## Seminar information archive

Seminar information archive ～05/25｜Today's seminar 05/26 | Future seminars 05/27～

#### thesis presentations

9:15-10:30 Room #122 (Graduate School of Math. Sci. Bldg.)

#### thesis presentations

12:45-14:00 Room #122 (Graduate School of Math. Sci. Bldg.)

#### thesis presentations

13:45-15:00 Room #126 (Graduate School of Math. Sci. Bldg.)

### 2020/01/29

#### Operator Algebra Seminars

13:00-14:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Horn's problem, polytope volumes and tensor product decompositions (English)

**Colin McSwiggen**(Brown Univ.)Horn's problem, polytope volumes and tensor product decompositions (English)

#### Operator Algebra Seminars

16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)

Stationary actions of higher rank lattices on von Neumann algebras (English)

**Cyril Houdayer**(Univ. Paris-Sud)Stationary actions of higher rank lattices on von Neumann algebras (English)

### 2020/01/28

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Existence problems for fibered links (JAPANESE)

**Nozomu Sekino**(The University of Tokyo)Existence problems for fibered links (JAPANESE)

[ Abstract ]

It is known that every connected orientable closed 3-manifold has a fibered knot. However, finding (and classifying) fibered links whose fiber surfaces are fixed homeomorphism type in a given 3-manifold is difficult in general. We give a criterion of a simple closed curve on a genus 2g Heegaard surface being a genus g fibered knot in terms of its Heegaard diagram. As an application, we can prove the non-existence of genus one fibered knots in some Seifert manifolds.

There is one generalization of fibered links, homologically fibered links. This requests that the complement of the "fiber surface" is a homologically product of a surface and an interval. We give a necessary and sufficient condition for a connected sums of lens spaces of having a homologically fibered link whose fiber surfaces are some fixed types as some algebraic equations.

It is known that every connected orientable closed 3-manifold has a fibered knot. However, finding (and classifying) fibered links whose fiber surfaces are fixed homeomorphism type in a given 3-manifold is difficult in general. We give a criterion of a simple closed curve on a genus 2g Heegaard surface being a genus g fibered knot in terms of its Heegaard diagram. As an application, we can prove the non-existence of genus one fibered knots in some Seifert manifolds.

There is one generalization of fibered links, homologically fibered links. This requests that the complement of the "fiber surface" is a homologically product of a surface and an interval. We give a necessary and sufficient condition for a connected sums of lens spaces of having a homologically fibered link whose fiber surfaces are some fixed types as some algebraic equations.

#### Tuesday Seminar on Topology

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Fibred cusp b-pseudodifferential operators and its applications (JAPANESE)

**Jun Watanabe**(The University of Tokyo)Fibred cusp b-pseudodifferential operators and its applications (JAPANESE)

[ Abstract ]

Melrose's b-calculus and its variants are important tools to study index problems on manifolds with singularities. In this talk, we introduce a new variant "fibred cusp b-calculus", which is a generalization of fibred cusp calculus of Mazzeo-Melrose and b-calculus of Melrose. We discuss the basic property of this calculus and give a relative index formula. As its application, we prove the index theorem for a Z/k manifold with boundary, which is a generalization of the mod k index theorem of Freed-Melrose.

Melrose's b-calculus and its variants are important tools to study index problems on manifolds with singularities. In this talk, we introduce a new variant "fibred cusp b-calculus", which is a generalization of fibred cusp calculus of Mazzeo-Melrose and b-calculus of Melrose. We discuss the basic property of this calculus and give a relative index formula. As its application, we prove the index theorem for a Z/k manifold with boundary, which is a generalization of the mod k index theorem of Freed-Melrose.

#### Lie Groups and Representation Theory

10:00-16:40 Room # (Graduate School of Math. Sci. Bldg.)

Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations (English)

TBA (English)

From Symmetry breaking toward holographic transform in representation theory (English)

Cells of Harish-Chandra modules

(English)

**Taito Tauchi**(The University of Tokyo) 10:00-11:00Relationship between orbit decomposition on the flag varieties and multiplicities of induced representations (English)

**Mikhail Kapranov**(Kavli IPMU) 11:20-12:20TBA (English)

**Michael Pevzner**(University of Reims) 14:00-15:00From Symmetry breaking toward holographic transform in representation theory (English)

**Leticia Barchini**(Oklahoma University) 15:40-16:40Cells of Harish-Chandra modules

(English)

### 2020/01/27

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Canonical measure and it’s applications

**Hajime Tsuji**(Sophia Univ.)Canonical measure and it’s applications

[ Abstract ]

The canonical measure is a natural generalization of K\”ahler-Einstein metrics to the case of projective manifolds with nonnegative Kodaira dimension. In this talk we consider the variation of canonical measures under projective deformations and give some applications.

The canonical measure is a natural generalization of K\”ahler-Einstein metrics to the case of projective manifolds with nonnegative Kodaira dimension. In this talk we consider the variation of canonical measures under projective deformations and give some applications.

#### Lie Groups and Representation Theory

9:30-16:30 Room # (Graduate School of Math. Sci. Bldg.)

TBA (English)

Regular Representations on Homogeneous Spaces (English)

The adapted hyper-K\"ahler structure on the tangent bundle of a Hermitian symmetric space (English)

UNIVERSAL NATURE OF THE HOROSPHERICAL TRANSFORM IN SYMMETRIC SPACES (English)

**Joseph Bernstein**(Tel Aviv and The University of Tokyo) 10:00-11:00TBA (English)

**Toshiyuki Kobayashi**(The University of Tokyo) 11:20-12:20Regular Representations on Homogeneous Spaces (English)

**Laura Geatti**(University of Roma) 14:00-15:00The adapted hyper-K\"ahler structure on the tangent bundle of a Hermitian symmetric space (English)

**Simon Gindikin**(Rutgers University) 15:30-16:30UNIVERSAL NATURE OF THE HOROSPHERICAL TRANSFORM IN SYMMETRIC SPACES (English)

### 2020/01/22

#### FMSP Lectures

17:00-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Complex principal type operators in inverse conductivity problem (ENGLISH)

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SamuliSiltanen.pdf

**Samuli Siltanen**(University of Helsinki)Complex principal type operators in inverse conductivity problem (ENGLISH)

[ Abstract ]

Stroke is a leading cause of death all around the world. There are two main types of stroke: ischemic (blood clot preventing blood flow to a part of the brain) and hemorrhagic (bleeding in the brain). The symptoms are the same, but treatments very different. A portable "stroke classifier" would be a life-saving equipment to have in ambulances, but so far it does not exist. Electrical Impedance Tomography (EIT) is a promising and harmless imaging method for stroke classification. In EIT one attempts to recover the electric conductivity inside a domain from electric boundary measurements. This is a nonlinear and ill-posed inverse problem. The so-called Complex Geometric Optics (CGO) solutions have proven to be a useful computational tool for reconstruction tasks in EIT. A new property of CGO solutions is presented, showing that a one-dimensional Fourier transform in the spectral variable provides a connection to parallel-beam Xray tomography of the conductivity. One of the consequences of this “nonlinear Fourier slice theorem” is a novel capability to recover inclusions within inclusions in EIT. In practical imaging, measurement noise causes strong blurring in the recovered profile functions. However, machine learning algorithms can be combined with the nonlinear PDE techniques in a fruitful way. As an example, simulated strokes are classified into hemorrhagic and ischemic using EIT measurements.

[ Reference URL ]Stroke is a leading cause of death all around the world. There are two main types of stroke: ischemic (blood clot preventing blood flow to a part of the brain) and hemorrhagic (bleeding in the brain). The symptoms are the same, but treatments very different. A portable "stroke classifier" would be a life-saving equipment to have in ambulances, but so far it does not exist. Electrical Impedance Tomography (EIT) is a promising and harmless imaging method for stroke classification. In EIT one attempts to recover the electric conductivity inside a domain from electric boundary measurements. This is a nonlinear and ill-posed inverse problem. The so-called Complex Geometric Optics (CGO) solutions have proven to be a useful computational tool for reconstruction tasks in EIT. A new property of CGO solutions is presented, showing that a one-dimensional Fourier transform in the spectral variable provides a connection to parallel-beam Xray tomography of the conductivity. One of the consequences of this “nonlinear Fourier slice theorem” is a novel capability to recover inclusions within inclusions in EIT. In practical imaging, measurement noise causes strong blurring in the recovered profile functions. However, machine learning algorithms can be combined with the nonlinear PDE techniques in a fruitful way. As an example, simulated strokes are classified into hemorrhagic and ischemic using EIT measurements.

http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_SamuliSiltanen.pdf

### 2020/01/21

#### Algebraic Geometry Seminar

15:30-17:00 Room #118 (Graduate School of Math. Sci. Bldg.)

(Few) rational curves on K3 surfaces (English)

**Matthias Schütt**(Universität Hannover)(Few) rational curves on K3 surfaces (English)

[ Abstract ]

Rational curves play a fundamental role for the structure of a K3 surface. I will first review the general theory before focussing on the case of low degree curves where joint work with S. Rams (Krakow) extends bounds of Miyaoka and Degtyarev. Time permitting, I will also discuss the special case of smooth rational curves as well as applications to Enriques surfaces.

Rational curves play a fundamental role for the structure of a K3 surface. I will first review the general theory before focussing on the case of low degree curves where joint work with S. Rams (Krakow) extends bounds of Miyaoka and Degtyarev. Time permitting, I will also discuss the special case of smooth rational curves as well as applications to Enriques surfaces.

#### Lie Groups and Representation Theory

14:00-16:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On Plancherel measure (English)

**Joseph Bernstein**(Tel Aviv University)On Plancherel measure (English)

### 2020/01/20

#### Numerical Analysis Seminar

16:50-18:20 Room #056 (Graduate School of Math. Sci. Bldg.)

Isogeometric Hierarchical Model Reduction: from analysis to patient-specific simulations (English)

**Yves A. B. C. Barbosa**(Politecnico di Milano)Isogeometric Hierarchical Model Reduction: from analysis to patient-specific simulations (English)

[ Abstract ]

In the field of hemodynamics, numerical models have evolved to account for the demands in speed and accuracy of modern diagnostic medicine. In this context, we studied in detail Hierarchical Model Reduction technique combined with Isogeometric Analysis (HigaMOD), a technique recently developed in [Perotto, Reali, Rusconi and Veneziani (2017)]. HigaMod is a reduction procedure used to downscale models when the phenomenon at hand presents a preferential direction of flow, e.g., when modelling the blood flow in arteries or the water flow in a channel network. The method showed a significant improvement in reducing the computational power and simulation time, while giving enough information to analyze the problem at hand.

Recently, we focused our work in solving the ADR problem and the Stokes problem in a patient-specific framework. Specifically, we evaluate the computational efficiency of HigaMod in simulating the blood flow in coronary arteries and cerebral arteries. The main goal is to assess the

mprovement that 1D enriched models can provide, with respect to traditional full models, when dealing with demanding 3D CFD simulations. The results obtained, even though preliminary, are promising [Brandes, Barbosa and Perotto (2019); Brandes, Barbosa, Perotto and Suito (2020)].

In the field of hemodynamics, numerical models have evolved to account for the demands in speed and accuracy of modern diagnostic medicine. In this context, we studied in detail Hierarchical Model Reduction technique combined with Isogeometric Analysis (HigaMOD), a technique recently developed in [Perotto, Reali, Rusconi and Veneziani (2017)]. HigaMod is a reduction procedure used to downscale models when the phenomenon at hand presents a preferential direction of flow, e.g., when modelling the blood flow in arteries or the water flow in a channel network. The method showed a significant improvement in reducing the computational power and simulation time, while giving enough information to analyze the problem at hand.

Recently, we focused our work in solving the ADR problem and the Stokes problem in a patient-specific framework. Specifically, we evaluate the computational efficiency of HigaMod in simulating the blood flow in coronary arteries and cerebral arteries. The main goal is to assess the

mprovement that 1D enriched models can provide, with respect to traditional full models, when dealing with demanding 3D CFD simulations. The results obtained, even though preliminary, are promising [Brandes, Barbosa and Perotto (2019); Brandes, Barbosa, Perotto and Suito (2020)].

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Diederich-Fornaess and Steinness indices for abstract CR manifolds

**Masanori Adachi**(Shizuoka Univ.)Diederich-Fornaess and Steinness indices for abstract CR manifolds

[ Abstract ]

The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).

The Diederich-Fornaes and Steinness indices are estimated for weakly pseudoconvex domains in complex manifolds in terms of the D'Angelo 1-form of the boundary CR manifolds. In particular, CR invariance of these indices is shown when the domain is Takeuchi 1-convex. This is a joint work with Jihun Yum (Pusan National University).

### 2020/01/16

#### Information Mathematics Seminar

16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)

Foundation of Quantum Annealing (Japanese)

**Hirotaka Irie**(DENSO CORPORATION/RIKEN iTHEMS)Foundation of Quantum Annealing (Japanese)

[ Abstract ]

Explanation of Quantum Annealing

Explanation of Quantum Annealing

### 2020/01/14

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

SO(3)-invariant G

**Ryohei Chihara**(The University of Tokyo)SO(3)-invariant G

_{2}-geometry (JAPANESE)
[ Abstract ]

Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G

Berger's classification of holonomy groups of Riemannian manifolds includes exceptional cases of the Lie groups G

_{2}and Spin(7). Many authors have studied G_{2}- and Spin(7)-manifolds with torus symmetry. In this talk, we generalize the celebrated examples due to Bryant and Salamon and study G_{2}-manifolds with SO(3)-symmetry. Such torsion-free G_{2}-structures are described as a dynamical system of SU(3)-structures on an SO(3)-fibration over a 3-manifold. As a main result, we reduce this system into a constrained Hamiltonian dynamical system on the cotangent bundle over the space of all Riemannian metrics on the 3-manifold. The Hamiltonian function is very similar to that of the Hamiltonian formulation of general relativity.#### Tuesday Seminar on Topology

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Algebraic entropy of sign-stable mutation loops (JAPANESE)

**Tsukasa Ishibashi**(The University of Tokyo)Algebraic entropy of sign-stable mutation loops (JAPANESE)

[ Abstract ]

Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.

We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

Since its discovery, the cluster algebra has been developed with friutful connections with other branches of mathematics, unifying several combinatorial operations as well as their positivity notions. A mutation loop induces several dynamical systems via cluster transformations, and they form a group which can be seen as a combinatorial generalization of the mapping class groups of marked surfaces.

We introduce a new property of mutation loops called the sign stability, with a focus on an asymptotic behavior of the iteration of the tropicalized cluster X-transformation. A sign-stable mutation loop has a numerical invariant which we call the "cluster stretch factor", in analogy with the stretch factor of a pseudo-Anosov mapping class on a marked surface. We compute the algebraic entropies of the cluster A- and X-transformations induced by a sign-stable mutation loop, and conclude that these two coincide with the logarithm of the cluster stretch factor. This talk is based on a joint work with Shunsuke Kano.

#### Tuesday Seminar of Analysis

16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Scattering near a two-cluster threshold (English)

**Erik Skibsted**(Aarhus University)Scattering near a two-cluster threshold (English)

[ Abstract ]

For a one-body Schr\"odinger operator with an attractive slowly decaying potential the scattering matrix is well-defined at the energy zero, and the structure of its singularities is well-studied. The usual (non-relativistic) model for the Hydrogen atom is a particular example of such Schr\"odinger operator.

Less is known on scattering at a two-cluster threshold of an $N$-body Schr\"odinger operator for which the effective interaction between the two bound clusters is attractive Coulombic. An example of interest is scattering at a two-cluster threshold of a neutral atom/molecule. We present results of an ongoing joint work with X.P. Wang on the subject, including a version of the Sommerfeld uniqueness result and its applications.

We shall also present general results on spectral theory at a two-cluster threshold (not requiring the effective interaction to be attractive Coulombic). This includes a general structure theorem on the bound and resonance states at the threshold as well as a resolvent expansion in weighted spaces above the threshold (under more restrictive conditions). Applications to scattering theory will be indicated.

For a one-body Schr\"odinger operator with an attractive slowly decaying potential the scattering matrix is well-defined at the energy zero, and the structure of its singularities is well-studied. The usual (non-relativistic) model for the Hydrogen atom is a particular example of such Schr\"odinger operator.

Less is known on scattering at a two-cluster threshold of an $N$-body Schr\"odinger operator for which the effective interaction between the two bound clusters is attractive Coulombic. An example of interest is scattering at a two-cluster threshold of a neutral atom/molecule. We present results of an ongoing joint work with X.P. Wang on the subject, including a version of the Sommerfeld uniqueness result and its applications.

We shall also present general results on spectral theory at a two-cluster threshold (not requiring the effective interaction to be attractive Coulombic). This includes a general structure theorem on the bound and resonance states at the threshold as well as a resolvent expansion in weighted spaces above the threshold (under more restrictive conditions). Applications to scattering theory will be indicated.

### 2020/01/10

#### Lectures

10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Nori motives over function fields and period functions (ENGLISH)

**Javier Fresan**(Ecole Polytechnique)Nori motives over function fields and period functions (ENGLISH)

[ Abstract ]

Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs

algebraic relations between period functions.

This last part is based on an ongoing work with Peter Jossen.

Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs

algebraic relations between period functions.

This last part is based on an ongoing work with Peter Jossen.

### 2020/01/09

#### Information Mathematics Seminar

16:50-18:35 Room #122 (Graduate School of Math. Sci. Bldg.)

Innovation and business administration of the manufacturing industry by AI/IoT (Japanese)

**Hiroshi Fujiwara**(BroadBand Tower, Inc.)Innovation and business administration of the manufacturing industry by AI/IoT (Japanese)

[ Abstract ]

Explanation on business administration of the manufacturing industry by AI/IoT

Explanation on business administration of the manufacturing industry by AI/IoT

#### Lectures

14:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Nori motives over function fields and period functions (ENGLISH)

**Javier Fresan**(Ecole Polytechnique)Nori motives over function fields and period functions (ENGLISH)

[ Abstract ]

Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs

algebraic relations between period functions.

This last part is based on an ongoing work with Peter Jossen.

Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs

algebraic relations between period functions.

This last part is based on an ongoing work with Peter Jossen.

#### Lectures

16:00-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Nori motives over function fields and period functions (ENGLISH)

**Javier Fresan**(Ecole Polytechnique)Nori motives over function fields and period functions (ENGLISH)

[ Abstract ]

Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs

algebraic relations between period functions.

This last part is based on an ongoing work with Peter Jossen.

Around twenty years ago, Nori introduced a tannakian category of mixed motives over a subfield of the complex numbers, thus giving the first unconditional construction of the motivic Galois group. In this series of lectures, I will first survey on Nori's theory and its relationship to other categories of motives. I will then explain how to extend his construction to functions fields and why the resulting tannakian group governs

algebraic relations between period functions.

This last part is based on an ongoing work with Peter Jossen.

### 2020/01/07

#### Tuesday Seminar on Topology

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Magnitude homology of crushable spaces (JAPANESE)

**Yasuhiko Asao**(The University of Tokyo)Magnitude homology of crushable spaces (JAPANESE)

[ Abstract ]

The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them has been revealed. In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

The magnitude homology and the blurred magnitude homology are novel notions of homology theory for general metric spaces coined by Leinster et al. They are expected to be dealt with in the context of Topological Data Analysis since its original idea is based on a kind of "persistence of points clouds". However, little property of them has been revealed. In this talk, we see that the blurred magnitude homology is trivial when a metric space is contractible by a distance decreasing homotopy. We use techniques from singular homology theory.

#### Tuesday Seminar on Topology

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)

**Tomohiro Asano**(The University of Tokyo)Intersection number estimate of rational Lagrangian immersions in cotangent bundles via microlocal sheaf theory (JAPANESE)

[ Abstract ]

Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

Guillermou associated sheaves to exact Lagrangian submanifolds in cotangent bundles and proved topological properties of the Lagrangian submanifolds. In this talk, I will give an estimate on the displacement energy of rational Lagrangian immersions in cotangent bundles with intersection number estimates via microlocal sheaf theory. This result overlaps with results by Chekanov, Liu, and Akaho via Floer theory. This is joint work with Yuichi Ike.

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