Text only print
Full screen print
Applied Analysis
Seminar information archive ～09/27｜Next seminar｜Future seminars 09/28～
| Date, time & place | Thursday 16:00 - 17:30 002Room #002 (Graduate School of Math. Sci. Bldg.) |
|---|
Seminar information archive
2019/04/25
16:00-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)
Matteo Muratori (Polytechnic University of Milan) 16:00-17:00
The porous medium equation on noncompact Riemannian manifolds with initial datum a measure
(English)
On sharp large deviations for the bridge of a general diffusion
(English)
Matteo Muratori (Polytechnic University of Milan) 16:00-17:00
The porous medium equation on noncompact Riemannian manifolds with initial datum a measure
(English)
[ Abstract ]
We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.
Maurizia Rossi (University of Pisa) 17:00-18:00We investigate existence and uniqueness of weak solutions of the Cauchy problem for the porous medium equation on Cartan-Hadamard manifolds. We show existence of solutions that take a finite Radon measure as initial datum, possibly sign-changing. We then prove uniqueness in the class of nonnegative solutions, upon assuming a quadratic lower bound on the Ricci curvature. Our result is "optimal" in the sense that any weak solution necessarily solves a Cauchy problem with initial datum a finite Radon measure. Moreover, as byproducts of the techniques we employ, we obtain some new results in potential analysis on manifolds, concerning the validity of a modified version of the mean-value inequality for superharmonic functions and related properties of potentials of positive Radon measures. Finally, we briefly discuss some work in progress regarding stability of the porous medium equation with respect to the Wasserstein distance, on Riemannian manifolds with Ricci curvature bounded below.
On sharp large deviations for the bridge of a general diffusion
(English)
[ Abstract ]
In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.
In this talk we provide sharp Large Deviation estimates for the probability of exit from a domain for the bridge of a d-dimensional general diffusion process X, as the conditioning time tends to 0. This kind of results is motivated by applications to numerical simulation. In particular we investigate the influence of the drift b of X. It turns out that the sharp asymptotics for the exit time probability are independent of the drift, provided b enjoyes a simple condition that is always satisfied in dimension 1. On the other hand, we show that the drift can be influential if this assumption is not satisfied. This talk is based on a joint work with P. Baldi and L. Caramellino.
2018/11/15
16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Nakao Hayashi (Osaka University)
Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)
Nakao Hayashi (Osaka University)
Inhomogeneous Dirichlet-boundary value problem for one dimensional nonlinear Schr\"{o}dinger equations (Japanese)
[ Abstract ]
We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.
We consider the inhomogeneous Dirichlet-boundary value problem for the cubic nonlinear Schr\"{o}dinger equations on the half line. We present sufficient conditions of initial and boundary data which ensure asymptotic behavior of small solutions to equations by using the classical energy method and factorization techniques.
2018/10/11
16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Takahito Kashiwabara (University of Tokyo)
(Japanese)
Takahito Kashiwabara (University of Tokyo)
(Japanese)
2018/10/04
16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Hiroko Yamamoto (University of Tokyo)
(Japanese)
Hiroko Yamamoto (University of Tokyo)
(Japanese)
2018/07/19
16:00-17:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Norihisa Ikoma (Keio University)
Uniqueness and nondegeneracy of ground states to scalar field equation involving critical Sobolev exponent
(Japanese)
Norihisa Ikoma (Keio University)
Uniqueness and nondegeneracy of ground states to scalar field equation involving critical Sobolev exponent
(Japanese)
[ Abstract ]
This talk is devoted to studying the uniqueness and nondegeneracy of ground states to a nonlinear scalar field equation on the whole space. The nonlinearity consists of two power functions, and their growths are subcritical and critical in the Sobolev sense respectively. Under some assumptions, it is known that the equation admits a positive radial ground state and other ground states are made from the positive radial one. We show that if the dimensions are greater than or equal to 5 and the frequency is sufficiently large, then the positive radial ground state is unique and nondegenerate. This is based on joint work with Takafumi Akahori (Shizuoka Univ.), Slim Ibrahim (Univ. of Victoria), Hiroaki Kikuchi (Tsuda Univ.) and Hayato Nawa (Meiji Univ.).
This talk is devoted to studying the uniqueness and nondegeneracy of ground states to a nonlinear scalar field equation on the whole space. The nonlinearity consists of two power functions, and their growths are subcritical and critical in the Sobolev sense respectively. Under some assumptions, it is known that the equation admits a positive radial ground state and other ground states are made from the positive radial one. We show that if the dimensions are greater than or equal to 5 and the frequency is sufficiently large, then the positive radial ground state is unique and nondegenerate. This is based on joint work with Takafumi Akahori (Shizuoka Univ.), Slim Ibrahim (Univ. of Victoria), Hiroaki Kikuchi (Tsuda Univ.) and Hayato Nawa (Meiji Univ.).
2018/05/24
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Eiji Yanagida (Tokyo Institute of Technology)
Sign-changing solutions for a one-dimensional semilinear parabolic problem (Japanese)
Eiji Yanagida (Tokyo Institute of Technology)
Sign-changing solutions for a one-dimensional semilinear parabolic problem (Japanese)
[ Abstract ]
This talk is concerned with a nonlinear parabolic equation on a bounded interval with the homogeneous Dirichlet or Neumann boundary condition. Under rather general conditions on the nonlinearity, we consider the blow-up and global existence of sign-changing solutions. It is shown that there exists a nonnegative integer $k$ such that the solution blows up in finite time if the initial value changes its sign at most $k$ times, whereas there exists a stationary solution with more than $k$ zeros. The proof is based on an intersection number argument combined with a topological method.
This talk is concerned with a nonlinear parabolic equation on a bounded interval with the homogeneous Dirichlet or Neumann boundary condition. Under rather general conditions on the nonlinearity, we consider the blow-up and global existence of sign-changing solutions. It is shown that there exists a nonnegative integer $k$ such that the solution blows up in finite time if the initial value changes its sign at most $k$ times, whereas there exists a stationary solution with more than $k$ zeros. The proof is based on an intersection number argument combined with a topological method.
2017/12/21
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
2017/12/14
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
I-Kun, Chen (Kyoto University)
Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain
(English)
I-Kun, Chen (Kyoto University)
Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain
(English)
[ Abstract ]
We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.
We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.
2017/07/13
16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)
2017/02/16
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Danielle Hilhorst (CNRS / University of Paris-Sud)
Diffusive and inviscid traveling wave solution of the Fisher-KPP equation
(ENGLISH)
Danielle Hilhorst (CNRS / University of Paris-Sud)
Diffusive and inviscid traveling wave solution of the Fisher-KPP equation
(ENGLISH)
[ Abstract ]
Our purpose is to study the limit of traveling wave solutions of the Fisher-KPP equation as the diffusion coefficient tends to zero. More precisely, we consider monotone traveling waves which connect the stable steady state to the unstable one. It is well known that there exists a positive constant c* such that there does not exist any traveling wave solution if c < c* and a unique (up to translation) monotone traveling wave solution of wave speed c for each c > c*.
We consider the corresponding inviscid ordinary differential equation where the diffusion coefficient is equal to zero and show that it possesses a unique traveling wave solution. We then fix c > 0 arbitrary and prove the convergence of the travelling wave of the parabolic equation with velocity c to that of the corresponding traveling wave solution of the inviscid problem.
Further research should involve a similar problem for monostable systems.
This is joint work with Yong Jung Kim.
Our purpose is to study the limit of traveling wave solutions of the Fisher-KPP equation as the diffusion coefficient tends to zero. More precisely, we consider monotone traveling waves which connect the stable steady state to the unstable one. It is well known that there exists a positive constant c* such that there does not exist any traveling wave solution if c < c* and a unique (up to translation) monotone traveling wave solution of wave speed c for each c > c*.
We consider the corresponding inviscid ordinary differential equation where the diffusion coefficient is equal to zero and show that it possesses a unique traveling wave solution. We then fix c > 0 arbitrary and prove the convergence of the travelling wave of the parabolic equation with velocity c to that of the corresponding traveling wave solution of the inviscid problem.
Further research should involve a similar problem for monostable systems.
This is joint work with Yong Jung Kim.
2016/10/27
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Fred Weissler (Universite Paris 13)
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 ]
2015/11/05
16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Henri Berestycki (EHESS)
The effect of a line with fast diffusion on Fisher-KPP propagation (ENGLISH)
Henri Berestycki (EHESS)
The effect of a line with fast diffusion on Fisher-KPP propagation (ENGLISH)
[ Abstract ]
I will present a system of equations describing the effect of inclusion of a line (the "road") with fast diffusion on biological invasions in the plane. Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain precise threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. I will discuss several further effects such as transport or reaction on the road. I will also discuss the influence of various parameters on the asymptotic behaviour of the invasion speed and shape. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.
I will present a system of equations describing the effect of inclusion of a line (the "road") with fast diffusion on biological invasions in the plane. Outside of the road, the propagation is of the classical Fisher-KPP type. We find that past a certain precise threshold for the ratio of diffusivity coefficients, the presence of the road enhances the speed of global propagation. I will discuss several further effects such as transport or reaction on the road. I will also discuss the influence of various parameters on the asymptotic behaviour of the invasion speed and shape. I report here on results from a series of joint works with Jean-Michel Roquejoffre and Luca Rossi.
2015/10/22
16:00-17:50 Room #002 (Graduate School of Math. Sci. Bldg.)
Hans-Otto Walther (University of Giessen)
(Part I) The semiflow of a delay differential equation on its solution manifold
(Part II) Shilnikov chaos due to state-dependent delay, by means of the fixed point index
(ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf
Hans-Otto Walther (University of Giessen)
(Part I) The semiflow of a delay differential equation on its solution manifold
(Part II) Shilnikov chaos due to state-dependent delay, by means of the fixed point index
(ENGLISH)
[ Abstract ]
(Part I) 16:00 - 16:50
The semiflow of a delay differential equation on its solution manifold
(Part II) 17:00 - 17:50
Shilnikov chaos due to state-dependent delay, by means of the fixed point index
(Part I)
The lecture surveys recent work on initial value problems for differential equations with variable delay. The focus is on differentiable solution operators.
The lecture explains why the theory for retarded functional differential equations which is familiar from monographs before the turn of the millenium fails in case of variable delay, discusses what has been achieved in this case, for autonomous and non-autonomous equations, with delays bounded and unbounded, and addresses open problems.
[detailed abstract]
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf
(Part II)
What can variability of a delay in a delay differential equation do to the dynamics? We find a bounded delay functional $d(\phi)$, with $d(\phi)=1$ on a neighborhood of $\phi=0$, such that the equation $x'(t)=-a x(t-d(x_t))$ has a solution which is homoclinic to $0$, with shift dynamics in its vicinity, whereas the linear equation $x'(t)=-a x(t-1)$ with constant time lag, for small solutions, is hyperbolic with 2-dimensional unstable space.
The proof involves regularity properties of the semiflow close to the homoclinic loop in the solution manifold and a generalization of a method due to Piotr Zgliczynsky which uses the fixed point index and a closing argument in order to establish shift dynamics when certain covering relations hold. (Joint work with Bernhard Lani-Wayda)
[detailed abstract]
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-2.pdf
[ Reference URL ](Part I) 16:00 - 16:50
The semiflow of a delay differential equation on its solution manifold
(Part II) 17:00 - 17:50
Shilnikov chaos due to state-dependent delay, by means of the fixed point index
(Part I)
The lecture surveys recent work on initial value problems for differential equations with variable delay. The focus is on differentiable solution operators.
The lecture explains why the theory for retarded functional differential equations which is familiar from monographs before the turn of the millenium fails in case of variable delay, discusses what has been achieved in this case, for autonomous and non-autonomous equations, with delays bounded and unbounded, and addresses open problems.
[detailed abstract]
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf
(Part II)
What can variability of a delay in a delay differential equation do to the dynamics? We find a bounded delay functional $d(\phi)$, with $d(\phi)=1$ on a neighborhood of $\phi=0$, such that the equation $x'(t)=-a x(t-d(x_t))$ has a solution which is homoclinic to $0$, with shift dynamics in its vicinity, whereas the linear equation $x'(t)=-a x(t-1)$ with constant time lag, for small solutions, is hyperbolic with 2-dimensional unstable space.
The proof involves regularity properties of the semiflow close to the homoclinic loop in the solution manifold and a generalization of a method due to Piotr Zgliczynsky which uses the fixed point index and a closing argument in order to establish shift dynamics when certain covering relations hold. (Joint work with Bernhard Lani-Wayda)
[detailed abstract]
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-2.pdf
http://fmsp.ms.u-tokyo.ac.jp/Walther-abstract-1.pdf
2015/07/16
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Yoshihiro Tonegawa (Tokyo Institute of Technology)
(Japanese)
Yoshihiro Tonegawa (Tokyo Institute of Technology)
(Japanese)
2015/06/11
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
2015/05/14
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Masahito Ohta (Tokyo University of Science)
Strong instability of standing waves for some nonlinear Schr\"odinger equations (Japanese)
Masahito Ohta (Tokyo University of Science)
Strong instability of standing waves for some nonlinear Schr\"odinger equations (Japanese)
2015/04/23
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Bernold Fiedler (Free University of Berlin)
The importance of being just late (ENGLISH)
Bernold Fiedler (Free University of Berlin)
The importance of being just late (ENGLISH)
[ Abstract ]
Delays are a ubiquitous nuisance in control. Delays increase finite-dimensional phase spaces to become infinite-dimensional. But, are delays all that bad?
Following an idea of Pyragas, we attempt noninvasive and model-independent stabilization of unstable p-periodic phenomena $u(t)$ by a friendly delay $r$ . Our feedback only evaluates differences $u(t-r)-u(t)$. When the time delay $r$ is chosen to be an integer multiple $np$ of the minimal period $p$, the difference and the feedback vanish alike: the control strategy becomes noninvasive on the target periodic orbit.
We survey promise and limitations of this idea, including applications and an example of delay control of delay equations.
The results are joint work with P. Hoevel, W. Just, I. Schneider, E. Schoell, H.-J. Wuensche, S. Yanchuk, and others. See also
http://dynamics.mi.fu-berlin.de/
Delays are a ubiquitous nuisance in control. Delays increase finite-dimensional phase spaces to become infinite-dimensional. But, are delays all that bad?
Following an idea of Pyragas, we attempt noninvasive and model-independent stabilization of unstable p-periodic phenomena $u(t)$ by a friendly delay $r$ . Our feedback only evaluates differences $u(t-r)-u(t)$. When the time delay $r$ is chosen to be an integer multiple $np$ of the minimal period $p$, the difference and the feedback vanish alike: the control strategy becomes noninvasive on the target periodic orbit.
We survey promise and limitations of this idea, including applications and an example of delay control of delay equations.
The results are joint work with P. Hoevel, W. Just, I. Schneider, E. Schoell, H.-J. Wuensche, S. Yanchuk, and others. See also
http://dynamics.mi.fu-berlin.de/
2015/01/22
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Arnaud Ducrot (University of Bordeaux)
On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data
(ENGLISH)
Arnaud Ducrot (University of Bordeaux)
On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data
(ENGLISH)
[ Abstract ]
In this talk we discuss the asymptotic behaviour of a multi-dimensional Fisher-KPP equation posed in an asymptotically homogeneous medium and supplemented together with a compactly supported initial datum. We derive precise estimates for the location of the front before proving the convergence of the solutions towards travelling front. In particular we show that the location of the front drastically depends on the rate at which the medium become homogeneous at infinity. Fast rate of convergence only changes the location by some constant while lower rate of convergence induces further logarithmic delay.
In this talk we discuss the asymptotic behaviour of a multi-dimensional Fisher-KPP equation posed in an asymptotically homogeneous medium and supplemented together with a compactly supported initial datum. We derive precise estimates for the location of the front before proving the convergence of the solutions towards travelling front. In particular we show that the location of the front drastically depends on the rate at which the medium become homogeneous at infinity. Fast rate of convergence only changes the location by some constant while lower rate of convergence induces further logarithmic delay.
2014/07/24
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Nagasawa Takeyuki (Saitama University)
Decomposition of the Mobius energy (JAPANESE)
Nagasawa Takeyuki (Saitama University)
Decomposition of the Mobius energy (JAPANESE)
2014/07/03
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Kazuhiro Ishige (Tohoku University)
Parabolic power concavity and parabolic boundary value problems (JAPANESE)
Kazuhiro Ishige (Tohoku University)
Parabolic power concavity and parabolic boundary value problems (JAPANESE)
2014/01/23
16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Thomas Giletti (Univ. of Lorraine at Nancy)
Inside dynamics of pushed and pulled fronts (ENGLISH)
Thomas Giletti (Univ. of Lorraine at Nancy)
Inside dynamics of pushed and pulled fronts (ENGLISH)
[ Abstract ]
Mathematical analysis of reaction-diffusion equations is a powerful tool in the understanding of dynamics of many real-life propagation phenomena. A feature of particular interest is the fact that dynamics and their underlying mechanisms vary greatly, depending on the choice of the nonlinearity in the reaction term. In this talk, we will discuss the pushed/pulled front terminology, based upon the role of each component of the front inside the whole propagating structure.
Mathematical analysis of reaction-diffusion equations is a powerful tool in the understanding of dynamics of many real-life propagation phenomena. A feature of particular interest is the fact that dynamics and their underlying mechanisms vary greatly, depending on the choice of the nonlinearity in the reaction term. In this talk, we will discuss the pushed/pulled front terminology, based upon the role of each component of the front inside the whole propagating structure.
2013/12/12
16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Yuki Yasuda (University of Tokyo (Department of Earth and Planetary Science))
A theoretical study on the spontaneous radiation of atmospheric gravity waves using the renormalization group method (JAPANESE)
Yuki Yasuda (University of Tokyo (Department of Earth and Planetary Science))
A theoretical study on the spontaneous radiation of atmospheric gravity waves using the renormalization group method (JAPANESE)
2013/11/14
16:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Danielle Hilhorst (Université de Paris-Sud / CNRS)
Singular limit of a damped wave equation with a bistable nonlinearity (ENGLISH)
Danielle Hilhorst (Université de Paris-Sud / CNRS)
Singular limit of a damped wave equation with a bistable nonlinearity (ENGLISH)
[ Abstract ]
We study the singular limit of a damped wave equation with
a bistable nonlinearity. In order to understand interfacial
phenomena, we derive estimates for the generation and the motion
of interfaces. We prove that steep interfaces are generated in
a short time and that their motion is governed by mean curvature
flow under the assumption that the damping is sufficiently strong.
To this purpose, we prove a comparison principle for the damped
wave equation and construct suitable sub- and super-solutions.
This is joint work with Mitsunori Nata.
We study the singular limit of a damped wave equation with
a bistable nonlinearity. In order to understand interfacial
phenomena, we derive estimates for the generation and the motion
of interfaces. We prove that steep interfaces are generated in
a short time and that their motion is governed by mean curvature
flow under the assumption that the damping is sufficiently strong.
To this purpose, we prove a comparison principle for the damped
wave equation and construct suitable sub- and super-solutions.
This is joint work with Mitsunori Nata.
2013/06/06
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Chang-Shou Lin (National Taiwan University)
The Geometry of Critical Points of Green functions On Tori (ENGLISH)
Chang-Shou Lin (National Taiwan University)
The Geometry of Critical Points of Green functions On Tori (ENGLISH)
[ Abstract ]
The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.
We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.
We can generalize our results to a sum of two Green functions.
The Green function of a torus can be expressed by elliptic functions or Jacobic theta functions. It is not surprising the geometry of its critical points would be involved with behaviors of those classical functions. Thus, the non-degeneracy of critical points gives rise to some inequality for elliptic functions. One of consequences of our analysis is to prove any saddle point is non-degenerate, i.e., the Hessian is negative.
We will also show that the number of the critical points of Green function in any torus is either three or five critical points. Furthermore, the moduli space of tori which Green function has five critical points is a simple-connected connected set. The proof of these results use a nonlinear PDE (mean field equation) and the formula for counting zeros of modular form. For a N torsion point,the related modular form is the Eisenstein series of weight one, which was discovered by Hecke (1926). Thus, our PDE method gives a deformation of those Eisenstein series and allows us to find the zeros of those Eisenstein series.
We can generalize our results to a sum of two Green functions.
2012/09/20
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Bernold Fiedler (Free University of Berlin)
Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes
(ENGLISH)
Bernold Fiedler (Free University of Berlin)
Fusco-Rocha meanders: from Temperley-Lieb algebras to black holes
(ENGLISH)
[ Abstract ]
Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutation. These permutation relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.
We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.
This is joint work with Juliette Hell, Brian Smith, Carlos Rocha, Pablo Castaneda, and Matthias Wolfrum.
Fusco and Rocha studied Neumann boundary value problems for ODEs of second order via a shooting approach. They introduced the notion of what we now call Sturm permutation. These permutation relate, on the one hand, to a special class of meandering curves as introduced by Arnol'd in a singularity context. On the other hand, their special class became central in the study of global attractors of parabolic PDEs of Sturm type.
We discuss relations of Fusco-Rocha meanders with further areas: the multiplicative and trace structure in Temperley-Lieb algebras, discrete versions of Cartesian billiards, and the problem of constructing initial conditions for black hole dynamics which satisfy the Einstein constraints. We also risk a brief glimpse at the long and meandric history of meander patterns themselves.
This is joint work with Juliette Hell, Brian Smith, Carlos Rocha, Pablo Castaneda, and Matthias Wolfrum.
