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Seminar information archive
Seminar information archive ～04/18｜Today's seminar 04/19 | Future seminars 04/20～
2019/11/27
Operator Algebra Seminars
17:15-18:45 Room #126 (Graduate School of Math. Sci. Bldg.)
Taro Sogabe (Kyoto University)
The homotopy groups of the automorphism groups of Cuntz-Toeplitz algebras
Taro Sogabe (Kyoto University)
The homotopy groups of the automorphism groups of Cuntz-Toeplitz algebras
Number Theory Seminar
17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)
Ahmed Abbes (CNRS & IHÉS)
The relative Hodge-Tate spectral sequence (ENGLISH)
Ahmed Abbes (CNRS & IHÉS)
The relative Hodge-Tate spectral sequence (ENGLISH)
[ Abstract ]
It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.
It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.
Operator Algebra Seminars
15:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Hokkaido Univ.)
(日本語)
Reiji Tomatsu (Hokkaido Univ.)
(日本語)
2019/11/26
Tuesday Seminar of Analysis
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
ASHIDA Sohei (Gakushuin University)
Accurate lower bounds for eigenvalues of electronic Hamiltonians (Japanese)
ASHIDA Sohei (Gakushuin University)
Accurate lower bounds for eigenvalues of electronic Hamiltonians (Japanese)
[ Abstract ]
Electronic Hamiltonians are differential operators depending on relative positions of nuclei as parameters. When we regard an eigenvalues of an electronic Hamiltonian as a function of relative positions of nuclei, minimum points correspond to shapes of molecules. Upper bounds for eigenvalues are obtained by variational methods. However, since the physical information as minimum points does not change when a reference point of energy changes, physical information can not be obtained by variational methods only. Combining lower and upper bounds physical information is obtained.
In this talk we discuss the Weinstein-Arnszajn intermediate problem methods for lower bounds of eigenvalues based on comparison of operators. A method for lower bounds of one-electronic Hamiltonians is also introduced. Some computations for two kinds of hydrogen molecule-ion are shown.
Electronic Hamiltonians are differential operators depending on relative positions of nuclei as parameters. When we regard an eigenvalues of an electronic Hamiltonian as a function of relative positions of nuclei, minimum points correspond to shapes of molecules. Upper bounds for eigenvalues are obtained by variational methods. However, since the physical information as minimum points does not change when a reference point of energy changes, physical information can not be obtained by variational methods only. Combining lower and upper bounds physical information is obtained.
In this talk we discuss the Weinstein-Arnszajn intermediate problem methods for lower bounds of eigenvalues based on comparison of operators. A method for lower bounds of one-electronic Hamiltonians is also introduced. Some computations for two kinds of hydrogen molecule-ion are shown.
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Marco De Renzi (Waseda University)
$2+1$-TQFTs from non-semisimple modular categories (ENGLISH)
Marco De Renzi (Waseda University)
$2+1$-TQFTs from non-semisimple modular categories (ENGLISH)
[ Abstract ]
Non-semisimple constructions have substantially generalized the standard approach of Witten, Reshetikhin, and Turaev to quantum topology, producing powerful invariants and TQFTs with unprecedented properties. We will explain how to use the theory of modified traces to renormalize Lyubashenko’s closed 3-manifold invariants coming from finite twist non-degenerate unimodular ribbon categories. Under the additional assumption of factorizability, our renormalized invariants extend to $2+1$-TQFTs, unlike Lyubashenko’s original ones. This general framework encompasses important examples of non-semisimple modular categories which were left out of previous non-semisimple TQFT constructions.
Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.
Non-semisimple constructions have substantially generalized the standard approach of Witten, Reshetikhin, and Turaev to quantum topology, producing powerful invariants and TQFTs with unprecedented properties. We will explain how to use the theory of modified traces to renormalize Lyubashenko’s closed 3-manifold invariants coming from finite twist non-degenerate unimodular ribbon categories. Under the additional assumption of factorizability, our renormalized invariants extend to $2+1$-TQFTs, unlike Lyubashenko’s original ones. This general framework encompasses important examples of non-semisimple modular categories which were left out of previous non-semisimple TQFT constructions.
Based on a joint work with Azat Gainutdinov, Nathan Geer, Bertrand Patureau, and Ingo Runkel.
Operator Algebra Seminars
15:00-17:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Hokkaido Univ.)
(日本語)
Reiji Tomatsu (Hokkaido Univ.)
(日本語)
PDE Real Analysis Seminar
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Dan Tiba (Institute of Mathematics of the Romanian Academy / Academy of Romanian Scientists)
A Hamiltonian approach with penalization in shape and topology optimization (English)
Dan Tiba (Institute of Mathematics of the Romanian Academy / Academy of Romanian Scientists)
A Hamiltonian approach with penalization in shape and topology optimization (English)
[ Abstract ]
General geometric optimization problems involve boundary and topology variations. This research area has already almost fifty years of history and very rich applications in computer aided industrial design. Recently, a new representation of manifolds, using iterated Hamiltonian systems, has been introduced in arbitrary dimension and co-dimension. Combining this technique with a penalization procedure for the boundary conditions, a comprehensive approximation method for optimal design problems associated to elliptic equations, is obtained. It reduces shape and topology optimization problems to optimal control problems, in a general setting. It enters the category of fixed domain methods in variable/unknown domain problems and it has consistent advantages at the computational level. It allows "free" changes of the boundary and/or the topology, during the iterations. This methodology, based on iterated Hamiltonian systems and implicit parametrizations, was also applied to nonlinear programming problems in arbitrary dimension.
General geometric optimization problems involve boundary and topology variations. This research area has already almost fifty years of history and very rich applications in computer aided industrial design. Recently, a new representation of manifolds, using iterated Hamiltonian systems, has been introduced in arbitrary dimension and co-dimension. Combining this technique with a penalization procedure for the boundary conditions, a comprehensive approximation method for optimal design problems associated to elliptic equations, is obtained. It reduces shape and topology optimization problems to optimal control problems, in a general setting. It enters the category of fixed domain methods in variable/unknown domain problems and it has consistent advantages at the computational level. It allows "free" changes of the boundary and/or the topology, during the iterations. This methodology, based on iterated Hamiltonian systems and implicit parametrizations, was also applied to nonlinear programming problems in arbitrary dimension.
2019/11/25
Operator Algebra Seminars
15:00-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Reiji Tomatsu (Hokkaido Univ.)
(日本語)
Reiji Tomatsu (Hokkaido Univ.)
(日本語)
2019/11/22
Discrete mathematical modelling seminar
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Adam Doliwa (University of Warmia and Mazury)
The Hopf algebra structure of coloured non-commutative symmetric functions
Adam Doliwa (University of Warmia and Mazury)
The Hopf algebra structure of coloured non-commutative symmetric functions
[ Abstract ]
The Hopf algebra of symmetric functions (Sym), especially its Schur function basis, plays an important role in the theory of KP hierarchy. The Hopf algebra of non-commutative symmetric functions (NSym) was introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. In my talk I would like to present its "A-coloured" version NSym_A and its graded dual - the Hopf algebra QSym_A of coloured quasi-symmetric functions. It turns out that these two algebras are both non-commutative and non-cocommutative (for |A|>1), and their product and coproduct operations allow for simple combinatorial meaning. I will also show how the structure of the poset of sentences over alphabet A (A-coloured compositions) gives rise to a description of the corresponding coloured version of the ribbon Schur basis of NSym_A.
The Hopf algebra of symmetric functions (Sym), especially its Schur function basis, plays an important role in the theory of KP hierarchy. The Hopf algebra of non-commutative symmetric functions (NSym) was introduced by Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon. In my talk I would like to present its "A-coloured" version NSym_A and its graded dual - the Hopf algebra QSym_A of coloured quasi-symmetric functions. It turns out that these two algebras are both non-commutative and non-cocommutative (for |A|>1), and their product and coproduct operations allow for simple combinatorial meaning. I will also show how the structure of the poset of sentences over alphabet A (A-coloured compositions) gives rise to a description of the corresponding coloured version of the ribbon Schur basis of NSym_A.
2019/11/21
Information Mathematics Seminar
16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)
Yasunari Suzuki (NTT Secure Platform Laboratories)
Quantum error correcting (Japanese)
Yasunari Suzuki (NTT Secure Platform Laboratories)
Quantum error correcting (Japanese)
[ Abstract ]
Explanation of quantum error correcting.
Explanation of quantum error correcting.
Logic
13:30-15:00 Room #156 (Graduate School of Math. Sci. Bldg.)
Kentaro Sato
Self-referential Theorems for Finitist Arithmetic
Kentaro Sato
Self-referential Theorems for Finitist Arithmetic
[ Abstract ]
The finitist logic excludes,on the syntax level, unbounded quantifiers
and accommodates only bounded quantifiers.
The following two self-referential theorems for arithmetic theories
over the finitist logic will be considered:
Tarski's impossibility of naive truth predicate and
Goedel's incompleteness theorem.
Particularly, it will be briefly explained that
(i) the naive truth theory over the finitist arithmetic with summation and multiplication
is consistent and proves its own consistency, and that
(ii) by the use of finitist arithmetic, the hierarchy of consistency strengths,
based on Goedel's second incompleteness theorem,
can be extended downward (to the area not reachable by first order predicate arithmetic).
This is a joint work with Jan Walker, and overlaps significantly with his doctoral dissertation.
The finitist logic excludes,on the syntax level, unbounded quantifiers
and accommodates only bounded quantifiers.
The following two self-referential theorems for arithmetic theories
over the finitist logic will be considered:
Tarski's impossibility of naive truth predicate and
Goedel's incompleteness theorem.
Particularly, it will be briefly explained that
(i) the naive truth theory over the finitist arithmetic with summation and multiplication
is consistent and proves its own consistency, and that
(ii) by the use of finitist arithmetic, the hierarchy of consistency strengths,
based on Goedel's second incompleteness theorem,
can be extended downward (to the area not reachable by first order predicate arithmetic).
This is a joint work with Jan Walker, and overlaps significantly with his doctoral dissertation.
2019/11/20
Number Theory Seminar
18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Vasudevan Srinivas (Tata Institute of Fundamental Research)
Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)
Vasudevan Srinivas (Tata Institute of Fundamental Research)
Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)
[ Abstract ]
For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.
For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.
Operator Algebra Seminars
16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)
Stefan Hollands (Univ. Leipzig)
Modular theory and entanglement in CFT
Stefan Hollands (Univ. Leipzig)
Modular theory and entanglement in CFT
2019/11/19
PDE Real Analysis Seminar
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Peter Topping (University of Warwick)
Starting Ricci flow with rough initial data (English)
Peter Topping (University of Warwick)
Starting Ricci flow with rough initial data (English)
[ Abstract ]
Ricci flow is a nonlinear PDE that is traditionally used to deform a manifold we would like to understand into a manifold we already understand. For example, Hamilton showed that a simply connected closed 3-manifold with positive Ricci curvature is deformed into a manifold of constant sectional curvature, thus allowing us to identify it as topologically a sphere.
In this talk we take a look at a different use of Ricci flow. We would like to exploit the regularising effect of parabolic PDE to turn a rough space into a smooth space by running the Ricci flow. In practice, this revolves around proving good a priori estimates on solutions, and taking unorthodox approaches to solving parabolic PDE. We will see some theory, first in 2D, then in higher dimension, and some applications.
Ricci flow is a nonlinear PDE that is traditionally used to deform a manifold we would like to understand into a manifold we already understand. For example, Hamilton showed that a simply connected closed 3-manifold with positive Ricci curvature is deformed into a manifold of constant sectional curvature, thus allowing us to identify it as topologically a sphere.
In this talk we take a look at a different use of Ricci flow. We would like to exploit the regularising effect of parabolic PDE to turn a rough space into a smooth space by running the Ricci flow. In practice, this revolves around proving good a priori estimates on solutions, and taking unorthodox approaches to solving parabolic PDE. We will see some theory, first in 2D, then in higher dimension, and some applications.
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Ramón Barral Lijó (Ritsumeikan University)
The smooth Gromov space and the realization problem (ENGLISH)
Ramón Barral Lijó (Ritsumeikan University)
The smooth Gromov space and the realization problem (ENGLISH)
[ Abstract ]
The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds. In this talk we will present the definition and basic properties of this space as well as two different applications: The first addresses the following classical problem in foliation theory.
Realization problem: what kind of manifolds can be leaves of compact foliations?
Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.
Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor. Moreover, we can assume that X has trivial holonomy and is transversely Cantor.
The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.
The n-dimensional smooth Gromov space consists of the pointed isometry classes of complete Riemannian n-manifolds. In this talk we will present the definition and basic properties of this space as well as two different applications: The first addresses the following classical problem in foliation theory.
Realization problem: what kind of manifolds can be leaves of compact foliations?
Our joint work with Álvarez López has produced the following solution in the context of foliated spaces.
Theorem. Every Riemannian manifold of bounded geometry is a leaf in a compact foliated space X endowed with a metric tensor. Moreover, we can assume that X has trivial holonomy and is transversely Cantor.
The second application is the recent research by Abert and Biringer on the subject of unimodular random Riemannian manifolds.
Tuesday Seminar of Analysis
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Wenjia Jing (Tsinghua University)
Quantitative homogenization for the Dirichlet problem of Stokes system in periodic perforated domain - a unified approach (English)
Wenjia Jing (Tsinghua University)
Quantitative homogenization for the Dirichlet problem of Stokes system in periodic perforated domain - a unified approach (English)
[ Abstract ]
We present a new unified approach for the quantitative homogenization of the Stokes system in periodically perforated domains, that is domains outside a periodic array of holes, with Dirichlet data at the boundary of the holes. The method is based on the (rescaled) cell-problem and is adaptive to the ratio between the typical distance and the typical side length of the holes; in particular, for the critical ratio identified by Cioranescu-Murat, we recover the “strange term from nowhere”termed by them, which, in the context of Stokes system, corresponds to the Brinkman’s law. An advantage of the method is that it can be systematically quantified using the periodic layer potential technique. We will also report some new correctors to the homogenization problem using this approach. The talk is based on joint work with Yong Lu and Christophe Prange.
We present a new unified approach for the quantitative homogenization of the Stokes system in periodically perforated domains, that is domains outside a periodic array of holes, with Dirichlet data at the boundary of the holes. The method is based on the (rescaled) cell-problem and is adaptive to the ratio between the typical distance and the typical side length of the holes; in particular, for the critical ratio identified by Cioranescu-Murat, we recover the “strange term from nowhere”termed by them, which, in the context of Stokes system, corresponds to the Brinkman’s law. An advantage of the method is that it can be systematically quantified using the periodic layer potential technique. We will also report some new correctors to the homogenization problem using this approach. The talk is based on joint work with Yong Lu and Christophe Prange.
2019/11/18
Seminar on Geometric Complex Analysis
10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)
Ken-ichi Yoshikawa (Kyoto Univ.)
j-invariant and Borcherds Phi-function (Japanese)
Ken-ichi Yoshikawa (Kyoto Univ.)
j-invariant and Borcherds Phi-function (Japanese)
[ Abstract ]
The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.
The j-invariant is a modular function on the complex upper half plane inducing an isomorphism between the moduli space of elliptic curves and the complex plane. Besides the j-invariant itself, the difference of j-invariants has also attracted some mathematicians. In this talk, I will explain a factorization of the difference of j-invariants in terms of Borcherds Phi-function, the automorphic form on the period domain for Enriques surfaces characterizing the discriminant divisor. This is a joint work with Shu Kawaguchi and Shigeru Mukai.
2019/11/14
Information Mathematics Seminar
16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)
Kosuke Mitarai (Graduate School of Engineering Science, Osaka University)
Algorithms for machine learning with quantum computer (Japanese)
Kosuke Mitarai (Graduate School of Engineering Science, Osaka University)
Algorithms for machine learning with quantum computer (Japanese)
[ Abstract ]
Explanation of the machine learning by quantum computer
Explanation of the machine learning by quantum computer
2019/11/12
Tuesday Seminar of Analysis
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
KAMIMOTO Shingo (Hiroshima University)
Mould expansion and resurgent structure (Japanese)
KAMIMOTO Shingo (Hiroshima University)
Mould expansion and resurgent structure (Japanese)
2019/11/08
Colloquium
15:30-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)
2019/11/07
Information Mathematics Seminar
16:50-18:35 Room #126 (Graduate School of Math. Sci. Bldg.)
Yuya O. Nakagawa (QunaSys Inc.)
Quantum chemistry calculations by quantum computers (Japanese)
Yuya O. Nakagawa (QunaSys Inc.)
Quantum chemistry calculations by quantum computers (Japanese)
[ Abstract ]
Explanation of quantum chemistry
Explanation of quantum chemistry
2019/11/06
Operator Algebra Seminars
16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)
Yuto Moriwaki (Univ. Tokyo)
Deformation of two-dimensional conformal field theory and vertex algebra
Yuto Moriwaki (Univ. Tokyo)
Deformation of two-dimensional conformal field theory and vertex algebra
2019/11/05
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Kiyonori Gomi (Tokyo Institute of Technology)
Magnitude homology of geodesic space (JAPANESE)
Kiyonori Gomi (Tokyo Institute of Technology)
Magnitude homology of geodesic space (JAPANESE)
[ Abstract ]
Magnitude is an invariant which counts `effective number of points' on a metric space. Its categorification is magnitude homology. This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton, and then for any metric spaces by Leinster and Shulman. The definition of the magnitude homology is easy, but its calculation is rather difficult. For example, the magnitude homology of the circle with geodesic metric was known partially. In my talk, I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption. In this result, the magnitude homology is described in terms of geodesics. Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.
Magnitude is an invariant which counts `effective number of points' on a metric space. Its categorification is magnitude homology. This notion is first formulated for metric spaces associated to simple graphs by Hepworth and Willerton, and then for any metric spaces by Leinster and Shulman. The definition of the magnitude homology is easy, but its calculation is rather difficult. For example, the magnitude homology of the circle with geodesic metric was known partially. In my talk, I will explain my result that fully determines the magnitude homology of any geodesic metric space subject to a certain non-branching assumption. In this result, the magnitude homology is described in terms of geodesics. Complete and connected Riemannian manifolds are examples of the geodesic metric spaces satisfying the assumption.
Tuesday Seminar of Analysis
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Ngô Quốc Anh (Vietnam National University, Hanoi / the University of Tokyo)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
Ngô Quốc Anh (Vietnam National University, Hanoi / the University of Tokyo)
Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space (English)
[ Abstract ]
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[\Delta^m u = \pm u^\alpha\] in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.
This talk concerns entire, non-trivial, non-negative solutions and/or entire, positive solutions to the simplest models of polyharmonic equations with power-type nonlinearity \[\Delta^m u = \pm u^\alpha\] in $\mathbb R^n$ with $n \geqslant 1$, $m \geqslant 1$, and $\alpha \in \mathbb R$. For small $m$, the above equations arise in many physical phenomena and applied mathematics. They also arise from several prescribing geometric curvture problems in conformal geometry such as the Yamabe problem, the scalar curvature problem, and the Q-curvature problem for the Paneitz operator. Higher-order cases also arise from the Q-curvature problem for the GJMS operator. In this talk, I will present a complete picture of the existence and non-existence of solutions to the above equations in the full rage of the parameters $n$, $m$, and $\alpha$. This is joint work with V.H. Nguyen, Q.H. Phan, and D. Ye.
2019/10/31
Applied Analysis
16:00-17:30 Room #128 (TBD) (Graduate School of Math. Sci. Bldg.)
Marius Ghergu (University College Dublin)
Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)
Marius Ghergu (University College Dublin)
Behaviour around the isolated singularity for solutions of some nonlinear elliptic inequalities and systems (English)
[ Abstract ]
We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).
We present some results on the behaviour around the isolated singularity for solutions of nonlinear elliptic inequalities driven by the Laplace operator. We derive optimal conditions that imply either a blow-up or the existence of pointwise bounds for solutions. We obtain that whenever a pointwise bound exists, then an optimal bound is given by the fundamental solution of the Laplace operator. This situation changes in case of systems of inequalities where other types of optimal bounds may occur. The approach relies on integral representation of solutions combined with various nonlinear potential estimates. Further extensions to the parabolic case will be presented. This talk is based on joint works with S. Taliaferro (Texas A&M University) and I. Verbitsky (Missouri University).
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