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Seminar information archive
Seminar information archive ～06/30｜Today's seminar 07/01 | Future seminars 07/02～
2023/09/14
Applied Analysis
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Michał Łasica (The Polish Academy of Sciences)
Bounds on the gradient of minimizers in variational denoising (English)
https://forms.gle/C39ZLdQNVHyVmJ4j8
Michał Łasica (The Polish Academy of Sciences)
Bounds on the gradient of minimizers in variational denoising (English)
[ Abstract ]
We consider minimization problem for a class of convex integral functionals composed of two terms:
-- a regularizing term of linear growth in the gradient,
-- and a fidelity term penalizing the distance from a given function.
To ensure that such functionals attain their minima, one needs to extend their domain to the BV space. In particular minimizers may exhibit jump discontinuities. I will discuss estimates on the gradient of minimizers in terms of the data, focusing on singular part of the gradient measure.
The talk is based on joint works with P. Rybka, Z. Grochulska and A. Chambolle.
[ Reference URL ]We consider minimization problem for a class of convex integral functionals composed of two terms:
-- a regularizing term of linear growth in the gradient,
-- and a fidelity term penalizing the distance from a given function.
To ensure that such functionals attain their minima, one needs to extend their domain to the BV space. In particular minimizers may exhibit jump discontinuities. I will discuss estimates on the gradient of minimizers in terms of the data, focusing on singular part of the gradient measure.
The talk is based on joint works with P. Rybka, Z. Grochulska and A. Chambolle.
https://forms.gle/C39ZLdQNVHyVmJ4j8
2023/09/07
Applied Analysis
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Samuel Mercer (Delft University of Technology)
Uniform Convergence of Gradient Flows on a Stack of Banach Spaces (English)
https://forms.gle/T8yWr2gDTYzj8vkE7
Samuel Mercer (Delft University of Technology)
Uniform Convergence of Gradient Flows on a Stack of Banach Spaces (English)
[ Abstract ]
Within this talk I will recall the classical result: Given a sequence of convex functionals on a Hilbert space, Gamma-convergence of this sequence implies uniform convergence on finite time-intervals for their gradient flows. I will then discuss a generalisation for this result. In particular our functionals are defined on a sequence of distinct Banach spaces that can be stacked together inside of a unifying space. We will study a kind of gradient flow for our functionals inside their respective Banach space and ask the following question. What structure is necessary within our unifying space to attain uniform convergence of gradient flows?
[ Reference URL ]Within this talk I will recall the classical result: Given a sequence of convex functionals on a Hilbert space, Gamma-convergence of this sequence implies uniform convergence on finite time-intervals for their gradient flows. I will then discuss a generalisation for this result. In particular our functionals are defined on a sequence of distinct Banach spaces that can be stacked together inside of a unifying space. We will study a kind of gradient flow for our functionals inside their respective Banach space and ask the following question. What structure is necessary within our unifying space to attain uniform convergence of gradient flows?
https://forms.gle/T8yWr2gDTYzj8vkE7
2023/08/30
Discrete mathematical modelling seminar
16:30-17:30 Room #370 (Graduate School of Math. Sci. Bldg.)
Joshua Capel (University of New South Wales)
Classical structure theory for second-order semi-degenerate super-integrable systems (English)
Joshua Capel (University of New South Wales)
Classical structure theory for second-order semi-degenerate super-integrable systems (English)
[ Abstract ]
Second-order superintegrable systems have long been a subject of interest in the study of integral systems, primarily due to their close connection with separation of variables. Among these, the 'non-degenerate' second-order systems on constant curvature spaces have been particularly well studies, and are mostly understood.
A non-degenerate system in n-dimensions comprises an (n+1)-dimensional vector space of potentials, along with 2n-1 second-order constants that commute with the Hamiltonian of the system.
In contrast, a semi-degenerate system is characterised by having fewer than n+1 parameters. In this talk, we will discuss the structure theory of truly n-dimensional potentials (meaning potentials that are not just restrictions of (n+1)-dimensional counterparts). We will see that the classical structure theory appears just as rich as the non-degenerate case.
This talk is joint work with Jeremy Nugent and Jonathan Kress.
Second-order superintegrable systems have long been a subject of interest in the study of integral systems, primarily due to their close connection with separation of variables. Among these, the 'non-degenerate' second-order systems on constant curvature spaces have been particularly well studies, and are mostly understood.
A non-degenerate system in n-dimensions comprises an (n+1)-dimensional vector space of potentials, along with 2n-1 second-order constants that commute with the Hamiltonian of the system.
In contrast, a semi-degenerate system is characterised by having fewer than n+1 parameters. In this talk, we will discuss the structure theory of truly n-dimensional potentials (meaning potentials that are not just restrictions of (n+1)-dimensional counterparts). We will see that the classical structure theory appears just as rich as the non-degenerate case.
This talk is joint work with Jeremy Nugent and Jonathan Kress.
2023/08/25
thesis presentations
13:00-14:15 Room #123 (Graduate School of Math. Sci. Bldg.)
KITAMURA Kan (Graduate School of Mathematical Sciences University of Tokyo)
Discrete quantum subgroups of quantum doubles
(量子ダブルの離散部分量子群について)
KITAMURA Kan (Graduate School of Mathematical Sciences University of Tokyo)
Discrete quantum subgroups of quantum doubles
(量子ダブルの離散部分量子群について)
2023/08/24
Tokyo-Nagoya Algebra Seminar
10:30-12:00 Online
Sota Asai (University of Tokyo)
TF equivalence on the real Grothendieck group (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Sota Asai (University of Tokyo)
TF equivalence on the real Grothendieck group (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/08/22
Tuesday Seminar of Analysis
16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Daniel Parra (Universidad de Santiago de Chile)
Towards a Levinson's Theorem for Discrete Magnetic operators on tubes under finite rank perturbations (English)
https://forms.gle/VBp4nSnYYKVpXFhB9
Daniel Parra (Universidad de Santiago de Chile)
Towards a Levinson's Theorem for Discrete Magnetic operators on tubes under finite rank perturbations (English)
[ Abstract ]
In this talk we study a family of magnetic Hamiltonians on discrete tubes under a finite rank perturbation supported on its border. We go into detail for the case of rank $2$ and show how the eigenvalues can be related to the scattering matrix to exhibit an index theorem in the tradition of Levison’s theorem. We then turn to the general case, discuss the different spectral scenarios that can occur and explain the C*-algebraic framework that could allow us to treat this case. This is an ongoing work with S. Richard (Nagoya), V. Austen (Nagoya) and A. Rennie (Wollongong).
[ Reference URL ]In this talk we study a family of magnetic Hamiltonians on discrete tubes under a finite rank perturbation supported on its border. We go into detail for the case of rank $2$ and show how the eigenvalues can be related to the scattering matrix to exhibit an index theorem in the tradition of Levison’s theorem. We then turn to the general case, discuss the different spectral scenarios that can occur and explain the C*-algebraic framework that could allow us to treat this case. This is an ongoing work with S. Richard (Nagoya), V. Austen (Nagoya) and A. Rennie (Wollongong).
https://forms.gle/VBp4nSnYYKVpXFhB9
2023/08/21
Classical Analysis
10:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Xiaomeng Xu (BICMR, China) 10:00-11:30
Stokes matrices of confluent hypergeometric systems and the isomonodromy deformation equations (ENGLISH)
Stokes matrices of quantum confluent hypergeometric systems and the representation of quantum groups (ENGLISH)
The WKB approximation of (quantum) confluent hypergeometric systems, Cauchy interlacing inequality and crystal basis (ENGLISH)
Xiaomeng Xu (BICMR, China) 10:00-11:30
Stokes matrices of confluent hypergeometric systems and the isomonodromy deformation equations (ENGLISH)
[ Abstract ]
This talk first gives an introduction to the Stokes matrices of a linear meromorphic system of Poncaré rank 1, and the associated nonlinear isomonodromy deformation equation. The nonlinear equation naturally arises from the theory of Frobenius manifolds, stability conditions, Poisson-Lie groups and so on, and can be seen as a higher rank generalizations of the sixth Painlevé equation. The talk then gives a parameterization of the asymptotics of the solutions of the isomonodromy equation at a critical point, the explicit formula of the monodromy/Stokes matrices of the linear problem via the parameterization, as well as a connection formula between two differential critical points. It can be seen as a generalization of Jimbo's work for the sixth Painlevé equation to a higher rank case. It is partially based on a joint work with Qian Tang.
Xiaomeng Xu (BICMR, China) 14:00-15:30This talk first gives an introduction to the Stokes matrices of a linear meromorphic system of Poncaré rank 1, and the associated nonlinear isomonodromy deformation equation. The nonlinear equation naturally arises from the theory of Frobenius manifolds, stability conditions, Poisson-Lie groups and so on, and can be seen as a higher rank generalizations of the sixth Painlevé equation. The talk then gives a parameterization of the asymptotics of the solutions of the isomonodromy equation at a critical point, the explicit formula of the monodromy/Stokes matrices of the linear problem via the parameterization, as well as a connection formula between two differential critical points. It can be seen as a generalization of Jimbo's work for the sixth Painlevé equation to a higher rank case. It is partially based on a joint work with Qian Tang.
Stokes matrices of quantum confluent hypergeometric systems and the representation of quantum groups (ENGLISH)
[ Abstract ]
This talk studies a quantum analog of Stokes matrices of confluent hypergeometric systems. It first gives an introduction to the Stokes phenomenon of an irregular Knizhnik–Zamolodchikov at a second order pole, associated to a regular semisimple element u and a representation $L(\lambda)$ of $gl_n$. It then shows that the Stokes matrices of the
irregular Knizhnik–Zamolodchikov equation define representation of $U_q(gl_n)$ on $L(\lambda)$. In then end, using the isomonodromy approach, it derives an explicit expression of the regularized limit of the Stokes matrices as the regular semisimple element u goes to the caterpillar point in the wonderful compactification.
Xiaomeng Xu (BICMR, China) 16:00-17:30This talk studies a quantum analog of Stokes matrices of confluent hypergeometric systems. It first gives an introduction to the Stokes phenomenon of an irregular Knizhnik–Zamolodchikov at a second order pole, associated to a regular semisimple element u and a representation $L(\lambda)$ of $gl_n$. It then shows that the Stokes matrices of the
irregular Knizhnik–Zamolodchikov equation define representation of $U_q(gl_n)$ on $L(\lambda)$. In then end, using the isomonodromy approach, it derives an explicit expression of the regularized limit of the Stokes matrices as the regular semisimple element u goes to the caterpillar point in the wonderful compactification.
The WKB approximation of (quantum) confluent hypergeometric systems, Cauchy interlacing inequality and crystal basis (ENGLISH)
[ Abstract ]
This talk studies the WKB approximation of the linear meromorphic systems of Poncaré rank 1 appearing in talk 1 and 2, via the isomonodromy approach. In the classical case, it unveils a relation between the WKB approximation, the Cauchy interlacing inequality and cluster algebras with the help of the spectral network; in the quantum case, motivated by the crystal limit of the quantum groups, it shows a relation between the WKB approximation and the gl_n-crystal structures. It is partially based on a joint work with
Anton Alekseev, Andrew Neitzke and Yan Zhou.
This talk studies the WKB approximation of the linear meromorphic systems of Poncaré rank 1 appearing in talk 1 and 2, via the isomonodromy approach. In the classical case, it unveils a relation between the WKB approximation, the Cauchy interlacing inequality and cluster algebras with the help of the spectral network; in the quantum case, motivated by the crystal limit of the quantum groups, it shows a relation between the WKB approximation and the gl_n-crystal structures. It is partially based on a joint work with
Anton Alekseev, Andrew Neitzke and Yan Zhou.
2023/08/08
Lectures
14:00-15:00 Room #123 (Graduate School of Math. Sci. Bldg.)
Dror Bar-Natan (University of Toronto)
Cars, Interchanges, Traffic Counters, and some Pretty Darned Good Knot Invariants (English)
http://www.math.toronto.edu/~drorbn/Talks/Tokyo-230808/
Dror Bar-Natan (University of Toronto)
Cars, Interchanges, Traffic Counters, and some Pretty Darned Good Knot Invariants (English)
[ Abstract ]
Reporting on joint work with Roland van der Veen, I'll tell you some stories about ρ_1, an easy to define, strong, fast to compute, homomorphic, and well-connected knot invariant. ρ_1 was first studied by Rozansky and Overbay, it is dominated by the coloured Jones polynomial (but it isn't lesser!), it has far-reaching generalizations, and I wish I understood it.
[ Reference URL ]Reporting on joint work with Roland van der Veen, I'll tell you some stories about ρ_1, an easy to define, strong, fast to compute, homomorphic, and well-connected knot invariant. ρ_1 was first studied by Rozansky and Overbay, it is dominated by the coloured Jones polynomial (but it isn't lesser!), it has far-reaching generalizations, and I wish I understood it.
http://www.math.toronto.edu/~drorbn/Talks/Tokyo-230808/
Lectures
15:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Dror Bar-Natan (University of Toronto)
Shifted Partial Quadratics, their Pushwards, and Signature Invariants for Tangles (English)
http://www.math.toronto.edu/~drorbn/Talks/Tokyo-230808/
Dror Bar-Natan (University of Toronto)
Shifted Partial Quadratics, their Pushwards, and Signature Invariants for Tangles (English)
[ Abstract ]
Following a general discussion of the computation of zombians of unfinished columbaria (with examples), I will tell you about my recent joint work with Jessica Liu on what we feel is the "textbook" extension of knot signatures to tangles, which for unknown reasons, is not in any of the textbooks that we know.
[ Reference URL ]Following a general discussion of the computation of zombians of unfinished columbaria (with examples), I will tell you about my recent joint work with Jessica Liu on what we feel is the "textbook" extension of knot signatures to tangles, which for unknown reasons, is not in any of the textbooks that we know.
http://www.math.toronto.edu/~drorbn/Talks/Tokyo-230808/
2023/08/07
Tokyo Probability Seminar
17:00-18:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Freddy Delbaen (Professor emeritus at ETH Zurich)
Approximation of Random Variables by Elements that are independent of a given sigma algebra (English)
Freddy Delbaen (Professor emeritus at ETH Zurich)
Approximation of Random Variables by Elements that are independent of a given sigma algebra (English)
[ Abstract ]
Given a square integrable m-dimensional random variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sub sigma algebra $\mathcal{A}$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal{A}$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory. The proof needs elements from transportation theory, a parametric version due to Dudley and Blackwell of the Skorohod theorem, selection theorems, … The problem also triggers other approximation problems. (joint work with C. Majumdar)
Given a square integrable m-dimensional random variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sub sigma algebra $\mathcal{A}$, we show that there exists another m-dimensional random variable $Y$, independent of $\mathcal{A}$ and minimising the $L^2$ distance to $X$. Such results have an importance to fairness and bias reduction in Artificial Intelligence, Machine Learning and Network Theory. The proof needs elements from transportation theory, a parametric version due to Dudley and Blackwell of the Skorohod theorem, selection theorems, … The problem also triggers other approximation problems. (joint work with C. Majumdar)
2023/07/31
Operator Algebra Seminars
16:45-18:15 Room #128 (Graduate School of Math. Sci. Bldg.)
Roberto Longo (Univ. Rome, ”Tor Vergata”)
Operator Algebras and Quantum Field Theory: introductory elements
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
Roberto Longo (Univ. Rome, ”Tor Vergata”)
Operator Algebras and Quantum Field Theory: introductory elements
[ Reference URL ]
https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm
2023/07/28
Algebraic Geometry Seminar
13:30-15:00 Room #ハイブリッド開催/002 (Graduate School of Math. Sci. Bldg.)
Shihoko Ishii (The University of Tokyo)
TBA
Shihoko Ishii (The University of Tokyo)
TBA
2023/07/21
Algebraic Geometry Seminar
13:30-15:00 Room #ハイブリッド開催/056 (Graduate School of Math. Sci. Bldg.)
Sho Ejiri (Osaka Metropolitan University)
The Demailly--Peternell--Schneider conjecture is true in positive characteristic
Sho Ejiri (Osaka Metropolitan University)
The Demailly--Peternell--Schneider conjecture is true in positive characteristic
[ Abstract ]
In 1993, Demailly, Peternell and Schneider conjectured that the Albanese morphism of a compact K\"{a}hler manifold with nef anti-canonical divisor is surjective. For smooth projective varieties of characteristic zero, the conjecture was verified by Zhang in 1996. In positive characteristic, the conjecture was solved under the assumption that the geometric generic fiber F of the Albanese morphism has only mild singularities. However, F may have bad singularities even if we restrict ourselves to the case when the anti-canonical divisor is nef. In this talk, we prove the conjecture in positive characteristic without any extra assumption. We also discuss properties of the Albanese morphism, such as flatness or local isotriviality. This talk is based on joint work with Zsolt Patakfalvi.
In 1993, Demailly, Peternell and Schneider conjectured that the Albanese morphism of a compact K\"{a}hler manifold with nef anti-canonical divisor is surjective. For smooth projective varieties of characteristic zero, the conjecture was verified by Zhang in 1996. In positive characteristic, the conjecture was solved under the assumption that the geometric generic fiber F of the Albanese morphism has only mild singularities. However, F may have bad singularities even if we restrict ourselves to the case when the anti-canonical divisor is nef. In this talk, we prove the conjecture in positive characteristic without any extra assumption. We also discuss properties of the Albanese morphism, such as flatness or local isotriviality. This talk is based on joint work with Zsolt Patakfalvi.
Colloquium
15:30-16:30 Room #大講義室(auditorium) (Graduate School of Math. Sci. Bldg.)
If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please apply from the form at [Reference URL].
Masahito Yamazaki (Kavli Institute for the Physics and Mathematics of the Uniiverse, the University of Tokyo)
Quantum Field Theory in Mathematics (JAPANESE)
https://forms.gle/igR5ZB5AwginXBt49
If you do not belong to Graduate School of Mathematical Sciences, the University of Tokyo, please apply from the form at [Reference URL].
Masahito Yamazaki (Kavli Institute for the Physics and Mathematics of the Uniiverse, the University of Tokyo)
Quantum Field Theory in Mathematics (JAPANESE)
[ Abstract ]
While quantum field theory has primarily been a theory in physics, it has also been a source of new ideas in mathematics, and has facilitated interactions between different branches of mathematics. There have also been many attempts to formulate quantum field theories themselves rigorously in mathematics. In this lecture we will discuss some examples of research in knot invariants and integrable models, to illustrate the impact of quantum field theories and string theory in modern mathematics.
[ Reference URL ]While quantum field theory has primarily been a theory in physics, it has also been a source of new ideas in mathematics, and has facilitated interactions between different branches of mathematics. There have also been many attempts to formulate quantum field theories themselves rigorously in mathematics. In this lecture we will discuss some examples of research in knot invariants and integrable models, to illustrate the impact of quantum field theories and string theory in modern mathematics.
https://forms.gle/igR5ZB5AwginXBt49
2023/07/19
thesis presentations
9:15-10:30 Room #123 (Graduate School of Math. Sci. Bldg.)
TSUTSUI Yuki (Graduate School of Mathematical Sciences University of Tokyo)
Graded modules associated with permissible C∞-divisors on tropical manifolds
(トロピカル多様体上の可容C∞因子に付随した次数付き加群)
TSUTSUI Yuki (Graduate School of Mathematical Sciences University of Tokyo)
Graded modules associated with permissible C∞-divisors on tropical manifolds
(トロピカル多様体上の可容C∞因子に付随した次数付き加群)
thesis presentations
10:30-11:45 Room #117 (Graduate School of Math. Sci. Bldg.)
TSUBOUCHI Shuntaro ( Graduate School of Mathematical Sciences University of Tokyo)
A regularity theory for perturbed singular elliptic and parabolic equations
(摂動特異楕円型および放物型方程式に対する正則性理論)
TSUBOUCHI Shuntaro ( Graduate School of Mathematical Sciences University of Tokyo)
A regularity theory for perturbed singular elliptic and parabolic equations
(摂動特異楕円型および放物型方程式に対する正則性理論)
thesis presentations
10:45-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)
KOBAYASHI Kenta (Graduate School of Mathematical Sciences University of Tokyo)
Elliptic genera of complete intersection Calabi-Yau 17-folds in F4-Grassmannians
(F4型グラスマン多様体内の17次元完全交叉カラビ・ヤウ多様体の楕円種数)
KOBAYASHI Kenta (Graduate School of Mathematical Sciences University of Tokyo)
Elliptic genera of complete intersection Calabi-Yau 17-folds in F4-Grassmannians
(F4型グラスマン多様体内の17次元完全交叉カラビ・ヤウ多様体の楕円種数)
2023/07/14
Tokyo-Nagoya Algebra Seminar
10:30-12:00 Online
Michael Wemyss (University of Glasgow)
Local Forms of Noncommutative Functions and Applications (English)
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Michael Wemyss (University of Glasgow)
Local Forms of Noncommutative Functions and Applications (English)
[ Abstract ]
This talk will explain how Arnold's results for commutative singularities can be extended into the noncommutative setting, with the main result being a classification of certain Jacobi algebras
arising from (complete) free algebras. This class includes finite dimensional Jacobi algebras, and also Jacobi algebras of GK dimension one, suitably interpreted. The surprising thing is that a classification should exist at all, and it is even more surprising that ADE enters.
I will spend most of my time explaining what the algebras are, what they classify, and how to intrinsically extract ADE information from them. At the end, I'll explain why I'm really interested in this problem, an update including results on different quivers, and the applications of the above classification to curve counting and birational geometry. This is joint work with Gavin Brown.
Meeting ID: 863 9598 8196
passcode: 423160
[ Reference URL ]This talk will explain how Arnold's results for commutative singularities can be extended into the noncommutative setting, with the main result being a classification of certain Jacobi algebras
arising from (complete) free algebras. This class includes finite dimensional Jacobi algebras, and also Jacobi algebras of GK dimension one, suitably interpreted. The surprising thing is that a classification should exist at all, and it is even more surprising that ADE enters.
I will spend most of my time explaining what the algebras are, what they classify, and how to intrinsically extract ADE information from them. At the end, I'll explain why I'm really interested in this problem, an update including results on different quivers, and the applications of the above classification to curve counting and birational geometry. This is joint work with Gavin Brown.
Meeting ID: 863 9598 8196
passcode: 423160
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/07/13
Information Mathematics Seminar
16:50-18:35 Room #128 (Graduate School of Math. Sci. Bldg.)
Tatsuaki Okamoto (NTT)
Fully Homorphic Encryption and Functional Encryption (Japanese)
Tatsuaki Okamoto (NTT)
Fully Homorphic Encryption and Functional Encryption (Japanese)
[ Abstract ]
Explanation of fully homorphic encryption and functional encryption
Explanation of fully homorphic encryption and functional encryption
Discrete mathematical modelling seminar
17:30-18:30 Online
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Galina Filipuk (University of Warsaw)
On the Painlevé XXV - Ermakov equation (English)
This seminar will be held using Zoom. If you wish to participate, please contact R. Willox by email.
Galina Filipuk (University of Warsaw)
On the Painlevé XXV - Ermakov equation (English)
[ Abstract ]
We study a nonlinear second order ordinary differential equation which we call the Ermakov-Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced to the Ermakov or the Painlevé XXV equation. The Ermakov-Painlevé XXV equation also arises from a generalized Riccati equation and the related third order linear differential equation via the Schwarzian derivative. Starting from the Riccati equation and the second-order element of the Riccati chain as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions. Two families of Bäcklund transformations, which link the linear and nonlinear equations under investigation, are obtained. Some analytical examples will be given and discussed.
The talk will be mainly based on the paper
S. Carillo, A. Chichurin, G. Filipuk, F. Zullo, Schwarzian derivative, Painleve XXV--Ermakov equation and Backlund transformations, accepted in Mathematische Nachrichten, https://doi.org/10.1002/mana.202200180, available at arXiv:2201.02267 [nlin.SI].
We study a nonlinear second order ordinary differential equation which we call the Ermakov-Painlevé XXV equation since under certain restrictions on its coefficients it can be reduced to the Ermakov or the Painlevé XXV equation. The Ermakov-Painlevé XXV equation also arises from a generalized Riccati equation and the related third order linear differential equation via the Schwarzian derivative. Starting from the Riccati equation and the second-order element of the Riccati chain as the simplest examples of linearizable equations, by introducing a suitable change of variables, it is shown how the Schwarzian derivative represents a key tool in the construction of solutions. Two families of Bäcklund transformations, which link the linear and nonlinear equations under investigation, are obtained. Some analytical examples will be given and discussed.
The talk will be mainly based on the paper
S. Carillo, A. Chichurin, G. Filipuk, F. Zullo, Schwarzian derivative, Painleve XXV--Ermakov equation and Backlund transformations, accepted in Mathematische Nachrichten, https://doi.org/10.1002/mana.202200180, available at arXiv:2201.02267 [nlin.SI].
2023/07/11
Tuesday Seminar of Analysis
16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)
Julian López-Gómez (Complutense University of Madrid)
Nodal solutions for a class of degenerate BVP’s (English)
https://forms.gle/S3VgMSWg9wUP69cY6
Julian López-Gómez (Complutense University of Madrid)
Nodal solutions for a class of degenerate BVP’s (English)
[ Abstract ]
In this talk we characterize the existence of nodal solutions for a generalized class of one-dimensional diffusive logistic type equations, including
\[−u''=\lambda u−a(x)u^3,\quad x∈[0,L],\]
under the boundary conditions $u(0)=u(L)=0$, where $\lambda$ is regarded as a bifurcation parameter, and the non-negative weight function $a(x)$ vanishes on some subinterval
\[ [\alpha,\beta]\subset (0,L)\]
with $\alpha<\beta$.
At a later stage, the general case when $a(x)$ vanishes on finitely many subintervals with the same length is analyzed. Finally, we construct some examples with classical non-degenerate weights, with $a(x)>0$ for all $x∈[0,L]$, where the BVP has an arbitrarily large number of solutions with one node in $(0,L)$. These are the first examples of this nature constructed in the literature.
References:
P. Cubillos, J. López-Gómez and A. Tellini, Multiplicity of nodal solutions in classical non-degenerate logistic equations, El. Res. Archive 30 (2022), 898—928.
J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Disc. Cont. Dyn. Syst. B 22 (2017), 923—946.
J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP’s, Top. Meth. Nonl. Anal. 49 (2017), 359—376.
J. López-Gómez and P. H. Rabinowitz, The estructure of the set of 1-node solutions for a class of degenerate BVP’s, J. Differential Equations 268 (2020), 4691—4732.
P. H. Rabinowitz, A note on a anonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9 (1971), 536—548.
[ Reference URL ]In this talk we characterize the existence of nodal solutions for a generalized class of one-dimensional diffusive logistic type equations, including
\[−u''=\lambda u−a(x)u^3,\quad x∈[0,L],\]
under the boundary conditions $u(0)=u(L)=0$, where $\lambda$ is regarded as a bifurcation parameter, and the non-negative weight function $a(x)$ vanishes on some subinterval
\[ [\alpha,\beta]\subset (0,L)\]
with $\alpha<\beta$.
At a later stage, the general case when $a(x)$ vanishes on finitely many subintervals with the same length is analyzed. Finally, we construct some examples with classical non-degenerate weights, with $a(x)>0$ for all $x∈[0,L]$, where the BVP has an arbitrarily large number of solutions with one node in $(0,L)$. These are the first examples of this nature constructed in the literature.
References:
P. Cubillos, J. López-Gómez and A. Tellini, Multiplicity of nodal solutions in classical non-degenerate logistic equations, El. Res. Archive 30 (2022), 898—928.
J. López-Gómez, M. Molina-Meyer and P. H. Rabinowitz, Global bifurcation diagrams of one-node solutions on a class of degenerate boundary value problems, Disc. Cont. Dyn. Syst. B 22 (2017), 923—946.
J. López-Gómez and P. H. Rabinowitz, Nodal solutions for a class of degenerate one dimensional BVP’s, Top. Meth. Nonl. Anal. 49 (2017), 359—376.
J. López-Gómez and P. H. Rabinowitz, The estructure of the set of 1-node solutions for a class of degenerate BVP’s, J. Differential Equations 268 (2020), 4691—4732.
P. H. Rabinowitz, A note on a anonlinear eigenvalue problem for a class of differential equations, J. Differential Equations 9 (1971), 536—548.
https://forms.gle/S3VgMSWg9wUP69cY6
2023/07/10
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Ken-Ichi Yoshikawa (Kyoto University)
Degenerations of Riemann surfaces and small eigenvalues of the Laplacian (日本語)
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A
Ken-Ichi Yoshikawa (Kyoto University)
Degenerations of Riemann surfaces and small eigenvalues of the Laplacian (日本語)
[ Abstract ]
In this talk, we consider a proper surjective holomorphic map from a smooth projective surface to a compact Riemann surface. Near a singular fiber, this is viewed as a one-parameter degeneration of compact Riemann surfaces. We fix a Kähler metric on the projective surface and consider the Kähler metric on the fibers induced from this metric. In this setting, for each regular fiber, we can consider the Laplacian acting on the functions on the fiber. It is known that for any k, the k-th eigenvalue of the Laplacian extends to a continuous function on the base curve. In particular, if the singular fiber is not irreducible, some eigenvalues of the Laplacian of the regular fiber converge to zero as the regular fiber approaches to the singular fiber. We call such eigenvalues small eigenvalues. In this talk, when the singular fiber is reduced, we will explain the asymptotic behavior of the product of all small eigenvalues of the Laplacian.
[ Reference URL ]In this talk, we consider a proper surjective holomorphic map from a smooth projective surface to a compact Riemann surface. Near a singular fiber, this is viewed as a one-parameter degeneration of compact Riemann surfaces. We fix a Kähler metric on the projective surface and consider the Kähler metric on the fibers induced from this metric. In this setting, for each regular fiber, we can consider the Laplacian acting on the functions on the fiber. It is known that for any k, the k-th eigenvalue of the Laplacian extends to a continuous function on the base curve. In particular, if the singular fiber is not irreducible, some eigenvalues of the Laplacian of the regular fiber converge to zero as the regular fiber approaches to the singular fiber. We call such eigenvalues small eigenvalues. In this talk, when the singular fiber is reduced, we will explain the asymptotic behavior of the product of all small eigenvalues of the Laplacian.
https://u-tokyo-ac-jp.zoom.us/meeting/register/tZEqceqsrTIjEtRxenSMdPogvCxlWzAogj5A
Tokyo Probability Seminar
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
松井 千尋 (東京大学大学院数理科学研究科)
孤立量子系の熱化と緩和 (日本語)
松井 千尋 (東京大学大学院数理科学研究科)
孤立量子系の熱化と緩和 (日本語)
2023/07/07
Tokyo-Nagoya Algebra Seminar
15:00-16:30 Online
Hideto Asashiba (Shizuoka University, Kyoto University, Osaka Metropolitan University)
クイバー表現のパーシステンス加群への応用: 区間加群による近似と分解 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
Hideto Asashiba (Shizuoka University, Kyoto University, Osaka Metropolitan University)
クイバー表現のパーシステンス加群への応用: 区間加群による近似と分解 (Japanese)
[ Reference URL ]
http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html
2023/07/06
Information Mathematics Seminar
16:50-18:35 Room #128 (Graduate School of Math. Sci. Bldg.)
Tatsuaki Okamoto (NTT)
Cryptographic protocols (Japanese)
Tatsuaki Okamoto (NTT)
Cryptographic protocols (Japanese)
[ Abstract ]
Explanation of cryptographic protocols
Explanation of cryptographic protocols
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