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Seminar information archive
Seminar information archive ～12/05｜Today's seminar 12/06 | Future seminars 12/07～
2015/03/24
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Mina Aganagic (University of California, Berkeley)
Knots and Mirror Symmetry (ENGLISH)
Mina Aganagic (University of California, Berkeley)
Knots and Mirror Symmetry (ENGLISH)
[ Abstract ]
I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.
I will describe two conjectures relating knot theory and mirror symmetry. One can associate, to every knot K, one a Calabi-Yau manifold Y(K), which depends on the homotopy type of the knot only. The first conjecture is that Y(K) arises by a generalization of SYZ mirror symmetry, as mirror to the conifold, O(-1)+O(-1)->P^1. The second conjecture is that topological string provides a quantization of Y(K) which leads to quantum HOMFLY invariants of the knot. The conjectures are based on joint work with C. Vafa and also with T.Ekholm, L. Ng.
Lie Groups and Representation Theory
18:00-19:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences)
A Gysin formula for Hall-Littlewood polynomials
Piotr Pragacz (Institute of Mathematics, Polish Academy of Sciences)
A Gysin formula for Hall-Littlewood polynomials
[ Abstract ]
Schubert calculus on Grassmannians is governed by Schur S-functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations.
We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory.
With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers").
We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions.
Schubert calculus on Grassmannians is governed by Schur S-functions, the one on Lagrangian Grassmannians by Schur Q-functions. There were several attempts to give a unifying approach to both situations.
We propose to use Hall-Littlewood symmetric polynomials. They appeared implicitly in Hall's study of the combinatorial lattice structure of finite abelian p-groups and in Green's calculations of the characters of GL(n) over finite fields; they appeared explicitly in the work of Littlewood on some problems in representation theory.
With the projection in a Grassmann bundle, there is associated its Gysin map, induced by pushing forward cycles (topologists call it "integration along fibers").
We state and prove a Gysin formula for HL-polynomials in these bundles. We discuss its two specializations, giving better insights to previously known formulas for Schur S- and P-functions.
2015/03/20
Numerical Analysis Seminar
13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Gadi Fibich (Tel Aviv University)
Asymmetric Auctions (English)
Gadi Fibich (Tel Aviv University)
Asymmetric Auctions (English)
[ Abstract ]
Auctions are central to the modern economy, both on-line and off-line. A fundamental result in auction theory is that when bidders are symmetric (identical), then under quite general conditions, all auctions are revenue equivalent. While it is known that this result does not hold when bidders are asymmetric, the effect of bidders' asymmetry is poorly understood, since asymmetric auctions are much harder to analyze.
In this talk I will discuss the mathematical theory of asymmetric auctions. I will focus on asymmetric first-price auctions, where the mathematical model is given by a nonstandard system of $n$ nonlinear ordinary differential equations, with $2n$ boundary conditions and a free boundary. I will present various analytic and numerical approaches for this system. Then I will present some recent results on asymptotic revenue equivalence of asymmetric auctions.
Joint work with A. Gavious and N. Gavish.
Auctions are central to the modern economy, both on-line and off-line. A fundamental result in auction theory is that when bidders are symmetric (identical), then under quite general conditions, all auctions are revenue equivalent. While it is known that this result does not hold when bidders are asymmetric, the effect of bidders' asymmetry is poorly understood, since asymmetric auctions are much harder to analyze.
In this talk I will discuss the mathematical theory of asymmetric auctions. I will focus on asymmetric first-price auctions, where the mathematical model is given by a nonstandard system of $n$ nonlinear ordinary differential equations, with $2n$ boundary conditions and a free boundary. I will present various analytic and numerical approaches for this system. Then I will present some recent results on asymptotic revenue equivalence of asymmetric auctions.
Joint work with A. Gavious and N. Gavish.
2015/03/19
FMSP Lectures
9:00-11:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)
Matthew Gursky (Univ. Nortre Dame) 9:00-9:50
Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)
https://sites.google.com/site/princetontokyo/mini-courses
Gábor Székelyhidi (Univ. Nortre Dame) 10:10-11:00
Hessian type equations on compact Kähler manifolds (ENGLISH)
https://sites.google.com/site/princetontokyo/mini-courses
Matthew Gursky (Univ. Nortre Dame) 9:00-9:50
Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)
[ Abstract ]
In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.
These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.
I will also explain some recent work which attempts to understand the moduli space of critical metrics.
[ Reference URL ]In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.
These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.
I will also explain some recent work which attempts to understand the moduli space of critical metrics.
https://sites.google.com/site/princetontokyo/mini-courses
Gábor Székelyhidi (Univ. Nortre Dame) 10:10-11:00
Hessian type equations on compact Kähler manifolds (ENGLISH)
[ Abstract ]
I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.
[ Reference URL ]I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.
https://sites.google.com/site/princetontokyo/mini-courses
2015/03/18
FMSP Lectures
9:00-11:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)
Matthew Gursky (Univ. Nortre Dame) 9:00-9:50
Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)
https://sites.google.com/site/princetontokyo/mini-courses
Gábor Székelyhidi (Univ. Nortre Dame) 10:10-11:00
Hessian type equations on compact Kähler manifolds (ENGLISH)
https://sites.google.com/site/princetontokyo/mini-courses
Matthew Gursky (Univ. Nortre Dame) 9:00-9:50
Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)
[ Abstract ]
In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.
These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.
I will also explain some recent work which attempts to understand the moduli space of critical metrics.
[ Reference URL ]In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.
These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.
I will also explain some recent work which attempts to understand the moduli space of critical metrics.
https://sites.google.com/site/princetontokyo/mini-courses
Gábor Székelyhidi (Univ. Nortre Dame) 10:10-11:00
Hessian type equations on compact Kähler manifolds (ENGLISH)
[ Abstract ]
I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.
[ Reference URL ]I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.
https://sites.google.com/site/princetontokyo/mini-courses
2015/03/17
FMSP Lectures
13:30-15:00, 15:30-17:30 Room #Balcony A, Kavli IPMU (Graduate School of Math. Sci. Bldg.)
Hiroshi Iritani (Kyoto University)
Toric mirror symmetry via shift operators (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Iritani.pdf
Hiroshi Iritani (Kyoto University)
Toric mirror symmetry via shift operators (ENGLISH)
[ Abstract ]
Recently, shift operator for equivariant quantum cohomology
has been introduced in the work of Braverman, Maulik, Okounkov and
Pandharipande. This can be viewed as an equivariant lift of the
Seidel representation, and intertwines quantum connections with
different equivariant parameters.
In this series of talks, I will explain that shift operators essentially
"reconstruct" mirrors of toric varieties. More precisely we obtain the
following from basic properties of shift operators:
1. Givental's mirror theorem.
2. Landau-Ginzburg potential and primitive form.
3. Extended I-functions.
We will also see that the Gamma integral structure arises as
a solution to the difference equation defined by shift operators.
[ Reference URL ]Recently, shift operator for equivariant quantum cohomology
has been introduced in the work of Braverman, Maulik, Okounkov and
Pandharipande. This can be viewed as an equivariant lift of the
Seidel representation, and intertwines quantum connections with
different equivariant parameters.
In this series of talks, I will explain that shift operators essentially
"reconstruct" mirrors of toric varieties. More precisely we obtain the
following from basic properties of shift operators:
1. Givental's mirror theorem.
2. Landau-Ginzburg potential and primitive form.
3. Extended I-functions.
We will also see that the Gamma integral structure arises as
a solution to the difference equation defined by shift operators.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Iritani.pdf
FMSP Lectures
9:00-11:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)
Matthew Gursky (Univ. Nortre Dame) 9:00-9:50
Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)
https://sites.google.com/site/princetontokyo/mini-courses
Gábor Székelyhidi (Univ. Nortre Dame) 10:10-11:00
Hessian type equations on compact Kähler manifolds (ENGLISH)
https://sites.google.com/site/princetontokyo/mini-courses
Matthew Gursky (Univ. Nortre Dame) 9:00-9:50
Critical metrics for quadratic Riemannian functionals in dimension four (ENGLISH)
[ Abstract ]
In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.
These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.
I will also explain some recent work which attempts to understand the moduli space of critical metrics.
[ Reference URL ]In these lectures I will give an overview of a proof of existence, via gluing methods, of metrics which are critical points of quadratic Riemannian functionals. This is a joint project with J. Viaclovsky.
These are functionals on the space of metrics which are given by integrals of quadratic polynomials in the curvature tensor. Our approach is to construct these metrics on connected sums of Einstein four-manifolds, specifically the Fubini-Study metric on CP2 and the product metric on S2 X S2. Using these metrics in various gluing configurations, toric-invariant critical metrics are found on connected sums for a specific quadratic functional, which depends on the global geometry of the factors.
I will also explain some recent work which attempts to understand the moduli space of critical metrics.
https://sites.google.com/site/princetontokyo/mini-courses
Gábor Székelyhidi (Univ. Nortre Dame) 10:10-11:00
Hessian type equations on compact Kähler manifolds (ENGLISH)
[ Abstract ]
I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.
[ Reference URL ]I will discuss a priori estimates for a general class of nonlinear equations on compact Kähler manifolds. This unifies and generalizes several previous works on specific equations, such as the complex Monge-Ampère, Hessian, and inverse Hessian equations.
https://sites.google.com/site/princetontokyo/mini-courses
2015/03/16
FMSP Lectures
13:30-15:00, 15:30-17:30 Room #Balcony A, Kavli IPMU (Graduate School of Math. Sci. Bldg.)
Hiroshi Iritani (Kyoto University)
Toric mirror symmetry via shift operators (ENGLISH)
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Iritani.pdf
Hiroshi Iritani (Kyoto University)
Toric mirror symmetry via shift operators (ENGLISH)
[ Abstract ]
Recently, shift operator for equivariant quantum cohomology
has been introduced in the work of Braverman, Maulik, Okounkov and
Pandharipande. This can be viewed as an equivariant lift of the
Seidel representation, and intertwines quantum connections with
different equivariant parameters.
In this series of talks, I will explain that shift operators essentially
"reconstruct" mirrors of toric varieties. More precisely we obtain the
following from basic properties of shift operators:
1. Givental's mirror theorem.
2. Landau-Ginzburg potential and primitive form.
3. Extended I-functions.
We will also see that the Gamma integral structure arises as
a solution to the difference equation defined by shift operators.
[ Reference URL ]Recently, shift operator for equivariant quantum cohomology
has been introduced in the work of Braverman, Maulik, Okounkov and
Pandharipande. This can be viewed as an equivariant lift of the
Seidel representation, and intertwines quantum connections with
different equivariant parameters.
In this series of talks, I will explain that shift operators essentially
"reconstruct" mirrors of toric varieties. More precisely we obtain the
following from basic properties of shift operators:
1. Givental's mirror theorem.
2. Landau-Ginzburg potential and primitive form.
3. Extended I-functions.
We will also see that the Gamma integral structure arises as
a solution to the difference equation defined by shift operators.
http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_Iritani.pdf
2015/03/13
Colloquium
14:00-15:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)
Takayuki Oda (Graduate School of Mathematical Sciences, University of Tokyo)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~takayuki/index-j.html
Takayuki Oda (Graduate School of Mathematical Sciences, University of Tokyo)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~takayuki/index-j.html
Colloquium
16:30-17:30 Room #大講義室 (Graduate School of Math. Sci. Bldg.)
Shigeo KUSUOKA (Graduate School of Mathematical Sciences, University of Tokyo)
(JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/teacher/kusuoka.html
Shigeo KUSUOKA (Graduate School of Mathematical Sciences, University of Tokyo)
(JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/teacher/kusuoka.html
Colloquium
15:10-16:10 Room #大講義室 (Graduate School of Math. Sci. Bldg.)
Yoichi Miyaoka (Graduate School of Mathematical Sciences, University of Tokyo)
(JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/teacher/miyaoka.html
Yoichi Miyaoka (Graduate School of Mathematical Sciences, University of Tokyo)
(JAPANESE)
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/teacher/miyaoka.html
2015/03/10
Tuesday Seminar on Topology
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Andrei Pajitnov (Univ. de Nantes)
Arnold conjecture, Floer homology,
and augmentation ideals of finite groups.
(ENGLISH)
Andrei Pajitnov (Univ. de Nantes)
Arnold conjecture, Floer homology,
and augmentation ideals of finite groups.
(ENGLISH)
[ Abstract ]
Let H be a generic time-dependent 1-periodic
Hamiltonian on a closed weakly monotone
symplectic manifold M. We construct a refined version
of the Floer chain complex associated to (M,H),
and use it to obtain new lower bounds for the number P(H)
of the 1-periodic orbits of the corresponding hamiltonian
vector field. We prove in particular that
if the fundamental group of M is finite
and solvable or simple, then P(H)
is not less than the minimal number
of generators of the fundamental group.
This is joint work with Kaoru Ono.
Let H be a generic time-dependent 1-periodic
Hamiltonian on a closed weakly monotone
symplectic manifold M. We construct a refined version
of the Floer chain complex associated to (M,H),
and use it to obtain new lower bounds for the number P(H)
of the 1-periodic orbits of the corresponding hamiltonian
vector field. We prove in particular that
if the fundamental group of M is finite
and solvable or simple, then P(H)
is not less than the minimal number
of generators of the fundamental group.
This is joint work with Kaoru Ono.
2015/02/24
thesis presentations
15:00-16:15 Room #128 (Graduate School of Math. Sci. Bldg.)
小池 祐太 (情報・システム研究機構 統計数理研究所)
Covariance Estimation from Ultra-High-Frequency Date（超高頻度データに対する共分散推定） (JAPANESE)
小池 祐太 (情報・システム研究機構 統計数理研究所)
Covariance Estimation from Ultra-High-Frequency Date（超高頻度データに対する共分散推定） (JAPANESE)
2015/02/23
Operator Algebra Seminars
16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Zhenghan Wang (Microsoft Research Station Q)
Classification of (2+1)-TQFTs and its applications to physics and quantum computation (English)
Zhenghan Wang (Microsoft Research Station Q)
Classification of (2+1)-TQFTs and its applications to physics and quantum computation (English)
2015/02/19
Infinite Analysis Seminar Tokyo
13:30-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Shunsuke Tsuji (Graduate School of Mathematical Sciences, the University of Tokyo) 13:30-15:00
Skein algebra and mapping class group (JAPANESE)
An extension of the LMO functor (JAPANESE)
Shunsuke Tsuji (Graduate School of Mathematical Sciences, the University of Tokyo) 13:30-15:00
Skein algebra and mapping class group (JAPANESE)
[ Abstract ]
We define some filtrations of skein modules and the skein algebra on an oriented surface, and define the completed skein modules and
the completed skein algebra of the surface with respect to these filtration. We give an explicit formula for the action of the Dehn
twists on the completed skein modules in terms of the action of the completed skein algebra of the surface. As an application, we describe the action of the Johnson kernel on the completed skein modules.
Yuta Nozaki (Graduate School of Mathematical Sciences, the University of Tokyo) 15:30-17:00We define some filtrations of skein modules and the skein algebra on an oriented surface, and define the completed skein modules and
the completed skein algebra of the surface with respect to these filtration. We give an explicit formula for the action of the Dehn
twists on the completed skein modules in terms of the action of the completed skein algebra of the surface. As an application, we describe the action of the Johnson kernel on the completed skein modules.
An extension of the LMO functor (JAPANESE)
[ Abstract ]
Cheptea, Habiro and Massuyeau introduced the LMO functor as an extension of the LMO invariant of closed 3-manifolds. The LMO functor is “the monoidal category of Lagrangian cobordisms between surfaces with at most one boundary component” to “the monoidal category of top-substantial Jacobi diagrams”. In this talk, we extend the LMO functor to the case of any number of boundary components. Moreover, we explain that the internal degree d part of the extension is a finite-type invariant of degree d.
Cheptea, Habiro and Massuyeau introduced the LMO functor as an extension of the LMO invariant of closed 3-manifolds. The LMO functor is “the monoidal category of Lagrangian cobordisms between surfaces with at most one boundary component” to “the monoidal category of top-substantial Jacobi diagrams”. In this talk, we extend the LMO functor to the case of any number of boundary components. Moreover, we explain that the internal degree d part of the extension is a finite-type invariant of degree d.
Seminar on Probability and Statistics
16:30-17:40 Room #052 (Graduate School of Math. Sci. Bldg.)
Dobrislav Dobrev (Board of Governors of the Federal Reserve System, Division of International Finance)
TBA
Dobrislav Dobrev (Board of Governors of the Federal Reserve System, Division of International Finance)
TBA
[ Abstract ]
TBA
TBA
2015/02/18
Number Theory Seminar
16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)
Piotr Achinger (University of California, Berkeley)
Wild ramification and $K(\pi, 1)$ spaces (English)
Piotr Achinger (University of California, Berkeley)
Wild ramification and $K(\pi, 1)$ spaces (English)
[ Abstract ]
A smooth variety in characteristic zero is Zariski-locally a $K(\pi,1)$ space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin's proof that $\ell$-adic cohomology agrees with singular cohomology over $\mathbb{C}$. The characteristic $p$ variant of this is not known --- we do not even know whether the affine plane is a $K(\pi, 1)$ in positive characteristic! I will show how to reduce this question to a ``Bertini-type’' statement regarding wild ramification of $\ell$-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank $1$ and speculate on how one might treat the general case.
A smooth variety in characteristic zero is Zariski-locally a $K(\pi,1)$ space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin's proof that $\ell$-adic cohomology agrees with singular cohomology over $\mathbb{C}$. The characteristic $p$ variant of this is not known --- we do not even know whether the affine plane is a $K(\pi, 1)$ in positive characteristic! I will show how to reduce this question to a ``Bertini-type’' statement regarding wild ramification of $\ell$-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank $1$ and speculate on how one might treat the general case.
Numerical Analysis Seminar
14:30-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Nao Hamamuki (Hokkaido University)
Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)
Nao Hamamuki (Hokkaido University)
Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations (日本語)
Numerical Analysis Seminar
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Toshio Fukushima (National Astronomical Observatory)
Precise and fast computation of elliptic integrals and elliptic functions (日本語)
Toshio Fukushima (National Astronomical Observatory)
Precise and fast computation of elliptic integrals and elliptic functions (日本語)
[ Abstract ]
Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, $K(m)$, $E(m)$, and $\Pi(n|m)$, (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$, (iii) Jacobian elliptic functions, $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $\mathrm{dn}(u|m)$, and $\mathrm{am}(u|m)$, (iv) the inverse functions of $K(m)$ and $E(m)$, $m_K(K)$ and $m_E(E)$, (v) the inverse of a general incomplete elliptic integral in Jacobi's form, $G(\mathrm{am}(u|m),n|m)$, with respect to $u$, and (vi) the partial derivatives of $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $dn(u|m)$, $E(\mathrm{am}(u|m)|m)$, and $\Pi(\mathrm{am}(u|m),n|m)$ with respect to $u$ and those of $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$ with respect to $\phi$. In order to avoid the information loss when $n\ll 1$ and/or $m \ll 1$, focused are the associate incomplete elliptc integrals defined as $B(\phi|m)=[E(\phi|m)-(1-m)F(\phi|m)]/m$, $D(\phi|m)=[F(\phi|m)-E(\phi|m)]/m$, and $J(\phi,n|m)=[\Pi(\phi,n|m)-F(\phi|m)]/n$, and their complete versions, $B(m)=[E(m)-(1-m)K(m)]/m$, $D(m)=[K(m)-E(m)]/m$, and $J(n|m)=[\Pi(n|m)-K(m)]/n$. The main techniques used are (i) the piecewise approximation for single variable functions as $K(m)$, and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to $u = F(\phi|m)$. The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for $K(m)$ and $E(m)$, (ii) 2.5 times faster than Bulirsch's cel for $\Pi(n|m)$, (iii) slightly faster than Bulirsch's el1 for $F(\phi|m)$, (iv) 3.5 times faster than Carlson's $R_D$ for $E(\phi|m)$, (v) 3.5 times faster than Carlson's $R_C$, $R_D$, $R_F$, and $R_J$ for $\Pi(\phi,n|m)$, and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, and $\mathrm{dn}(u|m)$.
Summarized is the recent progress of the methods to compute (i) Legendre's normal form complete elliptic integrals of all three kinds, $K(m)$, $E(m)$, and $\Pi(n|m)$, (ii) Legendre's normal form incomplete elliptic integrals of all three kinds, $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$, (iii) Jacobian elliptic functions, $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $\mathrm{dn}(u|m)$, and $\mathrm{am}(u|m)$, (iv) the inverse functions of $K(m)$ and $E(m)$, $m_K(K)$ and $m_E(E)$, (v) the inverse of a general incomplete elliptic integral in Jacobi's form, $G(\mathrm{am}(u|m),n|m)$, with respect to $u$, and (vi) the partial derivatives of $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, $dn(u|m)$, $E(\mathrm{am}(u|m)|m)$, and $\Pi(\mathrm{am}(u|m),n|m)$ with respect to $u$ and those of $F(\phi|m)$, $E(\phi|m)$, and $\Pi(\phi,n|m)$ with respect to $\phi$. In order to avoid the information loss when $n\ll 1$ and/or $m \ll 1$, focused are the associate incomplete elliptc integrals defined as $B(\phi|m)=[E(\phi|m)-(1-m)F(\phi|m)]/m$, $D(\phi|m)=[F(\phi|m)-E(\phi|m)]/m$, and $J(\phi,n|m)=[\Pi(\phi,n|m)-F(\phi|m)]/n$, and their complete versions, $B(m)=[E(m)-(1-m)K(m)]/m$, $D(m)=[K(m)-E(m)]/m$, and $J(n|m)=[\Pi(n|m)-K(m)]/n$. The main techniques used are (i) the piecewise approximation for single variable functions as $K(m)$, and (ii) the combination of repeated usage of the half and double argument transformations and the truncated Maclaurin series expansions with respect to $u = F(\phi|m)$. The new methods are of the full double precision accuracy without any chance of cancellation against small input arguments. They run significantly faster than the existing methods: (i) 2.5 times faster than Cody's Chebyshev polynomial approximations for $K(m)$ and $E(m)$, (ii) 2.5 times faster than Bulirsch's cel for $\Pi(n|m)$, (iii) slightly faster than Bulirsch's el1 for $F(\phi|m)$, (iv) 3.5 times faster than Carlson's $R_D$ for $E(\phi|m)$, (v) 3.5 times faster than Carlson's $R_C$, $R_D$, $R_F$, and $R_J$ for $\Pi(\phi,n|m)$, and (vi) 1.5 times faster than Bulirsch's \texttt{sncndn} for $\mathrm{sn}(u|m)$, $\mathrm{cn}(u|m)$, and $\mathrm{dn}(u|m)$.
2015/02/10
thesis presentations
9:30-10:45 Room #118 (Graduate School of Math. Sci. Bldg.)
三原 朋樹 (東京大学大学院数理科学研究科)
On a new geometric construction of a family of Galois representations associated to modular forms
（保型形式に付随するガロア表現の族の新たな幾何的構成について） (JAPANESE)
三原 朋樹 (東京大学大学院数理科学研究科)
On a new geometric construction of a family of Galois representations associated to modular forms
（保型形式に付随するガロア表現の族の新たな幾何的構成について） (JAPANESE)
thesis presentations
11:00-12:15 Room #118 (Graduate School of Math. Sci. Bldg.)
田中 雄一郎 (東京大学大学院数理科学研究科)
VISIBLE ACTIONS OF REDUCTIVE ALGEBRAIC GROUPS ON COMPLEX ALGEBRAIC VARIETIES（簡約代数群の複素代数多様体への可視的作用について） (JAPANESE)
田中 雄一郎 (東京大学大学院数理科学研究科)
VISIBLE ACTIONS OF REDUCTIVE ALGEBRAIC GROUPS ON COMPLEX ALGEBRAIC VARIETIES（簡約代数群の複素代数多様体への可視的作用について） (JAPANESE)
thesis presentations
13:00-14:15 Room #118 (Graduate School of Math. Sci. Bldg.)
中村 あかね (東京大学大学院数理科学研究科)
Autonomous limit of 4-dimensional Painlev´e-type equations and singular fibers of spectral curve fibrations（4次元Painlev´e 型方程式の自励極限とスペクトラル曲線ファイブレーションの特異ファイバー） (JAPANESE)
中村 あかね (東京大学大学院数理科学研究科)
Autonomous limit of 4-dimensional Painlev´e-type equations and singular fibers of spectral curve fibrations（4次元Painlev´e 型方程式の自励極限とスペクトラル曲線ファイブレーションの特異ファイバー） (JAPANESE)
thesis presentations
14:30-15:45 Room #118 (Graduate School of Math. Sci. Bldg.)
三田 史彦 (東京大学大学院数理科学研究科)
Fukaya categories and blow-ups（深谷圏とブローアップ） (JAPANESE)
三田 史彦 (東京大学大学院数理科学研究科)
Fukaya categories and blow-ups（深谷圏とブローアップ） (JAPANESE)
thesis presentations
11:00-12:15 Room #122 (Graduate School of Math. Sci. Bldg.)
中村 勇哉 (東京大学大学院数理科学研究科)
Studies on the minimal log discrepancies（極小ログ食い違い係数の研究） (JAPANESE)
中村 勇哉 (東京大学大学院数理科学研究科)
Studies on the minimal log discrepancies（極小ログ食い違い係数の研究） (JAPANESE)
thesis presentations
14:30-15:45 Room #122 (Graduate School of Math. Sci. Bldg.)
周 冠宇 (東京大学大学院数理科学研究科)
Numerical analysis of various domain-penalty and boundary-penalty methods（様々な領域処罰法および境界処罰法の数値解析） (JAPANESE)
周 冠宇 (東京大学大学院数理科学研究科)
Numerical analysis of various domain-penalty and boundary-penalty methods（様々な領域処罰法および境界処罰法の数値解析） (JAPANESE)
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