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Seminar information archive
Seminar information archive ～05/20｜Today's seminar 05/21 | Future seminars 05/22～
Colloquium
15:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)
GONGYO Yoshinori (Graduate School of Mathematical Sciences, The University of Tokyo)
Recent developments of MMP and around (JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/teacher/gongyo.html
GONGYO Yoshinori (Graduate School of Mathematical Sciences, The University of Tokyo)
Recent developments of MMP and around (JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/teacher/gongyo.html
2016/06/23
Classical Analysis
16:50-18:20 Room #118 (Graduate School of Math. Sci. Bldg.)
Shingo Kamimoto (Hiroshima University)
Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)
Shingo Kamimoto (Hiroshima University)
Resurgence of formal series solutions of nonlinear differential and difference equations (JAPANESE)
FMSP Lectures
15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Klaus Mainzer (Technische Universität München)
Complexity and Computability: Complex Dynamical Systems beyond Turing-Computability (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Mainzer.pdf
Klaus Mainzer (Technische Universität München)
Complexity and Computability: Complex Dynamical Systems beyond Turing-Computability (ENGLISH)
[ Abstract ]
The computational theory of complexity is founded by digital computing (e.g. Turing machine) which cannot fully grasp continuous concepts of mathematics. The mathematical theory of complex dynamical systems (with interdisciplinary applications in natural and economic sciences) is based on continuous concepts. Further on, there is an outstanding tradition in mathematics since Newton, Leibniz, Euler et al. with real algorithms in, e.g., numerical analysis. How can the gap between the digital and continuous world be mathematically overcome? The talk aims at mathematical and philosophical foundations and interdisciplinary applications of complex dynamical systems beyond Turing-computability.
[ Reference URL ]The computational theory of complexity is founded by digital computing (e.g. Turing machine) which cannot fully grasp continuous concepts of mathematics. The mathematical theory of complex dynamical systems (with interdisciplinary applications in natural and economic sciences) is based on continuous concepts. Further on, there is an outstanding tradition in mathematics since Newton, Leibniz, Euler et al. with real algorithms in, e.g., numerical analysis. How can the gap between the digital and continuous world be mathematically overcome? The talk aims at mathematical and philosophical foundations and interdisciplinary applications of complex dynamical systems beyond Turing-computability.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Mainzer.pdf
2016/06/21
Tuesday Seminar of Analysis
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
Masao HIROKAWA (Institute of Engineering, Hiroshima University)
On the Hepp-Lieb-Preparata quantum phase transition for the quantum Rabi model (Japanese)
Masao HIROKAWA (Institute of Engineering, Hiroshima University)
On the Hepp-Lieb-Preparata quantum phase transition for the quantum Rabi model (Japanese)
[ Abstract ]
In my talk, I would like to consider the quantum Rabi model from the point of the view of quantum phase transition. Preparata claims that the ground state of the matter coupled with light is dressed with some photons provided that the coupling strength grows large though those photons should primarily be emitted from the matter in the relaxation of quantum states, and then, that the perturbative ground state switches with a non-perturbative one (Hepp-Lieb-Preparata quantum phase transition). He finds this based on the mathematical structure of the Hepp-Lieb quantum phase transition. Yoshihara and others recently showed an experimental result for the quantum Rabi model in circuit QED. In the experiment, they succeeded in demonstrating the so-called deep-strong coupling regime and the ground state dressed with a photon. We consider the quantum Rabi model in the light of the Hepp-Lieb-Preparata quantum phase transition. Our research is among the studies with aspects of mathematical physics, and deals with the A2-term problem.
In my talk, I would like to consider the quantum Rabi model from the point of the view of quantum phase transition. Preparata claims that the ground state of the matter coupled with light is dressed with some photons provided that the coupling strength grows large though those photons should primarily be emitted from the matter in the relaxation of quantum states, and then, that the perturbative ground state switches with a non-perturbative one (Hepp-Lieb-Preparata quantum phase transition). He finds this based on the mathematical structure of the Hepp-Lieb quantum phase transition. Yoshihara and others recently showed an experimental result for the quantum Rabi model in circuit QED. In the experiment, they succeeded in demonstrating the so-called deep-strong coupling regime and the ground state dressed with a photon. We consider the quantum Rabi model in the light of the Hepp-Lieb-Preparata quantum phase transition. Our research is among the studies with aspects of mathematical physics, and deals with the A2-term problem.
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Noboru Ito (The University of Tokyo)
Spaces of chord diagrams of spherical curves (JAPANESE)
Noboru Ito (The University of Tokyo)
Spaces of chord diagrams of spherical curves (JAPANESE)
[ Abstract ]
In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.
In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.
Seminar on Probability and Statistics
13:00-15:00 Room #052 (Graduate School of Math. Sci. Bldg.)
Lorenzo Mercuri (University of Milan)
New Classes and Methods in YUIMA package
Lorenzo Mercuri (University of Milan)
New Classes and Methods in YUIMA package
[ Abstract ]
In this talk, we present three new classes recently introduced in YUIMA package.
These classes allow the user to manage three different problems:
・Construction of a multidimensional stochastic differential equation driven by a general multivariate Levy process. In particular we show how to define and then simulate a SDE driven by a multivariate Variance Gamma process.
・Definition and simulation of a functional of a general SDE.
・Definition and simulation of the integral of an object from the class yuima.model. In particular, we are able to evaluate Riemann Stieltjes integrals,deterministic integrals with random integrand and stochastic integrals.
Numerical examples are given in order to explain the new methods and classes.
In this talk, we present three new classes recently introduced in YUIMA package.
These classes allow the user to manage three different problems:
・Construction of a multidimensional stochastic differential equation driven by a general multivariate Levy process. In particular we show how to define and then simulate a SDE driven by a multivariate Variance Gamma process.
・Definition and simulation of a functional of a general SDE.
・Definition and simulation of the integral of an object from the class yuima.model. In particular, we are able to evaluate Riemann Stieltjes integrals,deterministic integrals with random integrand and stochastic integrals.
Numerical examples are given in order to explain the new methods and classes.
2016/06/20
Algebraic Geometry Seminar
16:30-18:00 Room #123 (Graduate School of Math. Sci. Bldg.)
De-Qi Zhang (National University of Singapore)
BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)
http://www.math.nus.edu.sg/~matzdq/
De-Qi Zhang (National University of Singapore)
BUILDING BLOCKS OF POLARIZED ENDOMORPHISMS OF NORMAL PROJECTIVE VARIETIES (English)
[ Abstract ]
An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.
We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.
As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.
Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.
Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.
This is a joint work with S. Meng.
[ Reference URL ]An endomorphism f of a normal projective variety X is polarized if f∗H ∼ qH for some ample Cartier divisor H and integer q > 1.
We first assert that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi- ́etale quotient of an abelian variety). Next we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one.
As a consequence, the building blocks of polarized endomorphisms are those of Q- abelian varieties and those of Fano varieties of Picard number one.
Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that a power of f acts as a scalar on the Neron-Severi group of X (modulo torsion) when X is smooth and rationally connected.
Partial answers about X being of Calabi-Yau type or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.
This is a joint work with S. Meng.
http://www.math.nus.edu.sg/~matzdq/
Algebraic Geometry Seminar
14:45-16:15 Room #123 (Graduate School of Math. Sci. Bldg.)
Zhixian Zhu (KIAS)
Fujita's freeness conjecture for 5-fold (English)
Zhixian Zhu (KIAS)
Fujita's freeness conjecture for 5-fold (English)
[ Abstract ]
Let X be a smooth projective variety of dimension n and L any ample line bundle. Fujita conjectured that the adjoint line bundle O(K_X+mL) is globally generated for any m greater or equal to dimX+1. By studying the singularity of pairs, we prove Fujita's freeness conjecture for smooth 5-folds.
Let X be a smooth projective variety of dimension n and L any ample line bundle. Fujita conjectured that the adjoint line bundle O(K_X+mL) is globally generated for any m greater or equal to dimX+1. By studying the singularity of pairs, we prove Fujita's freeness conjecture for smooth 5-folds.
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Shin-ichi Matsumura (Tohoku University)
A transcendental approach to injectivity theorems for log canonical pairs (JAPANESE)
Shin-ichi Matsumura (Tohoku University)
A transcendental approach to injectivity theorems for log canonical pairs (JAPANESE)
Operator Algebra Seminars
16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)
Yusuke Isono (RIMS, Kyoto Univ.)
Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence
Yusuke Isono (RIMS, Kyoto Univ.)
Bi-exact groups, strongly ergodic actions and group measure space type III factors with no central sequence
2016/06/14
Tuesday Seminar of Analysis
16:50-18:20 Room #126 (Graduate School of Math. Sci. Bldg.)
NIIKUNI, Hiroaki (Maebashi Institute of Technology)
Schr¥"odinger operators on a periodically broken zigzag carbon nanotube (Japanese)
NIIKUNI, Hiroaki (Maebashi Institute of Technology)
Schr¥"odinger operators on a periodically broken zigzag carbon nanotube (Japanese)
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Naohiko Kasuya (Aoyama Gakuin University)
Non-Kähler complex structures on R^4 (JAPANESE)
Naohiko Kasuya (Aoyama Gakuin University)
Non-Kähler complex structures on R^4 (JAPANESE)
[ Abstract ]
We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.
We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.
2016/06/13
Numerical Analysis Seminar
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Atsushi Suzuki (Osaka University)
Dissection : A direct solver with kernel detection for finite element matrices
(日本語)
Atsushi Suzuki (Osaka University)
Dissection : A direct solver with kernel detection for finite element matrices
(日本語)
[ Abstract ]
Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.
By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.
This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).
Large-scale sparse matrices are solved in finite element analyses of elasticity and/or flow problems. In some cases, the matrix may be singular, e.g. due to pressure ambiguity of the Navier-Stokes equations, or due to rigid body movements of sub-domain elasticity problems by a domain decomposition method. Therefore, it is better the linear solver understands rank-deficiency of the matrix.
By assuming the matrix is factorized into LDU form with a symmetric partial permutation, and by introducing a threshold to postpone factorization for pseudo null pivots, solvability of the last Schur complement matrix will be examined. Usual procedure for rank-deficiency problem is based on computation of eigenvalues or singular values and an introduced threshold determines the null space. However, developed new algorithm in DOI:10.1002/nme.4729 is based on computation of residuals combined with orthogonal projections onto supposed image spaces and there is no necessary to introduce a threshold for understanding zero value in floating point. The algorithm uses higher precision arithmetic, e.g. quadruple precision, to distinguish numerical round-off errors that occurred during factorization of the whole sparse matrix from ones during the kernel detection procedure itself.
This is joint work with François-Xavier Roux (LJLL, UPMC/ONERA).
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Yuki Tokushige (Research Institute for Mathematical Sciences, Kyoto University)
Jump processes on boudaries of random trees
Yuki Tokushige (Research Institute for Mathematical Sciences, Kyoto University)
Jump processes on boudaries of random trees
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Masanori Adachi (Tokyo University of Science)
(JAPANESE)
Masanori Adachi (Tokyo University of Science)
(JAPANESE)
Operator Algebra Seminars
16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)
Liang Kong (Univ. New Hampshire/Harvard Univ.)
Lattices models for topological orders and boundary-bulk duality
Liang Kong (Univ. New Hampshire/Harvard Univ.)
Lattices models for topological orders and boundary-bulk duality
2016/06/08
Number Theory Seminar
16:00-18:30 Room #16:00-17:00は002, 17:30-18:30は056 (Graduate School of Math. Sci. Bldg.)
Bruno Kahn (Institut de mathématiques de Jussieu-Paris Rive Gauche) 16:00-17:00
Torsion order of smooth projective surfaces (English)
Local and global geometric structures of perfectoid Shimura varieties (English)
Bruno Kahn (Institut de mathématiques de Jussieu-Paris Rive Gauche) 16:00-17:00
Torsion order of smooth projective surfaces (English)
[ Abstract ]
To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbb{Q}$-universally trivial one can associate its torsion order ${\mathrm{Tor}}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that ${\mathrm{Tor}}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.
Xu Shen (Morningside Center of Mathematics) 17:30-18:30To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbb{Q}$-universally trivial one can associate its torsion order ${\mathrm{Tor}}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that ${\mathrm{Tor}}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.
Local and global geometric structures of perfectoid Shimura varieties (English)
[ Abstract ]
In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.
In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.
2016/06/07
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Kenta Hayano (Keio University)
Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)
Kenta Hayano (Keio University)
Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)
[ Abstract ]
In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).
In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).
2016/06/06
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Shin Kikuta (Kogakuin University)
(JAPANESE)
Shin Kikuta (Kogakuin University)
(JAPANESE)
Operator Algebra Seminars
16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)
Yuhei Suzuki (Chiba Univ.)
Minimal ambient nuclear $C^*$-algebras
Yuhei Suzuki (Chiba Univ.)
Minimal ambient nuclear $C^*$-algebras
2016/06/03
Geometry Colloquium
13:00-14:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Takeo Nishinou (Rikkyo University)
On a construction of holomorphic disks (Japanese)
Takeo Nishinou (Rikkyo University)
On a construction of holomorphic disks (Japanese)
[ Abstract ]
Recent study of algebraic and symplectic geometry revealed that holomorphic disks play an important role in several situations, deforming the classical geometry in some sense. In this talk we give a construction of holomorphic disks based on deformation theory, mainly on certain algebraic surfaces.
Recent study of algebraic and symplectic geometry revealed that holomorphic disks play an important role in several situations, deforming the classical geometry in some sense. In this talk we give a construction of holomorphic disks based on deformation theory, mainly on certain algebraic surfaces.
Geometry Colloquium
15:00-16:30 Room #118 (Graduate School of Math. Sci. Bldg.)
Makoto Miura (KIAS)
Caldero's toric degenerations and mirror symmetry (Japanese)
Makoto Miura (KIAS)
Caldero's toric degenerations and mirror symmetry (Japanese)
[ Abstract ]
In this talk, we explain some basic facts on toric degenerations of Fano varieties. In particular, we focus on the toric degenerations of Schubert varieties proposed by Caldero, where we use the string parametrizations of Lusztig--Kashiwara's dual canonical basis. As an application, we introduce a conjectural mirror construction of a linear section Calabi--Yau 3-fold in an orthogonal Grassmannian OG(2,7). This talk is based on joint works with Daisuke Inoue and Atsushi Ito.
In this talk, we explain some basic facts on toric degenerations of Fano varieties. In particular, we focus on the toric degenerations of Schubert varieties proposed by Caldero, where we use the string parametrizations of Lusztig--Kashiwara's dual canonical basis. As an application, we introduce a conjectural mirror construction of a linear section Calabi--Yau 3-fold in an orthogonal Grassmannian OG(2,7). This talk is based on joint works with Daisuke Inoue and Atsushi Ito.
2016/06/01
Mathematical Biology Seminar
16:30-17:30 Room #128演習室 (Graduate School of Math. Sci. Bldg.)
Xiao Dongyuan (Graduate School of Mathematical Sciences, The University of Tokyo)
A variational problem associated with the minimal speed of traveling waves for the spatially
periodic KPP equation (ENGLISH)
Xiao Dongyuan (Graduate School of Mathematical Sciences, The University of Tokyo)
A variational problem associated with the minimal speed of traveling waves for the spatially
periodic KPP equation (ENGLISH)
[ Abstract ]
We consider a spatially periodic KPP equation of the form
$$u_t=u_{xx}+b(x)u(1-u).$$
This equation is motivated by a model in mathematical ecology describing the invasion of an alien species into spatially periodic habitat. We deal with the following variational problem:
$$\underset{b\in A_i}{\mbox{Maximize}}\ \ c^*(b),\ i=1,2,$$
where $c^*(b)$ denotes the minimal speed of the traveling wave of the above equation, and sets $A_1$, $A_2$ are defined by
$$A_1:=\{b\ |\ \int_0^Lb=\alpha L,||b||_{\infty}\le h \},$$
$$A_2:=\{b\ |\ \int_0^Lb^2=\beta L\},$$
with $h>\alpha>0$ and $\beta>0$ being arbitrarily given constants. It is known that $c^*(b)$ is given by the principal eigenvalue $k(\lambda,b)$ associated with the one-dimensional elliptic operator under the periodic boundary condition:
$$-L_{\lambda,b}\psi=-\frac{d^2}{dx^2}\psi-2\lambda\frac{d}{dx}\psi-(b(x)
+\lambda^2)\psi\ \ (x\in\mathbb{R}/L\mathbb{Z}).$$
It is important to note that, in one-dimensional reaction-diffusion equations, the minimal speed $c^*(b)$ coincides with the so-called spreading speed. The notion of spreading speed was introduced in mathematical ecology to describe how fast the invading species expands its territory. In other words, our goal is to find an optimal coefficient $b(x)$ that gives the fastest spreading speed under certain given constraints and to study the properties of such $b(x)$.
We consider a spatially periodic KPP equation of the form
$$u_t=u_{xx}+b(x)u(1-u).$$
This equation is motivated by a model in mathematical ecology describing the invasion of an alien species into spatially periodic habitat. We deal with the following variational problem:
$$\underset{b\in A_i}{\mbox{Maximize}}\ \ c^*(b),\ i=1,2,$$
where $c^*(b)$ denotes the minimal speed of the traveling wave of the above equation, and sets $A_1$, $A_2$ are defined by
$$A_1:=\{b\ |\ \int_0^Lb=\alpha L,||b||_{\infty}\le h \},$$
$$A_2:=\{b\ |\ \int_0^Lb^2=\beta L\},$$
with $h>\alpha>0$ and $\beta>0$ being arbitrarily given constants. It is known that $c^*(b)$ is given by the principal eigenvalue $k(\lambda,b)$ associated with the one-dimensional elliptic operator under the periodic boundary condition:
$$-L_{\lambda,b}\psi=-\frac{d^2}{dx^2}\psi-2\lambda\frac{d}{dx}\psi-(b(x)
+\lambda^2)\psi\ \ (x\in\mathbb{R}/L\mathbb{Z}).$$
It is important to note that, in one-dimensional reaction-diffusion equations, the minimal speed $c^*(b)$ coincides with the so-called spreading speed. The notion of spreading speed was introduced in mathematical ecology to describe how fast the invading species expands its territory. In other words, our goal is to find an optimal coefficient $b(x)$ that gives the fastest spreading speed under certain given constraints and to study the properties of such $b(x)$.
2016/05/31
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Benoît Guerville-Ballé (Tokyo Gakugei University)
A linking invariant for algebraic curves (ENGLISH)
Benoît Guerville-Ballé (Tokyo Gakugei University)
A linking invariant for algebraic curves (ENGLISH)
[ Abstract ]
We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).
We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Kiwamu Watanabe (Saitama University)
A Characterization of Symplectic Grassmannians (JAPANESE)
Kiwamu Watanabe (Saitama University)
A Characterization of Symplectic Grassmannians (JAPANESE)
[ Abstract ]
In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.
In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).
In the series of their works, J. M. Hwang and N. Mok have been developing the theory of Varieties of Minimal Rational Tangents (VMRT for short). In this direction, the results of Mok and J. Hong-Hwang allow us to recognize a homogeneous Fano manifold X of Picard number one by looking at its VMRT at a general point. This characterization works for all rational homogeneous manifolds of Picard number one whenever the VMRT is rational homogeneous, which is always the case except for the short root cases; namely for symplectic Grassmannians, and for two varieties of type F*4*.
In this talk we show that, if we impose that the VMRT is the expected one at every point of the variety, we may still characterize symplectic Grassmannians. This is a joint work with G. Occhetta and L. E. Sola Conde (arXiv:1604.06867).
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