials are based on the joint work with Anton Evseev.

### 2014/04/14

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Alternative proof of the geometric vrsion of Lemma on logarithmic derivatives (JAPANESE)

**Katsutoshi Yamanoi**(Tokyo Institute of Technology)Alternative proof of the geometric vrsion of Lemma on logarithmic derivatives (JAPANESE)

### 2014/04/10

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Deformations of homogeneous Cayley cone submanifolds (JAPANESE)

**Kotaro Kawai**( University of Tokyo)Deformations of homogeneous Cayley cone submanifolds (JAPANESE)

[ Abstract ]

A Cayley submanifold is a minimal submanifold in a Spin(7)-manifold, and is a special class of calibrated submanifolds introduced by Harvey and Lawson. The deformation of calibrated submanifolds is first studied by Mclean. He studied the compact case, and many people try to generalize it to noncompact cases (conical case, asymptotically conical case etc.). In general, the moduli space of deformations of a Cayley cone is known not to be smooth. In this talk, we focus on the homogeneous Cayley cones in R^8, and study their deformation spaces explicitly.

A Cayley submanifold is a minimal submanifold in a Spin(7)-manifold, and is a special class of calibrated submanifolds introduced by Harvey and Lawson. The deformation of calibrated submanifolds is first studied by Mclean. He studied the compact case, and many people try to generalize it to noncompact cases (conical case, asymptotically conical case etc.). In general, the moduli space of deformations of a Cayley cone is known not to be smooth. In this talk, we focus on the homogeneous Cayley cones in R^8, and study their deformation spaces explicitly.

### 2014/04/09

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Index and determnant of n-tuples of commuting operators (ENGLISH)

**Ryszard Nest**(Univ. Copenhagen)Index and determnant of n-tuples of commuting operators (ENGLISH)

### 2014/04/08

#### Tuesday Seminar on Topology

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

On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)

**Hidetoshi Masai**(The University of Tokyo)On the number of commensurable fibrations on a hyperbolic 3-manifold. (JAPANESE)

[ Abstract ]

By work of Thurston, it is known that if a hyperbolic fibred

$3$-manifold $M$ has Betti number greater than 1, then

$M$ admits infinitely many distinct fibrations.

For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,

the number of fibrations on $M$ that are commensurable in the sense of

Calegari-Sun-Wang to $\\omega$ is known to be finite.

In this talk, we prove that the number can be arbitrarily large.

By work of Thurston, it is known that if a hyperbolic fibred

$3$-manifold $M$ has Betti number greater than 1, then

$M$ admits infinitely many distinct fibrations.

For any fibration $\\omega$ on a hyperbolic $3$-manifold $M$,

the number of fibrations on $M$ that are commensurable in the sense of

Calegari-Sun-Wang to $\\omega$ is known to be finite.

In this talk, we prove that the number can be arbitrarily large.

#### Seminar on Probability and Statistics

13:00-14:10 Room #052 (Graduate School of Math. Sci. Bldg.)

Parametric estimation in fractional Ornstein-Uhlenbeck process (ENGLISH)

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/00.html

**Alexandre Brouste**(Universite du Maine, France)Parametric estimation in fractional Ornstein-Uhlenbeck process (ENGLISH)

[ Abstract ]

Several statistical models that imply the fractional Ornstein-Uhlenbeck (fOU) process will be presented: direct observations of the process or partial observations in an additive independent noise, continuous observations or discrete observations. In this different settings, we exhibit large sample (or high-frequency) asymptotic properties of the estimators (maximum likelihood estimator, quadratic variation based estimator, moment estimator, …) for all parameters of interest of the fOU. We also illustrate our results with the R package yuima.

[ Reference URL ]Several statistical models that imply the fractional Ornstein-Uhlenbeck (fOU) process will be presented: direct observations of the process or partial observations in an additive independent noise, continuous observations or discrete observations. In this different settings, we exhibit large sample (or high-frequency) asymptotic properties of the estimators (maximum likelihood estimator, quadratic variation based estimator, moment estimator, …) for all parameters of interest of the fOU. We also illustrate our results with the R package yuima.

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2014/00.html

### 2014/03/19

#### Classical Analysis

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

Discrete Schlesinger Equations and Difference Painlevé Equations (ENGLISH)

**Anton Dzhamay**(University of Northern Colorado)Discrete Schlesinger Equations and Difference Painlevé Equations (ENGLISH)

[ Abstract ]

The theory of Schlesinger equations describing isomonodromic

dynamic on the space of matrix coefficients of a Fuchsian system

w.r.t.~continuous deformations is well-know. In this talk we consider

a discrete version of this theory. Discrete analogues of Schlesinger

deformations are Schlesinger transformations that shift the eigenvalues

of the coefficient matrices by integers. By discrete Schlesinger equations

we mean the evolution equations on the matrix coefficients describing

such transformations. We derive these equations, show how they can be

split into the evolution equations on the space of eigenvectors of the

coefficient matrices, and explain how to write the latter equations in

the discrete Hamiltonian form. We also consider some reductions of those

equations to the difference Painlevé equations, again in complete parallel

to the differential case.

This is a joint work with H. Sakai (the University of Tokyo) and

T.Takenawa (Tokyo Institute of Marine Science and Technology).

The theory of Schlesinger equations describing isomonodromic

dynamic on the space of matrix coefficients of a Fuchsian system

w.r.t.~continuous deformations is well-know. In this talk we consider

a discrete version of this theory. Discrete analogues of Schlesinger

deformations are Schlesinger transformations that shift the eigenvalues

of the coefficient matrices by integers. By discrete Schlesinger equations

we mean the evolution equations on the matrix coefficients describing

such transformations. We derive these equations, show how they can be

split into the evolution equations on the space of eigenvectors of the

coefficient matrices, and explain how to write the latter equations in

the discrete Hamiltonian form. We also consider some reductions of those

equations to the difference Painlevé equations, again in complete parallel

to the differential case.

This is a joint work with H. Sakai (the University of Tokyo) and

T.Takenawa (Tokyo Institute of Marine Science and Technology).

### 2014/03/14

#### GCOE Seminars

16:00-16:50 Room #118 (Graduate School of Math. Sci. Bldg.)

A new finite difference scheme based on staggered grids for Navier Stokes equations (ENGLISH)

**Kazufumi Ito**(North Carolina State Univ.)A new finite difference scheme based on staggered grids for Navier Stokes equations (ENGLISH)

[ Abstract ]

We develop a new method that uses the staggered grid only for the pressure node, i.e., the pressure gird is the center of the square cell and the velocities are at the node. The advantage of the proposed method compared to the standard staggered grid methods is that it is very straight forward to treat the boundary conditions for the velocity field, the fluid structure interaction, and to deal with the multiphase flow using the immersed interface methods. We present our analysis and numerical tests.

We develop a new method that uses the staggered grid only for the pressure node, i.e., the pressure gird is the center of the square cell and the velocities are at the node. The advantage of the proposed method compared to the standard staggered grid methods is that it is very straight forward to treat the boundary conditions for the velocity field, the fluid structure interaction, and to deal with the multiphase flow using the immersed interface methods. We present our analysis and numerical tests.

#### GCOE Seminars

17:00-17:50 Room #118 (Graduate School of Math. Sci. Bldg.)

Efficient Domain Decomposition Methods for a Class of Linear and Nonlinear Inverse Problems (ENGLISH)

**Jun Zou**(The Chinese University of Hong Kong)Efficient Domain Decomposition Methods for a Class of Linear and Nonlinear Inverse Problems (ENGLISH)

[ Abstract ]

In this talk we shall present several new domain decomposition methods for solving some linear and nonlinear inverse problems. The motivations and derivations of the methods will be discussed, and numerical experiments will be demonstrated.

In this talk we shall present several new domain decomposition methods for solving some linear and nonlinear inverse problems. The motivations and derivations of the methods will be discussed, and numerical experiments will be demonstrated.

### 2014/03/13

#### Lectures

10:15-11:45 Room #470 (Graduate School of Math. Sci. Bldg.)

Almost sure triviality of the $C^1$-centralizer of random circle diffeomorphisms with periodic points (ENGLISH)

**Michele Triestino**(Ecole Normale Superieure de Lyon)Almost sure triviality of the $C^1$-centralizer of random circle diffeomorphisms with periodic points (ENGLISH)

[ Abstract ]

By the end of the 80s, Malliavin and Shavgulidze introduced a measure on the space of C^1 circle diffeomorphisms which carries many interesting features. Perhaps the most interesting aspect is that it can be considered as an analog of the Haar measure for the group Diff^1_+(S^1).

The nature of this measure has been mostly investigated in connection to representation theory.

For people working in dynamical systems, the MS measure offers a way to quantify dynamical phenomena: for example, which is the probability that a random diffeomorphism is irrational? Even if this question have occupied my mind for a long time, it remains still unanswered, as many other interesting ones. However, it is possible to understand precisely what are the typical features of a diffeomorphism with periodic points.

By the end of the 80s, Malliavin and Shavgulidze introduced a measure on the space of C^1 circle diffeomorphisms which carries many interesting features. Perhaps the most interesting aspect is that it can be considered as an analog of the Haar measure for the group Diff^1_+(S^1).

The nature of this measure has been mostly investigated in connection to representation theory.

For people working in dynamical systems, the MS measure offers a way to quantify dynamical phenomena: for example, which is the probability that a random diffeomorphism is irrational? Even if this question have occupied my mind for a long time, it remains still unanswered, as many other interesting ones. However, it is possible to understand precisely what are the typical features of a diffeomorphism with periodic points.

#### GCOE Seminars

17:00-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

STABILITY IN THE OBSTACLE PROBLEM FOR A SHALLOW SHELL (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/bm02.pdf

**Bernadette Miara**(Univ. Paris-Est)STABILITY IN THE OBSTACLE PROBLEM FOR A SHALLOW SHELL (ENGLISH)

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/bm02.pdf

### 2014/03/12

#### Lectures

10:15-11:45 Room #470 (Graduate School of Math. Sci. Bldg.)

Invariant distributions for circle diffeomorphisms of

irrational rotation number and low regularity (ENGLISH)

**Michele Triestino**(Ecole Normale Superieure de Lyon)Invariant distributions for circle diffeomorphisms of

irrational rotation number and low regularity (ENGLISH)

[ Abstract ]

The main inspiration of this joint work with Andrés Navas is the beautiful result of Ávila and Kocsard: if f is a C^\\infty circle diffeomorphism of irrational rotation number, then the unique invariant probability measure is also the unique (up to rescaling) invariant distribution.

Using conceptual geometric arguments (Hahn-Banach...), we investigate the uniqueness of invariant distributions for C^1 circle diffeomorphisms of irrational rotation number, with particular attention to sharp regularity.

We prove that If the diffeomorphism is C^{1+bv}, then there is a unique invariant distribution of order 1. On the other side, examples by Douady and Yoccoz, and by Kodama and Matsumoto exhibit differentiable dynamical systems for which the uniqueness does not hold.

The main inspiration of this joint work with Andrés Navas is the beautiful result of Ávila and Kocsard: if f is a C^\\infty circle diffeomorphism of irrational rotation number, then the unique invariant probability measure is also the unique (up to rescaling) invariant distribution.

Using conceptual geometric arguments (Hahn-Banach...), we investigate the uniqueness of invariant distributions for C^1 circle diffeomorphisms of irrational rotation number, with particular attention to sharp regularity.

We prove that If the diffeomorphism is C^{1+bv}, then there is a unique invariant distribution of order 1. On the other side, examples by Douady and Yoccoz, and by Kodama and Matsumoto exhibit differentiable dynamical systems for which the uniqueness does not hold.

#### Mathematical Biology Seminar

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

When size does matter: Ontogenetic symmetry and asymmetry in energetics

(ENGLISH)

http://staff.science.uva.nl/~aroos/

**Andre M. de Roos**(University of Amsterdam)When size does matter: Ontogenetic symmetry and asymmetry in energetics

(ENGLISH)

[ Abstract ]

Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

[ Reference URL ]Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

http://staff.science.uva.nl/~aroos/

#### GCOE Seminars

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

When size does matter: Ontogenetic symmetry and asymmetry in energetics (ENGLISH)

http://staff.science.uva.nl/~aroos/

**Andre M. de Roos**(University of Amsterdam)When size does matter: Ontogenetic symmetry and asymmetry in energetics (ENGLISH)

[ Abstract ]

Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

[ Reference URL ]Body size (≡ biomass) is the dominant determinant of population dynamical processes such as giving birth or dying in almost all species, with often drastically different behaviour occurring in different parts of the growth trajectory, while the latter is largely determined by food availability at the different life stages. This leads to the question under what conditions unstructured population models, formulated in terms of total population biomass, still do a fair job. To contribute to answering this question we first analyze the conditions under which a size-structured model collapses to a dynamically equivalent unstructured one in terms of total biomass. The only biologically meaningful case where this occurs is when body size does not affect any of the population dynamic processes, this is the case if and only if the mass-specific ingestion rate, the mass-specific biomass production and the mortality rate of the individuals are independent of size, a condition to which we refer as “ontogenetic symmetry”. Intriguingly, under ontogenetic symmetry the equilibrium biomass-body size spectrum is proportional to 1/size, a form that has been conjectured for marine size spectra and subsequently has been used as prior assumption in theoretical papers dealing with the latter. As a next step we consider an archetypical class of models in which reproduction takes over from growth upon reaching an adult body size, in order to determine how quickly discrepancies from ontogenetic symmetry lead to relevant novel population dynamical phenomena. The phenomena considered are biomass overcompensation, when additional imposed mortality leads, rather unexpectedly, to an increase in the equilibrium biomass of either the juveniles or the adults (a phenomenon with potentially big consequences for predators of the species), and the occurrence of two types of size-structure driven oscillations, juvenile-driven cycles with separated extended cohorts, and adult-driven cycles in which periodically a front of relatively steeply decreasing frequencies moves up the size distribution. A small discrepancy from symmetry can already lead to biomass overcompensation; size-structure driven cycles only occur for somewhat larger discrepancies.

http://staff.science.uva.nl/~aroos/

### 2014/03/11

#### GCOE Seminars

17:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Inverse problem for the waves : stability and convergence matters (ENGLISH)

**Lucie Baudouin**(LAAS-CNRS, equipe MAC)Inverse problem for the waves : stability and convergence matters (ENGLISH)

[ Abstract ]

This talk aims to present some recent works in collaboration with Maya de Buhan, Sylvain Ervedoza and Axel Osses regarding an inverse problem for the wave equation. More specifically, we study the determination of the potential in a wave equation with given Dirichlet boundary data from a measurement of the flux of the solution on a part of the boundary. On the one hand, we will focus on the question of convergence of the space semi-discrete inverse problems toward their continuous counterpart. Several uniqueness and stability results are available in the literature about the continuous setting of the inverse problem of determination of a potential in the wave equation. In particular, we can mention a Lipschitz stability result under a classical geometric condition obtained by Imanuvilov and Yamamoto, and a logarithmic stability result obtained by Bellassoued when the observation measurement is made on an arbitrary part of the boundary. In both situations, we can design a numerical process for which convergence results are proved. The analysis we conduct is based on discrete Carleman estimates, either for the hyperbolic or for the elliptic operator, in which case we shall use a result of Boyer, Hubert and Le Rousseau. On the other hand, still considering the same inverse problem, we will present a new reconstruction algorithm of the potential. The design and convergence of the algorithm are based on the Carleman estimates for the waves previously used to prove the Lipschitz stability. We will finally give some simple illustrative numerical simulations for 1-d problems.

This talk aims to present some recent works in collaboration with Maya de Buhan, Sylvain Ervedoza and Axel Osses regarding an inverse problem for the wave equation. More specifically, we study the determination of the potential in a wave equation with given Dirichlet boundary data from a measurement of the flux of the solution on a part of the boundary. On the one hand, we will focus on the question of convergence of the space semi-discrete inverse problems toward their continuous counterpart. Several uniqueness and stability results are available in the literature about the continuous setting of the inverse problem of determination of a potential in the wave equation. In particular, we can mention a Lipschitz stability result under a classical geometric condition obtained by Imanuvilov and Yamamoto, and a logarithmic stability result obtained by Bellassoued when the observation measurement is made on an arbitrary part of the boundary. In both situations, we can design a numerical process for which convergence results are proved. The analysis we conduct is based on discrete Carleman estimates, either for the hyperbolic or for the elliptic operator, in which case we shall use a result of Boyer, Hubert and Le Rousseau. On the other hand, still considering the same inverse problem, we will present a new reconstruction algorithm of the potential. The design and convergence of the algorithm are based on the Carleman estimates for the waves previously used to prove the Lipschitz stability. We will finally give some simple illustrative numerical simulations for 1-d problems.

### 2014/03/10

#### GCOE Seminars

13:30-15:00 Room #123 (Graduate School of Math. Sci. Bldg.)

The two-dimensional random walk in an isotropic random environment (ENGLISH)

**Erwin Bolthausen**(University of Zurich)The two-dimensional random walk in an isotropic random environment (ENGLISH)

#### Lectures

15:15-16:45 Room #123 (Graduate School of Math. Sci. Bldg.)

Energy fluctuations in the disordered harmonic chain (ENGLISH)

**Marielle Simon**(ENS Lyon, UMPA)Energy fluctuations in the disordered harmonic chain (ENGLISH)

[ Abstract ]

We study the energy diffusion in the disordered harmonic chain of oscillators: the usual Hamiltonian dynamics is provided with random masses and perturbed by a degenerate energy conserving noise. After rescaling space and time diffusively, we prove that energy fluctuations evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and an equivalent definition through the Green-Kubo formula. Since the model is non gradient, and the perturbation is very degenerate, the standard Varadhan's approach is reviewed under new perspectives.

We study the energy diffusion in the disordered harmonic chain of oscillators: the usual Hamiltonian dynamics is provided with random masses and perturbed by a degenerate energy conserving noise. After rescaling space and time diffusively, we prove that energy fluctuations evolve following an infinite dimensional linear stochastic differential equation driven by the linearized heat equation. We also give variational expressions for the thermal diffusivity and an equivalent definition through the Green-Kubo formula. Since the model is non gradient, and the perturbation is very degenerate, the standard Varadhan's approach is reviewed under new perspectives.

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