## Seminar information archive

Seminar information archive ～08/18｜Today's seminar 08/19 | Future seminars 08/20～

#### Number Theory Seminar

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Fargues' conjecture in the GL_2-case (ENGLISH)

**Ildar Gaisin**(University of Tokyo)Fargues' conjecture in the GL_2-case (ENGLISH)

[ Abstract ]

Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.

Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.

#### Number Theory Seminar

17:10-18:10 Room #002 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Noriyuki Abe**(University of Tokyo)(JAPANESE)

### 2018/04/17

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Global existence of a strong solution to the Navier-Stokes equations in a curved thin domain (Japanese)

**Tatsu-Hiko Miura**(The University of Tokyo)Global existence of a strong solution to the Navier-Stokes equations in a curved thin domain (Japanese)

[ Abstract ]

In this talk, we consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a closed surface.

We establish the global-in-time existence of a strong solution for large data when the width of the thin domain is very small.

A key idea is to decompose a three-dimensional vector field into the average part which is almost two-dimensional and the residual part to which we can apply Poincaré type inequalities.

Such decomposition enables us to derive a good estimate for the inner product of the inertia term and the viscous term, which is essential for our arguments.

In this talk, we consider the Navier-Stokes equations with Navier's slip boundary conditions in a three-dimensional curved thin domain around a closed surface.

We establish the global-in-time existence of a strong solution for large data when the width of the thin domain is very small.

A key idea is to decompose a three-dimensional vector field into the average part which is almost two-dimensional and the residual part to which we can apply Poincaré type inequalities.

Such decomposition enables us to derive a good estimate for the inner product of the inertia term and the viscous term, which is essential for our arguments.

#### Algebraic Geometry Seminar

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

SNC log symplectic structures on Fano products (English/Japanese)

**Katsuhiko Okumura**(Waseda Univ. )SNC log symplectic structures on Fano products (English/Japanese)

[ Abstract ]

In 2014, Lima and Pereira gave a characterization of the even-dimensional projective space in terms of log symplectic Poisson structures. After that Pym gave an another more algebraic proof. In this talk, we will extend the result of Lima and Pereira to the case that the variety is a product of Fano varieties with the cyclic Picard group. This will be proved by extending Pym's proof. As a corollary, we will obtain a characterization of the projective space of all dimensions.

In 2014, Lima and Pereira gave a characterization of the even-dimensional projective space in terms of log symplectic Poisson structures. After that Pym gave an another more algebraic proof. In this talk, we will extend the result of Lima and Pereira to the case that the variety is a product of Fano varieties with the cyclic Picard group. This will be proved by extending Pym's proof. As a corollary, we will obtain a characterization of the projective space of all dimensions.

#### Numerical Analysis Seminar

16:50-18:20 Room #002 (Graduate School of Math. Sci. Bldg.)

Introduction to Machine learning and its application to Medical diagnosis (Japanese)

**Yoshiki Sugitani**(Tohoku University)Introduction to Machine learning and its application to Medical diagnosis (Japanese)

#### Tuesday Seminar on Topology

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

Tight contact structures on Seifert surface complements and knot invariants (ENGLISH)

**Tamás Kálmán**(Tokyo Institute of Technology)Tight contact structures on Seifert surface complements and knot invariants (ENGLISH)

[ Abstract ]

In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and used Honda's method to enumerate those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with `hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.

In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and used Honda's method to enumerate those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with `hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.

### 2018/04/16

#### Seminar on Geometric Complex Analysis

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

Metrics on a closed surface which maximize the first eigenvalue of the Laplacian (JAPANESE)

**Shin Nayatani**(Nagoya University)Metrics on a closed surface which maximize the first eigenvalue of the Laplacian (JAPANESE)

[ Abstract ]

In this talk, I will focus on recent progress on metrics that maximize the first eigenvalue of the Laplacian (under area normalization) on a closed surface. First, I introduce Hersch-Yang-Yau's inequality (1970, 1980), which was the starting point of the above problem. This is an inequality indicating that the first eigenvalue (precisely, the product of it with the area) is bounded from above by a constant depending only on the genus of the surface. Then I will outline the recent progress on the existence problem for maximizing metrics together with the relation with minimal surfaces in the sphere. Finally, I will discuss Jacobson-Levitin-Nadirashvili-Nigam-Polterovich's conjecture, which explicitly predicts maximizing metrics in the case of genus two, and the affirmative resolution of it (joint work with Toshihiro Shoda).

In this talk, I will focus on recent progress on metrics that maximize the first eigenvalue of the Laplacian (under area normalization) on a closed surface. First, I introduce Hersch-Yang-Yau's inequality (1970, 1980), which was the starting point of the above problem. This is an inequality indicating that the first eigenvalue (precisely, the product of it with the area) is bounded from above by a constant depending only on the genus of the surface. Then I will outline the recent progress on the existence problem for maximizing metrics together with the relation with minimal surfaces in the sphere. Finally, I will discuss Jacobson-Levitin-Nadirashvili-Nigam-Polterovich's conjecture, which explicitly predicts maximizing metrics in the case of genus two, and the affirmative resolution of it (joint work with Toshihiro Shoda).

### 2018/04/11

#### Number Theory Seminar

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

Non-abelian cohomology and Diophantine geometry (ENGLISH)

**Minhyong Kim**(University of Oxford)Non-abelian cohomology and Diophantine geometry (ENGLISH)

[ Abstract ]

This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.

This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.

### 2018/04/10

#### Tuesday Seminar on Topology

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

On the Morse-Novikov number for 2-knots (JAPANESE)

**Hisaaki Endo**(Tokyo Institute of Technology)On the Morse-Novikov number for 2-knots (JAPANESE)

[ Abstract ]

Pajitnov, Rudolph and Weber defined the Morse-Novikov number for classical links and studied their undamental properties in 2001. This invariant has been investigated in relation to (twisted) Alexander polynomials and the (twisted) Novikov homology. In this talk, we define the Morse-Novikov number for 2-knots and show its several properties. In particular, we describe its relations to motion pictures and spin constructions for 2-knots. This talk is based on joint works with Andrei Pajitnov (Nantes University).

Pajitnov, Rudolph and Weber defined the Morse-Novikov number for classical links and studied their undamental properties in 2001. This invariant has been investigated in relation to (twisted) Alexander polynomials and the (twisted) Novikov homology. In this talk, we define the Morse-Novikov number for 2-knots and show its several properties. In particular, we describe its relations to motion pictures and spin constructions for 2-knots. This talk is based on joint works with Andrei Pajitnov (Nantes University).

### 2018/04/09

#### Algebraic Geometry Seminar

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

Adjoint forms on algebraic varieties (English)

**Luca Rizzi**(Udine)Adjoint forms on algebraic varieties (English)

[ Abstract ]

The so called adjoint theory was introduced by A. Collino and G.P. Pirola in the case of smooth algebraic curves and then generalized by G.P. Pirola and F. Zucconi in the case of smooth algebraic varieties of arbitrary dimension.

The main idea of this theory is to study particular differential forms, called adjoint forms, on an algebraic variety to obtain information on the infinitesimal deformations of the variety itself.

The natural context for the application of this theory is given by Torelli-type problems, in particular infinitesimal Torelli problems.

The so called adjoint theory was introduced by A. Collino and G.P. Pirola in the case of smooth algebraic curves and then generalized by G.P. Pirola and F. Zucconi in the case of smooth algebraic varieties of arbitrary dimension.

The main idea of this theory is to study particular differential forms, called adjoint forms, on an algebraic variety to obtain information on the infinitesimal deformations of the variety itself.

The natural context for the application of this theory is given by Torelli-type problems, in particular infinitesimal Torelli problems.

#### Algebraic Geometry Seminar

13:30-15:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Commuting nilpotents, punctual Hilbert schemes and jet bundles (ENGLISH)

**David Hyeon**(Seoul National University)Commuting nilpotents, punctual Hilbert schemes and jet bundles (ENGLISH)

[ Abstract ]

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via two different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

Pairs of commuting nilpotent matrices have been extensively studied, especially from the view point of quivers, but the space of commuting nilpotents modulo simultaneous conjugation has not received any attention at all despite its moduli theory flavor. I will explain how a 'moduli space' can be constructed via two different methods and demonstrate many interesting properties of the space:

- It is isomorphic to an open subscheme of a punctual Hilbert scheme.

- Over the field of complex numbers, it is diffeomorphic to a direct sum of twisted tangent bundles over a projective space.

- It is isomorphic to a bundle of regular jets.

- It gives examples of affine space bundles that are not vector bundles.

This is a joint work with W. Haboush (Illinois) and G. Bérczi (Zurich).

### 2018/04/06

#### Colloquium

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

### 2018/04/03

#### Tuesday Seminar on Topology

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

Chain level loop bracket and pseudo-holomorphic disks (JAPANESE)

**Kei Irie**(The University of Tokyo)Chain level loop bracket and pseudo-holomorphic disks (JAPANESE)

[ Abstract ]

Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This idea is due to Fukaya, who also pointed out its important consequences in symplectic topology. In this talk I will explain how to rigorously carry out this idea. Our argument is based on a string topology chain model previously introduced by the speaker, and theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.

Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This idea is due to Fukaya, who also pointed out its important consequences in symplectic topology. In this talk I will explain how to rigorously carry out this idea. Our argument is based on a string topology chain model previously introduced by the speaker, and theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.

#### Infinite Analysis Seminar Tokyo

15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Screening Operators and Parabolic inductions for W-algebras

(ENGLISH)

**Naoki Genra**(RIMS, Kyoto U.)Screening Operators and Parabolic inductions for W-algebras

(ENGLISH)

[ Abstract ]

(Affine) W-algebras are the family of vertex algebras defined by

Drinfeld-Sokolov reductions. We introduce the free field realizations of

W-algebras by the Wakimoto representations of affine Lie algebras, which

we call the Wakimoto representations of W-algebras. Then W-algebras may be

described as the intersections of the kernels of the screening operators.

As applications, the parabolic inductions for W-algebras are obtained.

This is motivated by results of Premet and Losev on finite W-algebras. In

A-types, this becomes a chiralization of coproducts by Brundan-Kleshchev.

In BCD-types, we also have analogue results in special cases.

(Affine) W-algebras are the family of vertex algebras defined by

Drinfeld-Sokolov reductions. We introduce the free field realizations of

W-algebras by the Wakimoto representations of affine Lie algebras, which

we call the Wakimoto representations of W-algebras. Then W-algebras may be

described as the intersections of the kernels of the screening operators.

As applications, the parabolic inductions for W-algebras are obtained.

This is motivated by results of Premet and Losev on finite W-algebras. In

A-types, this becomes a chiralization of coproducts by Brundan-Kleshchev.

In BCD-types, we also have analogue results in special cases.

### 2018/03/30

#### Tuesday Seminar on Topology

15:00-16:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)

**Matteo Felder**(University of Geneva)Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)

[ Abstract ]

The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.

The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.

#### Tuesday Seminar on Topology

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

Goldman-Turaev formality in genus 0 from the KZ connection (ENGLISH)

**Florian Naef**(Massachusetts Institute of Technology)Goldman-Turaev formality in genus 0 from the KZ connection (ENGLISH)

[ Abstract ]

Using the intersection and self-intersection of loops on a surface one can define the Goldman-Turaev Lie bialgebra. By earlier joint work with A. Alekseev, N. Kawazumi and Y. Kuno it is known that the linearization problem of the Goldman-Turaev Lie bialgebra is closely related to the Kashiwara-Vergne problem and hence to Drinfeld associators. It turns out that in the case of a genus 0 surface and over the field of complex numbers there is a very direct and explicit proof of the formality of the Goldman-Turaev Lie bialgebra using the monodromy of the Knizhnik-Zamolodchikov connection. This is joint work with Anton Alekseev.

Using the intersection and self-intersection of loops on a surface one can define the Goldman-Turaev Lie bialgebra. By earlier joint work with A. Alekseev, N. Kawazumi and Y. Kuno it is known that the linearization problem of the Goldman-Turaev Lie bialgebra is closely related to the Kashiwara-Vergne problem and hence to Drinfeld associators. It turns out that in the case of a genus 0 surface and over the field of complex numbers there is a very direct and explicit proof of the formality of the Goldman-Turaev Lie bialgebra using the monodromy of the Knizhnik-Zamolodchikov connection. This is joint work with Anton Alekseev.

### 2018/03/27

#### Infinite Analysis Seminar Tokyo

15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Schramm-Loewner evolutions and Liouville field theory (JAPANESE)

**Yoshiki Fukusumi**(The University of Tokyo, The Institute for Solid State Physics)Schramm-Loewner evolutions and Liouville field theory (JAPANESE)

[ Abstract ]

Schramm-Loewner evolutions (SLEs) are stochastic processes driven by Brownian motions which preserves conformal invariance. They describe the cluster boundaries associated with the minimal models of the conformal field theory, including the Ising model and the percolation as typical examples. The correlation functions of such models remarkably satisfy the martingale condition. We briefly review some known results. Then we analyse the time reversing procedure of Schramm Loewner evolutions and its relation to Liouville field theory or 2d pure gravity. We can get martingale observables by the calculation of the correlation functions of Liouville field theory without matter.

Schramm-Loewner evolutions (SLEs) are stochastic processes driven by Brownian motions which preserves conformal invariance. They describe the cluster boundaries associated with the minimal models of the conformal field theory, including the Ising model and the percolation as typical examples. The correlation functions of such models remarkably satisfy the martingale condition. We briefly review some known results. Then we analyse the time reversing procedure of Schramm Loewner evolutions and its relation to Liouville field theory or 2d pure gravity. We can get martingale observables by the calculation of the correlation functions of Liouville field theory without matter.

### 2018/03/26

#### FMSP Lectures

10:00-12:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Geometric Recursion (ENGLISH)

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

**Jørgen Ellegaard Andersen**(Aarhus University)Geometric Recursion (ENGLISH)

[ Abstract ]

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work

presented is joint with G. Borot and N. Orantin.

[ Reference URL ]Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work

presented is joint with G. Borot and N. Orantin.

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

### 2018/03/23

#### FMSP Lectures

10:00-12:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Geometric Recursion (ENGLISH)

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

**Jørgen Ellegaard Andersen**(Aarhus University)Geometric Recursion (ENGLISH)

[ Abstract ]

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the

Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work presented is joint with G. Borot and N. Orantin.

[ Reference URL ]Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the

Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmüller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. We shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmüller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work presented is joint with G. Borot and N. Orantin.

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

### 2018/03/19

#### Mathematical Biology Seminar

17:00-18:00 Room #509 (Graduate School of Math. Sci. Bldg.)

Large deviation theory for age-structured population dynamics

**Yuki Sugiyama**(Institute of Industrial Science, The University of Tokyo)Large deviation theory for age-structured population dynamics

[ Abstract ]

Control of population growth is ubiquitous problem in many fields. In the context of medical treatment, we attempt to diminish the growing speed of a cell population composed of cancer cells or pathogens by using antibiotics or some special therapies. In terms of evolutional biology, to survive in a fluctuating environment, cells maximize (optimize) their population growth by exploiting a risk hedge strategy for adaptation to the fluctuation. Recent development of experimental devices enables us to measure a big size lineage data that describes a growing cell population. In this study, by using these lineage data, we analyze a behavior of the population growth. Here, a structure of statistical mechanics using the large deviation theory plays an important role. As a results, we reveal that the population growth rate is given by the Legendre transform of the large deviation function for the semi-Markov process that describes a stochastic switch of cell types in the time evolution. Furthermore, by using this structure, we show that responses of the population growth rate with respect to an environmental change can be evaluated by statistics on a retrospective lineage path.

Control of population growth is ubiquitous problem in many fields. In the context of medical treatment, we attempt to diminish the growing speed of a cell population composed of cancer cells or pathogens by using antibiotics or some special therapies. In terms of evolutional biology, to survive in a fluctuating environment, cells maximize (optimize) their population growth by exploiting a risk hedge strategy for adaptation to the fluctuation. Recent development of experimental devices enables us to measure a big size lineage data that describes a growing cell population. In this study, by using these lineage data, we analyze a behavior of the population growth. Here, a structure of statistical mechanics using the large deviation theory plays an important role. As a results, we reveal that the population growth rate is given by the Legendre transform of the large deviation function for the semi-Markov process that describes a stochastic switch of cell types in the time evolution. Furthermore, by using this structure, we show that responses of the population growth rate with respect to an environmental change can be evaluated by statistics on a retrospective lineage path.

### 2018/03/15

#### Seminar on Probability and Statistics

16:00-17:10 Room #052 (Graduate School of Math. Sci. Bldg.)

On Hypotheses testing for discretely observed SDE (Joint work with Alessandro De Gregorio, University of Rome)

**Stefano Iacus**(University of Milan)On Hypotheses testing for discretely observed SDE (Joint work with Alessandro De Gregorio, University of Rome)

[ Abstract ]

In this talk we consider parametric hypotheses testing for discretely observed ergodic diffusion processes. We present the different test statistics proposed in literature and recall their asymptotic properties. We also compare the empirical performance of different tests in the case of small sample sizes.

In this talk we consider parametric hypotheses testing for discretely observed ergodic diffusion processes. We present the different test statistics proposed in literature and recall their asymptotic properties. We also compare the empirical performance of different tests in the case of small sample sizes.

### 2018/03/13

#### Lectures

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

### 2018/03/12

#### Lie Groups and Representation Theory

15:00-16:30 Room #126 (Graduate School of Math. Sci. Bldg.)

Plethysms and Kronecker coefficients in geometric complexity theory

**Christian Ikenmeyer**(Max-Planck-Institut fur Informatik)Plethysms and Kronecker coefficients in geometric complexity theory

[ Abstract ]

Research on Kronecker coefficients and plethysms gained significant momentum when the topics were connected with geometric complexity theory, an approach towards computational complexity lower bounds via algebraic geometry and representation theory. This talk is about several recent results that were obtained with geometric complexity theory as motivation, namely the NP-hardness of deciding the positivity of Kronecker coefficients and an inequality between rectangular Kronecker coefficients and plethysm coefficients. While the proof of the former statement is mainly combinatorial, the proof of the latter statement interestingly uses insights from algebraic complexity theory. As far as we know algebraic complexity theory has never been used before to prove an inequality between representation theoretic multiplicities.

Research on Kronecker coefficients and plethysms gained significant momentum when the topics were connected with geometric complexity theory, an approach towards computational complexity lower bounds via algebraic geometry and representation theory. This talk is about several recent results that were obtained with geometric complexity theory as motivation, namely the NP-hardness of deciding the positivity of Kronecker coefficients and an inequality between rectangular Kronecker coefficients and plethysm coefficients. While the proof of the former statement is mainly combinatorial, the proof of the latter statement interestingly uses insights from algebraic complexity theory. As far as we know algebraic complexity theory has never been used before to prove an inequality between representation theoretic multiplicities.

### 2018/03/10

#### Colloquium

11:00-12:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Hitoshi ARAI**(Univ. Tokyo)(JAPANESE)

#### Colloquium

13:00-14:00 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

(JAPANESE)

**Akito FUTAKI**(Univ. Tokyo)(JAPANESE)

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