## Seminar information archive

Seminar information archive ～02/06｜Today's seminar 02/07 | Future seminars 02/08～

#### Logic

13:30-15:00 Room #156 (Graduate School of Math. Sci. Bldg.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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.)

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).

#### FMSP Lectures

13:00-15:05 Room #002 (Graduate School of Math. Sci. Bldg.)

Topic on minimal submanifolds (6/6) (ENGLISH)

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

**Chung-jun Tsai**(National Taiwan University)Topic on minimal submanifolds (6/6) (ENGLISH)

[ Abstract ]

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

[ Reference URL ]The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

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

#### Information Mathematics Seminar

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

Foundation of quanutm computing (Japanese)

**Yasunari Suzuki**(NTT Secure Platform Laboratories)Foundation of quanutm computing (Japanese)

[ Abstract ]

Explanation of quanutm computing.

Explanation of quanutm computing.

### 2019/10/30

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

A Tannakian perspective on rigid analytic geometry (English)

**Andrew Macpherson**(IPMU)A Tannakian perspective on rigid analytic geometry (English)

[ Abstract ]

Raynaud's conception of analytic geometry contends that the category of analytic spaces over a non-Archimedean field is a (suitably "geometric") localisation of the category of formal schemes over the ring of integers at a class of modifications "along the central fibre". Unfortunately, as with all existing presentations of non-Archimedean geometry, this viewpoint is confounded by a proliferation of technical difficulties if one does not impose absolute finiteness conditions on the formal schemes under consideration.

I will argue that by combining Raynaud's idea with a Tannakian perspective which prioritises the module category, we can obtain a reasonable framework for rigid analytic geometry with no absolute finiteness hypotheses whatsoever, but which has descent for finitely presented modules.

Raynaud's conception of analytic geometry contends that the category of analytic spaces over a non-Archimedean field is a (suitably "geometric") localisation of the category of formal schemes over the ring of integers at a class of modifications "along the central fibre". Unfortunately, as with all existing presentations of non-Archimedean geometry, this viewpoint is confounded by a proliferation of technical difficulties if one does not impose absolute finiteness conditions on the formal schemes under consideration.

I will argue that by combining Raynaud's idea with a Tannakian perspective which prioritises the module category, we can obtain a reasonable framework for rigid analytic geometry with no absolute finiteness hypotheses whatsoever, but which has descent for finitely presented modules.

#### Lie Groups and Representation Theory

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On holographic transform (English)

**Quentin Labriet**(Reims University)On holographic transform (English)

[ Abstract ]

In representation theory, decomposing the restriction of a given representation $¥pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context, one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $¥pi¥vert_{G'}$ and its irreducible

components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform.

I will illustrate this construction by two examples :

- the diagonal case where one considers the restriction problem for $¥pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)¥times SL(2,R)$ and $G'=SL(2,R)$.

- the conformal case for the restriction of a scalar valued holomorphic discrete series representation $¥pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$.

I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.

In representation theory, decomposing the restriction of a given representation $¥pi$ of a Lie group $G$ to an appropriate subgroup $G'$ is an important issue referred to as a branching law. In this context, one can define symmetry breaking operators, as $G'$-intertwining operators between the restriction $¥pi¥vert_{G'}$ and its irreducible

components. Going in the opposite direction gives rise to holographic operators and the notion of holographic transform.

I will illustrate this construction by two examples :

- the diagonal case where one considers the restriction problem for $¥pi$ being an outer product of two holomorphic discrete series representations, $G=SL(2,R)¥times SL(2,R)$ and $G'=SL(2,R)$.

- the conformal case for the restriction of a scalar valued holomorphic discrete series representation $¥pi$ of $G=SO(2,n)$ to $G'=SO(2,n-1)$.

I will then explain different methods for an explicit construction of such holographic operators in these cases, and present some of my results and open problems in this direction.

### 2019/10/29

#### Tuesday Seminar on Topology

17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Strong stability of minimal submanifolds (ENGLISH)

**Chung-Jun Tsai**(National Taiwan University)Strong stability of minimal submanifolds (ENGLISH)

[ Abstract ]

It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario. Besides a strong local uniqueness property, a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow. We will also discuss some famous local (complete, non-compact) models. This is based on a joint work with Mu-Tao Wang.

It is well known that the distance function to a totally geodesic submanifold of a negatively curved ambient manifold is a convex function. One can identify a strong stability condition on minimal submanifolds that generalizes the above scenario. Besides a strong local uniqueness property, a strongly stable minimal submanifold is also Lipschitz stable under the mean curvature flow. We will also discuss some famous local (complete, non-compact) models. This is based on a joint work with Mu-Tao Wang.

### 2019/10/28

#### Seminar on Geometric Complex Analysis

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

On Kiyoshi Oka's unpublished papers 1943 (Japanese)

**Junjiro Noguchi**(Univ. of Tokyo)On Kiyoshi Oka's unpublished papers 1943 (Japanese)

### 2019/10/25

#### Colloquium

15:30-16:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Arithmeticity of discrete subgroups (英語)

**Yves Benoist**( CNRS, Paris-Sud)Arithmeticity of discrete subgroups (英語)

[ Abstract ]

By a theorem of Borel and Harish-Chandra,

an arithmetic group in a semisimple Lie group is a lattice.

Conversely, by a celebrated theorem of Margulis,

in a higher rank semisimple Lie group G

any irreducible lattice is an arithmetic group.

The aim of this lecture is to survey an

arithmeticity criterium for discrete subgroups

which are not assumed to be lattices.

This criterium, obtained with Miquel,

generalizes works of Selberg and Hee Oh

and solves a conjecture of Margulis. It says:

a discrete irreducible Zariski-dense subgroup

of G that intersects cocompactly at least one

horospherical subgroup of G is an arithmetic group.

By a theorem of Borel and Harish-Chandra,

an arithmetic group in a semisimple Lie group is a lattice.

Conversely, by a celebrated theorem of Margulis,

in a higher rank semisimple Lie group G

any irreducible lattice is an arithmetic group.

The aim of this lecture is to survey an

arithmeticity criterium for discrete subgroups

which are not assumed to be lattices.

This criterium, obtained with Miquel,

generalizes works of Selberg and Hee Oh

and solves a conjecture of Margulis. It says:

a discrete irreducible Zariski-dense subgroup

of G that intersects cocompactly at least one

horospherical subgroup of G is an arithmetic group.

### 2019/10/24

#### FMSP Lectures

13:00-15:05 Room #002 (Graduate School of Math. Sci. Bldg.)

Topic on minimal submanifolds (5/6) (ENGLISH)

**Chung-jun Tsai**(National Taiwan University)Topic on minimal submanifolds (5/6) (ENGLISH)

[ Abstract ]

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

The main theme of these lectures will be theory about minimal submanifolds, which are higher dimensional generalizations of geodesics. A naive motivation is that one tries to understand the geometry from its special submanifolds (minimal, etc.).

For minimal submanifolds, the equations are no longer ODEs, but elliptic PDEs. This increases the difficulties. The study are very good examples for the application of methods from PDEs and calculus of variations. We will try to explain some important results in this theory, which stimulate many of the researches today.

Here are some specific materials we plan to cover: Simon’s work based on the second variational formula, Sacks - Uhlenback theorem on the existence of minimal 2-spheres, the theory of stable minimal hypersurfaces by Schoen-Simon-Yau.

#### Logic

13:30-15:00 Room #156 (Graduate School of Math. Sci. Bldg.)

Generalizations of Bennet's results about partially conservative sentences (JAPANESE)

**Yuya Okawa**(Chiba University)Generalizations of Bennet's results about partially conservative sentences (JAPANESE)

#### Applied Analysis

16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)

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