## Seminar information archive

Seminar information archive ～05/26｜Today's seminar 05/27 | Future seminars 05/28～

#### Algebraic Geometry Seminar

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On classification of prime Q-Fano 3-folds with only 1/2(1,1,1)-singularities and of genus less than 2

**Hiromichi Takagi**(The University of Tokyo)On classification of prime Q-Fano 3-folds with only 1/2(1,1,1)-singularities and of genus less than 2

[ Abstract ]

I classified prime Q-Fano threefolds with only 1/2(1,1,1)-singularities and of genus greater than 1 (2002, Nagoya Math. J.).

In this talk, I will explain how the method in that paper can be extended to the case of genus less than 2. The method is so called two ray game. By this method, I can classify the possibilities of such Q-Fano's. The classification is not yet completed since constructions of examples in certain cases are difficult. I will also explain some pretty examples in this talk.

I classified prime Q-Fano threefolds with only 1/2(1,1,1)-singularities and of genus greater than 1 (2002, Nagoya Math. J.).

In this talk, I will explain how the method in that paper can be extended to the case of genus less than 2. The method is so called two ray game. By this method, I can classify the possibilities of such Q-Fano's. The classification is not yet completed since constructions of examples in certain cases are difficult. I will also explain some pretty examples in this talk.

### 2018/01/25

#### FMSP Lectures

15:00-16:30 Room #002 (Graduate School of Math. Sci. Bldg.)

NUMERICAL ANALYSIS, COBORDISM OF MANIFOLDS AND MONODROMY. (ENGLISH)

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

**Norbert A'Campo**(University of Basel)NUMERICAL ANALYSIS, COBORDISM OF MANIFOLDS AND MONODROMY. (ENGLISH)

[ Abstract ]

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

[ Reference URL ]http://fmsp.ms.u-tokyo.ac.jp/FMSPLectures_ACampo_abst.pdf

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

### 2018/01/23

#### Tuesday Seminar on Topology

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

An invariant of 3-manifolds via homology cobordisms (JAPANESE)

**Yuta Nozaki**(The University of Tokyo)An invariant of 3-manifolds via homology cobordisms (JAPANESE)

[ Abstract ]

For a closed 3-manifold X, we consider the topological invariant defined as the minimal integer g such that X is obtained as the closure of a homology cobordism over a surface of genus g. We prove that the invariant equals one for every lens space, which is contrast to the fact that some lens spaces do not admit any open book decomposition whose page is a surface of genus one. The proof is based on the Chebotarev density theorem and binary quadratic forms in number theory.

For a closed 3-manifold X, we consider the topological invariant defined as the minimal integer g such that X is obtained as the closure of a homology cobordism over a surface of genus g. We prove that the invariant equals one for every lens space, which is contrast to the fact that some lens spaces do not admit any open book decomposition whose page is a surface of genus one. The proof is based on the Chebotarev density theorem and binary quadratic forms in number theory.

#### Tuesday Seminar on Topology

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

Wrapping projections and decompositions of Keinian groups (JAPANESE)

**Junha Tanaka**(The University of Tokyo)Wrapping projections and decompositions of Keinian groups (JAPANESE)

[ Abstract ]

Let $S$ be a closed surface of genus $g ¥geq 2$. The deformation space $AH(S)$ consists of (conjugacy classes of) discrete faithful representations $\rho:\pi_{1}(S) \to PSL_{2}(\mathbb{C})$.

McMullen, and Bromberg and Holt showed that $AH(S)$ can self-bump, that is, the interior of $AH(S)$ has the self-intersecting closure.

Both of them demonstrated the existence of self-bumping under the exisetence of a non-trivial wrapping projections from an algebraic limits to a geometric limits which wraps an annulus cusp into a torus cusp.

In this talk, given a representation $\rho$ at the boundary of $AH(S)$, we characterize a wrapping projection to a geometric limit associated to $\rho$, by the information of the actions of decomposed Kleinian groups of the image of $\rho$.

Let $S$ be a closed surface of genus $g ¥geq 2$. The deformation space $AH(S)$ consists of (conjugacy classes of) discrete faithful representations $\rho:\pi_{1}(S) \to PSL_{2}(\mathbb{C})$.

McMullen, and Bromberg and Holt showed that $AH(S)$ can self-bump, that is, the interior of $AH(S)$ has the self-intersecting closure.

Both of them demonstrated the existence of self-bumping under the exisetence of a non-trivial wrapping projections from an algebraic limits to a geometric limits which wraps an annulus cusp into a torus cusp.

In this talk, given a representation $\rho$ at the boundary of $AH(S)$, we characterize a wrapping projection to a geometric limit associated to $\rho$, by the information of the actions of decomposed Kleinian groups of the image of $\rho$.

### 2018/01/22

#### Seminar on Geometric Complex Analysis

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

Recent topics on the study of the Gauss images of minimal surfaces

**Yu Kawakami**(Kanazawa University)Recent topics on the study of the Gauss images of minimal surfaces

[ Abstract ]

In this talk, we give a survey of recent advances on the study of the images of the Gauss maps of complete minimal surfaces in Euclidean space.

In this talk, we give a survey of recent advances on the study of the images of the Gauss maps of complete minimal surfaces in Euclidean space.

#### Operator Algebra Seminars

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

On Cauchy noise loss in a stochastic parameter optimization of random matrices

**Tomohiro Hayase**(Univ. Tokyo)On Cauchy noise loss in a stochastic parameter optimization of random matrices

#### Tokyo Probability Seminar

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

(JAPANESE)

**Makiko Sasada**(Graduate School of Mathematical Science, the University of Tokyo)(JAPANESE)

### 2018/01/17

#### Number Theory Seminar

18:00-19:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On the vanishing of cohomology for certain Shimura varieties (ENGLISH)

**Ana Caraiani**(Imperial College)On the vanishing of cohomology for certain Shimura varieties (ENGLISH)

[ Abstract ]

I will prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^\infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.

I will prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^\infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.

#### Discrete mathematical modelling seminar

17:00-18:45 Room #056 (Graduate School of Math. Sci. Bldg.)

Quantum matter bounce with a dark energy expanding phase (ENGLISH)

Mass of the vacuum: a Newtonian perspective (ENGLISH)

**Samuel Colin**(CBPF, Rio de Janeiro, Brasil) 17:00-17:50Quantum matter bounce with a dark energy expanding phase (ENGLISH)

[ Abstract ]

The ``matter bounce'' is an alternative scenario to inflationary cosmology, according to which the universe undergoes a contraction, followed by an expansion, the bounce occurring when the quantum effects become important. In my talk, I will show that such a scenario can be unambiguously analyzed in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics. More specifically, I will apply the pilot-wave theory to a Wheeler-DeWitt equation obtained from the quantization of a simple classical mini-superspace model, and show that there are numerical solutions describing bouncing universes with many desirable physical features. For example, one solution contains a dark energy phase during the expansion, without the need to postulate the existence of a cosmological constant in the classical action.

This work was done in collaboration with Nelson Pinto-Neto (CBPF, Rio de Janeiro, Brasil). Further details available at https://arxiv.org/abs/1706.03037.

The ``matter bounce'' is an alternative scenario to inflationary cosmology, according to which the universe undergoes a contraction, followed by an expansion, the bounce occurring when the quantum effects become important. In my talk, I will show that such a scenario can be unambiguously analyzed in the de Broglie-Bohm pilot-wave interpretation of quantum mechanics. More specifically, I will apply the pilot-wave theory to a Wheeler-DeWitt equation obtained from the quantization of a simple classical mini-superspace model, and show that there are numerical solutions describing bouncing universes with many desirable physical features. For example, one solution contains a dark energy phase during the expansion, without the need to postulate the existence of a cosmological constant in the classical action.

This work was done in collaboration with Nelson Pinto-Neto (CBPF, Rio de Janeiro, Brasil). Further details available at https://arxiv.org/abs/1706.03037.

**Thomas Durt**(Aix Marseille Université, Centrale Marseille, Institut Fresnel) 17:50-18:40Mass of the vacuum: a Newtonian perspective (ENGLISH)

[ Abstract ]

One could believe that special relativity forces us to totally renounce to the idea of an aether, but the aether reappears in general relativity which teaches us that space-time is structured by the local metrics. It also reappears in quantum field theory which teaches us that even at zero temperature space is filled by the quantum vacuum energy. Finally, aether reappears in modern astronomy where it was necessary to introduce ill-defined concepts such as dark matter and dark energy in order to explain apparent deviations from Newtonian dynamics (at the level of galactic rotation curves).

Newton dynamics being the unique limit of general relativistic dynamics in the classical regime, dark matter and dark energy can be seen as an ultimate, last chance strategy, aimed at reconciling the predictions of general relativity with astronomical data.

In our talk we shall describe a simple model, derived in the framework of Newtonian dynamics, aimed at explaining puzzling astronomical observations realized at the level of the solar system (Pioneer anomaly) and at the galactic scale (rotation curves), without adopting ad hoc hypotheses about the existence of dark matter and/or dark energy.

The basic idea is that Newtonian gravity is modified due to the presence of a (negative) density, everywhere in space, of mass-energy.

One could believe that special relativity forces us to totally renounce to the idea of an aether, but the aether reappears in general relativity which teaches us that space-time is structured by the local metrics. It also reappears in quantum field theory which teaches us that even at zero temperature space is filled by the quantum vacuum energy. Finally, aether reappears in modern astronomy where it was necessary to introduce ill-defined concepts such as dark matter and dark energy in order to explain apparent deviations from Newtonian dynamics (at the level of galactic rotation curves).

Newton dynamics being the unique limit of general relativistic dynamics in the classical regime, dark matter and dark energy can be seen as an ultimate, last chance strategy, aimed at reconciling the predictions of general relativity with astronomical data.

In our talk we shall describe a simple model, derived in the framework of Newtonian dynamics, aimed at explaining puzzling astronomical observations realized at the level of the solar system (Pioneer anomaly) and at the galactic scale (rotation curves), without adopting ad hoc hypotheses about the existence of dark matter and/or dark energy.

The basic idea is that Newtonian gravity is modified due to the presence of a (negative) density, everywhere in space, of mass-energy.

### 2018/01/16

#### Tuesday Seminar on Topology

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

On adequacy and the crossing number of satellite knots (JAPANESE)

**Jimenez Pascual Adrian**(The University of Tokyo)On adequacy and the crossing number of satellite knots (JAPANESE)

[ Abstract ]

It has always been difficult to prove results regarding the (minimal) crossing number of knots. In particular, apparently easy problems such as knowing the crossing number of the connected sum of knots, or bounding the crossing number of satellite knots have been conjectured through decades, yet still remain open. Focusing on this latter problem, in this talk I will prove that the crossing number of a satellite knot is bounded from below by the crossing number of its companion, when the companion is adequate.

It has always been difficult to prove results regarding the (minimal) crossing number of knots. In particular, apparently easy problems such as knowing the crossing number of the connected sum of knots, or bounding the crossing number of satellite knots have been conjectured through decades, yet still remain open. Focusing on this latter problem, in this talk I will prove that the crossing number of a satellite knot is bounded from below by the crossing number of its companion, when the companion is adequate.

#### Tuesday Seminar on Topology

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

A new relationship between the dilatation of pseudo-Anosov braids and fixed point theory (JAPANESE)

**Yumehito Kawashima**(The University of Tokyo)A new relationship between the dilatation of pseudo-Anosov braids and fixed point theory (JAPANESE)

[ Abstract ]

A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this talk we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.

A relation between the dilatation of pseudo-Anosov braids and fixed point theory was studied by Ivanov. In this talk we reveal a new relationship between the above two subjects by showing a formula for the dilatation of pseudo-Anosov braids by means of the representations of braid groups due to B. Jiang and H. Zheng.

#### FMSP Lectures

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

Large data global solutions for the shallow water system in one space dimension

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_180116.pdf

Introduction to the maximal Lp-regularity and its applications to the quasi-linear parabolic equations

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_180116.pdf

**Federico Pasqualotto**(Princeton) -Large data global solutions for the shallow water system in one space dimension

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_180116.pdf

**Naoto Kaziwara**(U. Tokyo) -Introduction to the maximal Lp-regularity and its applications to the quasi-linear parabolic equations

[ Reference URL ]

http://fmsp.ms.u-tokyo.ac.jp/FMSP_180116.pdf

### 2018/01/15

#### Seminar on Geometric Complex Analysis

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

Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds

**Shinya Akagawa**(Osaka University)Vanishing theorems of $L^2$-cohomology groups on Hessian manifolds

[ Abstract ]

A Hessian manifold is a Riemannian manifold whose metric is locally given by the Hessian of a function with respect to flat coordinates. In this talk, we discuss vanishing theorems of $L^2$-cohomology groups on complete Hessian Manifolds coupled with flat line bundles. In particular, we obtain stronger vanishing theorems on regular convex cones with the Cheng-Yau metrics. Further we show that the Cheng-Yau metrics on regular convex cones give rise to harmonic maps to the positive symmetric matrices.

A Hessian manifold is a Riemannian manifold whose metric is locally given by the Hessian of a function with respect to flat coordinates. In this talk, we discuss vanishing theorems of $L^2$-cohomology groups on complete Hessian Manifolds coupled with flat line bundles. In particular, we obtain stronger vanishing theorems on regular convex cones with the Cheng-Yau metrics. Further we show that the Cheng-Yau metrics on regular convex cones give rise to harmonic maps to the positive symmetric matrices.

### 2017/12/26

#### Algebraic Geometry Seminar

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

K-stability of log Fano hyperplane arrangements (English)

**Kento Fujita**(RIMS)K-stability of log Fano hyperplane arrangements (English)

[ Abstract ]

We completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

We completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

### 2017/12/25

#### Operator Algebra Seminars

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

Kazhdan's property (T) and semidefinite programming

**Narutaka Ozawa**(RIMS, Kyoto University)Kazhdan's property (T) and semidefinite programming

### 2017/12/21

#### Applied Analysis

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

#### Mathematical Biology Seminar

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Theoretical approaches to understand eco-evolutionary feedbacks

**Masato Yamamichi**(Department of General Systems Studies, The University of Tokyo)Theoretical approaches to understand eco-evolutionary feedbacks

### 2017/12/19

#### Numerical Analysis Seminar

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

#### Tuesday Seminar on Topology

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

Deformation of holomorphic quadratic differentials and its applications (JAPANESE)

**Hideki Miyachi**(Osaka university)Deformation of holomorphic quadratic differentials and its applications (JAPANESE)

[ Abstract ]

Quadratic differentials are standard and important objects in Teichmuller theory. The deformation space (moduli space) of the quadratic differentials is applied to many fields of mathematics. In this talk, I will develop the deformation of quadratic differentials. Indeed, following pioneer works by A. Douady, J. Hubbard, H. Masur and W. Veech, we describe the infinitesimal deformations in the odd (co)homology groups on the double covering spaces defined from the square roots of the quadratic differentials. We formulate the decomposition theorem for the infinitesimal deformations with keeping in mind of the induced deformation of the moduli of underlying complex structures. As applications, we obtain the Levi form of the Teichmuller distance, and an alternate proof of the Krushkal formula on the pluricomplex Green function on the Teichmuller space.

Quadratic differentials are standard and important objects in Teichmuller theory. The deformation space (moduli space) of the quadratic differentials is applied to many fields of mathematics. In this talk, I will develop the deformation of quadratic differentials. Indeed, following pioneer works by A. Douady, J. Hubbard, H. Masur and W. Veech, we describe the infinitesimal deformations in the odd (co)homology groups on the double covering spaces defined from the square roots of the quadratic differentials. We formulate the decomposition theorem for the infinitesimal deformations with keeping in mind of the induced deformation of the moduli of underlying complex structures. As applications, we obtain the Levi form of the Teichmuller distance, and an alternate proof of the Krushkal formula on the pluricomplex Green function on the Teichmuller space.

### 2017/12/18

#### Seminar on Geometric Complex Analysis

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

Gradient flow of the Ding functional

**Tomoyuki Hisamoto**(Nagoya University)Gradient flow of the Ding functional

[ Abstract ]

This is a joint work with T. Collins and R. Takahashi. We introduce the flow in the title to study the stability of a Fano manifold. The first result is the long-time existence of the flow. In the stable case it then converges to the Kähler-Einstein metric. In general the flow is expected to produce the optimally destabilizing degeneration of a Fano manifold. We confirm this expectation in the toric case.

This is a joint work with T. Collins and R. Takahashi. We introduce the flow in the title to study the stability of a Fano manifold. The first result is the long-time existence of the flow. In the stable case it then converges to the Kähler-Einstein metric. In general the flow is expected to produce the optimally destabilizing degeneration of a Fano manifold. We confirm this expectation in the toric case.

#### Operator Algebra Seminars

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

Isomorphism and Morita equivalence classes for crossed product of irrational rotation algebras by cyclic subgroups of $SL_2({\mathbb Z})$ (English)

**Zhuofeng He**(Univ. Tokyo)Isomorphism and Morita equivalence classes for crossed product of irrational rotation algebras by cyclic subgroups of $SL_2({\mathbb Z})$ (English)

### 2017/12/14

#### Algebraic Geometry Seminar

15:30-17:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Algebraic curves and modular forms of low degree (English)

**Gerard van der Geer**(Universiteit van Amsterdam)Algebraic curves and modular forms of low degree (English)

[ Abstract ]

For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

For genus 2 and 3 modular forms are intimately connected with the moduli of curves of genus 2 and 3. We give an explicit way to describe such modular forms for genus 2 and 3 using invariant theory and give some applications. This is based on joint work with Fabien Clery and Carel Faber.

#### Applied Analysis

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

Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

(English)

**I-Kun, Chen**(Kyoto University)Regularity for diffuse reflection boundary problem to the stationary linearized Boltzmann equation in a convex domain

(English)

[ Abstract ]

We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.

We consider the diffuse reflection boundary problem for the linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or Maxwellian molecular gases in a $C^2$ strictly convex bounded domain. We obtain a pointwise estimate for the derivative of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. Velocity averaging effect for stationary solutions as well as observations in geometry are used in this research.

#### Algebraic Geometry Seminar

10:30-12:00 Room #123 (Graduate School of Math. Sci. Bldg.)

Perfectoid test ideals (English)

**Linquan Ma**(University of Utah)Perfectoid test ideals (English)

[ Abstract ]

Inspired by the recent solution of the direct summand conjecture

of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed

characteristic. As an application, we obtain a uniform bound on the growth

of symbolic powers in regular local rings of mixed characteristic analogous

to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal

characteristic. This is joint work with Karl Schwede.

Inspired by the recent solution of the direct summand conjecture

of Andre and Bhatt, we introduce perfectoid multiplier/test ideals in mixed

characteristic. As an application, we obtain a uniform bound on the growth

of symbolic powers in regular local rings of mixed characteristic analogous

to results of Ein--Lazarsfeld--Smith and Hochster--Huneke in equal

characteristic. This is joint work with Karl Schwede.

#### Mathematical Biology Seminar

13:00-16:40 Room #126 (Graduate School of Math. Sci. Bldg.)

Mathematical analysis for HBV model and HBV-HDV coinfection model (ENGLISH)

**Xu Yaya**15:40-16:10Mathematical analysis for HBV model and HBV-HDV coinfection model (ENGLISH)

[ Abstract ]

The hepatitis beta virus (HBV) and hepatitis delta viurs (HDV)

are two common forms of viral hepatitis. However HDV is dependent

on coinfection with HBV since replication of HDV requires the hepati-

tis B surface antigen (HBsAg) which can only been produced by HBV.

Here we start with analyzing HBV only model, the dynamics between

healthy cells, HBV infected cells and free HBV.We show that a postive

equilbrium exsits and it's globally asmptotically stable for R0 > 1, an

infection free equilibrium is globally asymptotically stable for R0 < 1.

Then we introduce HDV to form a coinfection model which contains

three more variables, HDV infected cells, coinfected cells and free HDV.

Additionally, we investigate two coinfection models, one without and

one with treatment by oral drugs which are valid for HBV only. We

consider several durgs with variable eciencies. As a result, compari-

son of model simulations indicate that treatment is necessary to taking

contiously for choric infection.

The hepatitis beta virus (HBV) and hepatitis delta viurs (HDV)

are two common forms of viral hepatitis. However HDV is dependent

on coinfection with HBV since replication of HDV requires the hepati-

tis B surface antigen (HBsAg) which can only been produced by HBV.

Here we start with analyzing HBV only model, the dynamics between

healthy cells, HBV infected cells and free HBV.We show that a postive

equilbrium exsits and it's globally asmptotically stable for R0 > 1, an

infection free equilibrium is globally asymptotically stable for R0 < 1.

Then we introduce HDV to form a coinfection model which contains

three more variables, HDV infected cells, coinfected cells and free HDV.

Additionally, we investigate two coinfection models, one without and

one with treatment by oral drugs which are valid for HBV only. We

consider several durgs with variable eciencies. As a result, compari-

son of model simulations indicate that treatment is necessary to taking

contiously for choric infection.

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