## Seminar information archive

Seminar information archive ～05/23｜Today's seminar 05/24 | Future seminars 05/25～

### 2017/04/11

#### Number Theory Seminar

17:30-18:30 Room #117 (Graduate School of Math. Sci. Bldg.)

The geometric Satake equivalence in mixed characteristic (ENGLISH)

**Peter Scholze**(University of Bonn)The geometric Satake equivalence in mixed characteristic (ENGLISH)

[ Abstract ]

In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.

In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.

#### Numerical Analysis Seminar

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

Some issues in the Lagrange-Galerkin method and solutions: computability, dependence on the viscosity and inflow boundary conditions (日本語)

**Shinya Uchiumi**(Waseda University)Some issues in the Lagrange-Galerkin method and solutions: computability, dependence on the viscosity and inflow boundary conditions (日本語)

#### Tuesday Seminar on Topology

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

Homotopy Lie algebroids (ENGLISH)

**Alexander Voronov**(University of Minnesota)Homotopy Lie algebroids (ENGLISH)

[ Abstract ]

A well-known result of A. Vaintrob [Vai97] characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic dg-manifolds, building on the approach of D. Roytenberg [Roy99]. This extends naturally to the homotopy Lie case and leads to the notion of L

A well-known result of A. Vaintrob [Vai97] characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic dg-manifolds, building on the approach of D. Roytenberg [Roy99]. This extends naturally to the homotopy Lie case and leads to the notion of L

_{∞}-bialgebroids and L_{∞}-morphisms between them.### 2017/04/10

#### Seminar on Geometric Complex Analysis

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

Slice theorem for CR structures near the sphere and its applications

**Kengo Hirachi**(The University of Tokyo)Slice theorem for CR structures near the sphere and its applications

[ Abstract ]

We formulate a slice theorem for CR structures by following Bland-Duchamp and give some applications to the rigidity theorems.

We formulate a slice theorem for CR structures by following Bland-Duchamp and give some applications to the rigidity theorems.

#### Operator Algebra Seminars

16:45-18:15 Room #118 (Graduate School of Math. Sci. Bldg.)

Reconstruction of the Bost-Connes groupoid from K-theoretic data (English)

**Yosuke Kubota**(Riken)Reconstruction of the Bost-Connes groupoid from K-theoretic data (English)

### 2017/04/06

#### Mathematical Biology Seminar

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

(JAPANESE)

**Shinji Nakaoka**(JAPANESE)

#### Mathematical Biology Seminar

17:00-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

#### Colloquium of mathematical sciences and society

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

### 2017/03/30

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Logarithmic ramifications via pull-back to curves (English)

**Haoyu Hu**(University of Tokyo)Logarithmic ramifications via pull-back to curves (English)

[ Abstract ]

Let X be a smooth variety over a perfect field of characteristic p>0, D a strict normal crossing divisor of X, U the complement of D in X, j:U—>X the canonical injection, and F a locally constant and constructible sheaf of F_l-modules on U (l is a prime number different from p). Using Abbes and Saito’s logarithmic ramification theory, we define a Swan divisor SW(j_!F), which supported on D. Let i:C-->X be a quasi-finite morphism from a smooth curve C to X. Following T. Saito’s idea, we compare the pull-back of SW(j_!F) to C with the Swan divisor of the pull-back of j_!F to C. It answers an expectation of Esnault and Kerz and generalizes the same result of Barrientos for rank 1 sheaves. As an application, we obtain a lower semi-continuity property for Swan divisors of an l-adic sheaf on a smooth fibration, which gives a generalization of Deligne and Laumon’s lower semi-continuity property of Swan conductors of l-adic sheaves on relative curves to higher relative dimensions. This application is a supplement of the semi-continuity of total dimension of vanishing cycles due to T. Saito and the lower semi-continuity of total dimension divisors due to myself and E. Yang.

Let X be a smooth variety over a perfect field of characteristic p>0, D a strict normal crossing divisor of X, U the complement of D in X, j:U—>X the canonical injection, and F a locally constant and constructible sheaf of F_l-modules on U (l is a prime number different from p). Using Abbes and Saito’s logarithmic ramification theory, we define a Swan divisor SW(j_!F), which supported on D. Let i:C-->X be a quasi-finite morphism from a smooth curve C to X. Following T. Saito’s idea, we compare the pull-back of SW(j_!F) to C with the Swan divisor of the pull-back of j_!F to C. It answers an expectation of Esnault and Kerz and generalizes the same result of Barrientos for rank 1 sheaves. As an application, we obtain a lower semi-continuity property for Swan divisors of an l-adic sheaf on a smooth fibration, which gives a generalization of Deligne and Laumon’s lower semi-continuity property of Swan conductors of l-adic sheaves on relative curves to higher relative dimensions. This application is a supplement of the semi-continuity of total dimension of vanishing cycles due to T. Saito and the lower semi-continuity of total dimension divisors due to myself and E. Yang.

### 2017/03/22

#### FMSP Lectures

13:00- Room #117 (Graduate School of Math. Sci. Bldg.)

Lecture 1: Derived symplectic varieties and the Darboux theorem.

Lecture 2: The moduli of anti-canonically marked del Pezzo surfaces. (ENGLISH)

[ Reference URL ]

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

**Ian Grojnowski**(University of Cambridge)Lecture 1: Derived symplectic varieties and the Darboux theorem.

Lecture 2: The moduli of anti-canonically marked del Pezzo surfaces. (ENGLISH)

[ Reference URL ]

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

### 2017/03/21

#### Colloquium

14:40-15:40 Room #大講義室 (Graduate School of Math. Sci. Bldg.)

A tour around microlocal analysis and algebraic analysis (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kiyoomi/index.html

**Kiyoomi Kataoka**(Graduate School of Mathematical Sciences, The University of Tokyo)A tour around microlocal analysis and algebraic analysis (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~kiyoomi/index.html

#### Colloquium

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

40 years along with stochastic analysis --- Motivated by statistical physics problems --- (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~funaki/

**Tadahisa Funaki**(Graduate School of Mathematical Sciences, The University of Tokyo)40 years along with stochastic analysis --- Motivated by statistical physics problems --- (JAPANESE)

[ Reference URL ]

http://www.ms.u-tokyo.ac.jp/~funaki/

### 2017/03/10

#### Tuesday Seminar on Topology

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

Satake compactifications and metric Schottky problems (ENGLISH)

**Lizhen Ji**(University of Michigan)Satake compactifications and metric Schottky problems (ENGLISH)

[ Abstract ]

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:

(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,

(2) Arithmetic locally symmetric spaces Γ \ G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.

There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g. In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:

(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,

(2) Arithmetic locally symmetric spaces Γ \ G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.

There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g. In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.

#### Lie Groups and Representation Theory

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

Satake compactifications and metric Schottky problems (English)

**Lizhen Ji**(University of Michigan, USA)Satake compactifications and metric Schottky problems (English)

[ Abstract ]

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:

(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,

(2) Arithmetic locally symmetric spaces \Gamma \ G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.

There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g.

In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.

The quotient of the Poincare upper half plane by the modular group SL(2, Z) is a basic locally symmetric space and also the moduli space of compact Riemann surfaces of genus 1, and it admits two important classes of generalization:

(1) Moduli spaces M_g of compact Riemann surfaces of genus g>1,

(2) Arithmetic locally symmetric spaces \Gamma \ G/K such as the Siegel modular variety A_g, which is also the moduli of principally polarized abelian varieties of dimension g.

There have been a lot of fruitful work to explore the similarity between these two classes of spaces, and there is also a direct interaction between them through the Jacobian (or period) map J: M_g --> A_g.

In this talk, I will discuss some results along these lines related to the Stake compactifications and the Schottky problems on understanding the image J(M_g) in A_g from the metric perspective.

### 2017/03/08

#### Tuesday Seminar on Topology

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

Action of the Long-Moody Construction on Polynomial Functors (ENGLISH)

**Arthur Soulié**(Université de Strasbourg)Action of the Long-Moody Construction on Polynomial Functors (ENGLISH)

[ Abstract ]

In 2016, Randal-Williams and Wahl proved homological stability with certain twisted coefficients for different families of groups, in particular the one of braid groups. In fact, they obtain the stability for coefficients given by functors satisfying polynomial conditions. We only know few examples of such functors. Among them, we have the functor given by the unreduced Burau representations. In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. This construction complexifies in a sense the initial representation: for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. In this talk, I will present this construction from a functorial point of view. I will explain that the construction of Long and Moody defines an endofunctor, called the Long-Moody functor, between a suitable category of functors. Then, after defining strong polynomial functors in this context, I will prove that the Long-Moody functor increases by one the degree of strong polynomiality of a strong polynomial functor. Thus, the Long-Moody construction will provide new examples of twisted coefficients entering in the framework of Randal-Williams and Wahl.

In 2016, Randal-Williams and Wahl proved homological stability with certain twisted coefficients for different families of groups, in particular the one of braid groups. In fact, they obtain the stability for coefficients given by functors satisfying polynomial conditions. We only know few examples of such functors. Among them, we have the functor given by the unreduced Burau representations. In 1994, Long and Moody gave a construction on representations of braid groups which associates a representation of Bn with a representation of Bn+1. This construction complexifies in a sense the initial representation: for instance, starting from a dimension one representation, one obtains the unreduced Burau representation. In this talk, I will present this construction from a functorial point of view. I will explain that the construction of Long and Moody defines an endofunctor, called the Long-Moody functor, between a suitable category of functors. Then, after defining strong polynomial functors in this context, I will prove that the Long-Moody functor increases by one the degree of strong polynomiality of a strong polynomial functor. Thus, the Long-Moody construction will provide new examples of twisted coefficients entering in the framework of Randal-Williams and Wahl.

### 2017/03/07

#### Seminar on Probability and Statistics

14:00-15:30 Room #052 (Graduate School of Math. Sci. Bldg.)

Nonparametric change-point analysis of volatility

**Markus Bibinger**(Humboldt-Universität zu Berlin)Nonparametric change-point analysis of volatility

[ Abstract ]

We develop change-point methods for statistics of high-frequency data. The main interest is in the stochastic volatility process of an Itô semi-martingale, the latter being discretely observed over a fixed time horizon. For a local change-point problem under high-frequency asymptotics, we construct a minimax-optimal test to discriminate continuous volatility paths from paths comprising changes. The key example is identification of volatility jumps. We prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we study a different global change-point problem to identify changes in the regularity of the volatility process. In particular, this allows to infer changes in the Hurst parameter of a fractional stochastic volatility process. We establish an asymptotic minimax-optimal test for this problem.

We develop change-point methods for statistics of high-frequency data. The main interest is in the stochastic volatility process of an Itô semi-martingale, the latter being discretely observed over a fixed time horizon. For a local change-point problem under high-frequency asymptotics, we construct a minimax-optimal test to discriminate continuous volatility paths from paths comprising changes. The key example is identification of volatility jumps. We prove weak convergence of the test statistic under the hypothesis to an extreme value distribution. Moreover, we study a different global change-point problem to identify changes in the regularity of the volatility process. In particular, this allows to infer changes in the Hurst parameter of a fractional stochastic volatility process. We establish an asymptotic minimax-optimal test for this problem.

### 2017/03/06

#### Seminar on Geometric Complex Analysis

10:00-11:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Projective and c-projective metric geometries: why they are so similar (ENGLISH)

**Vladimir Matveev**(University of Jena)Projective and c-projective metric geometries: why they are so similar (ENGLISH)

[ Abstract ]

I will show an unexpected application of the standard techniques of integrable systems in projective and c-projective geometry (I will explain what they are and why they were studied). I will show that c-projectively equivalent metrics on a closed manifold generate a commutative isometric $\mathbb{R}^k$-action on the manifold. The quotients of the metrics w.r.t. this action are projectively equivalent, and the initial metrics can be uniquely reconstructed by the quotients. This gives an almost algorithmic way to obtain results in c-projective geometry starting with results in much better developed projective geometry. I will give many application of this algorithmic way including local description, proof of Yano-Obata conjecture for metrics of arbitrary signature, and describe the topology of closed manifolds admitting strictly nonproportional c-projectively equivalent metrics.

Most results are parts of two projects: one is joint with D. Calderbank, M. Eastwood and K. Neusser, and another is joint with A. Bolsinov and S. Rosemann.

I will show an unexpected application of the standard techniques of integrable systems in projective and c-projective geometry (I will explain what they are and why they were studied). I will show that c-projectively equivalent metrics on a closed manifold generate a commutative isometric $\mathbb{R}^k$-action on the manifold. The quotients of the metrics w.r.t. this action are projectively equivalent, and the initial metrics can be uniquely reconstructed by the quotients. This gives an almost algorithmic way to obtain results in c-projective geometry starting with results in much better developed projective geometry. I will give many application of this algorithmic way including local description, proof of Yano-Obata conjecture for metrics of arbitrary signature, and describe the topology of closed manifolds admitting strictly nonproportional c-projectively equivalent metrics.

Most results are parts of two projects: one is joint with D. Calderbank, M. Eastwood and K. Neusser, and another is joint with A. Bolsinov and S. Rosemann.

### 2017/02/24

#### Colloquium of mathematical sciences and society

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

### 2017/02/23

#### FMSP Lectures

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

### 2017/02/20

#### Tuesday Seminar on Topology

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

The Verlinde formula for Higgs bundles (ENGLISH)

**Jørgen Ellegaard Andersen**(Aarhus University)The Verlinde formula for Higgs bundles (ENGLISH)

[ Abstract ]

In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a one parameter family of 2D TQFT's, encoded in a one parameter family of Frobenius algebras, which we will construct.

In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a one parameter family of 2D TQFT's, encoded in a one parameter family of Frobenius algebras, which we will construct.

### 2017/02/16

#### Applied Analysis

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

Diffusive and inviscid traveling wave solution of the Fisher-KPP equation

(ENGLISH)

**Danielle Hilhorst**(CNRS / University of Paris-Sud)Diffusive and inviscid traveling wave solution of the Fisher-KPP equation

(ENGLISH)

[ Abstract ]

Our purpose is to study the limit of traveling wave solutions of the Fisher-KPP equation as the diffusion coefficient tends to zero. More precisely, we consider monotone traveling waves which connect the stable steady state to the unstable one. It is well known that there exists a positive constant c* such that there does not exist any traveling wave solution if c < c* and a unique (up to translation) monotone traveling wave solution of wave speed c for each c > c*.

We consider the corresponding inviscid ordinary differential equation where the diffusion coefficient is equal to zero and show that it possesses a unique traveling wave solution. We then fix c > 0 arbitrary and prove the convergence of the travelling wave of the parabolic equation with velocity c to that of the corresponding traveling wave solution of the inviscid problem.

Further research should involve a similar problem for monostable systems.

This is joint work with Yong Jung Kim.

Our purpose is to study the limit of traveling wave solutions of the Fisher-KPP equation as the diffusion coefficient tends to zero. More precisely, we consider monotone traveling waves which connect the stable steady state to the unstable one. It is well known that there exists a positive constant c* such that there does not exist any traveling wave solution if c < c* and a unique (up to translation) monotone traveling wave solution of wave speed c for each c > c*.

We consider the corresponding inviscid ordinary differential equation where the diffusion coefficient is equal to zero and show that it possesses a unique traveling wave solution. We then fix c > 0 arbitrary and prove the convergence of the travelling wave of the parabolic equation with velocity c to that of the corresponding traveling wave solution of the inviscid problem.

Further research should involve a similar problem for monostable systems.

This is joint work with Yong Jung Kim.

### 2017/02/13

#### Seminar on Geometric Complex Analysis

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

A Characterization of regular points by Ohsawa-Takegoshi Extension Theorem (ENGLISH)

**Qi'an Guan**(Peking University)A Characterization of regular points by Ohsawa-Takegoshi Extension Theorem (ENGLISH)

[ Abstract ]

In this talk, we will present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related Ohsawa-Takegoshi extension theorem holds. We also present a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

This is joint work with Dr. Zhenqian Li.

In this talk, we will present that the germ of a complex analytic set at the origin in $\mathbb{C}^n$ is regular if and only if the related Ohsawa-Takegoshi extension theorem holds. We also present a necessary condition of the $L^2$ extension of bounded holomorphic sections from singular analytic sets.

This is joint work with Dr. Zhenqian Li.

#### Tokyo Probability Seminar

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

Sharp interface limit for stochastically perturbed mass conserving Allen-Cahn equation

**Satoshi Yokoyama**(Graduate school of mathematical sciences, the university of Tokyo)Sharp interface limit for stochastically perturbed mass conserving Allen-Cahn equation

### 2017/02/10

#### Algebraic Geometry Seminar

14:00-15:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Stability theory of a klt singularity II (English)

**Chenyang Xu**(Beijing International Center of Mathematics Research)Stability theory of a klt singularity II (English)

[ Abstract ]

In higher dimensional geometry, it has been known that from many perspectives a log terminal singularity is a local analogue of Fano varieties. Many statements of Fano varieties have a counterpart for log terminal singularities. One central topic on the geometry of a Fano variety is its stability which in particular reflects whether the Fano variety carries a canonical metric. In the talks, we will discuss a series of recent works started by Chi Li, and then by Harold Blum, Yuchen Liu and myself, in which we want to establish an algebro-geometric stability theory of a fixed log terminal singularity. Inspired by the study from differential geometry, (e.g. metric tangent cone, Sasakian-Einstein metric), for any log terminal singularity, we investigate the valuation which has the minimal normalized volume. Our goal is to prove various properties of this valuation which enable us to degenerate the singularity to a K-semistable T-singularity (with a torus action) in the Sasakian-Einstein sense.

In higher dimensional geometry, it has been known that from many perspectives a log terminal singularity is a local analogue of Fano varieties. Many statements of Fano varieties have a counterpart for log terminal singularities. One central topic on the geometry of a Fano variety is its stability which in particular reflects whether the Fano variety carries a canonical metric. In the talks, we will discuss a series of recent works started by Chi Li, and then by Harold Blum, Yuchen Liu and myself, in which we want to establish an algebro-geometric stability theory of a fixed log terminal singularity. Inspired by the study from differential geometry, (e.g. metric tangent cone, Sasakian-Einstein metric), for any log terminal singularity, we investigate the valuation which has the minimal normalized volume. Our goal is to prove various properties of this valuation which enable us to degenerate the singularity to a K-semistable T-singularity (with a torus action) in the Sasakian-Einstein sense.

### 2017/02/09

#### Discrete mathematical modelling seminar

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

Growth of degrees of lattice equations and its signatures over finite fields (ENGLISH)

**Dinh Tran**(University of New South Wales, Sydney, Australia)Growth of degrees of lattice equations and its signatures over finite fields (ENGLISH)

[ Abstract ]

We study growth of degrees of autonomous and non-autonomous lattice equations, some of which are known to be integrable. We present a conjecture that helps us to prove polynomial growth of a certain class of equations including $Q_V$ and its non-autonomous generalization. In addition, we also study growth of degrees of several non-integrable equations. Exponential growth of degrees of these equations is also proved subject to a conjecture. Our technique is to determine the ambient degree growth of the equations and a conjectured growth of their common factors at each vertex, allowing the true degree growth to be found. Moreover, our results can also be used for mappings obtained as periodic reductions of integrable lattice equations. We also study signatures of growth of degrees of lattice equations over finite fields.

We study growth of degrees of autonomous and non-autonomous lattice equations, some of which are known to be integrable. We present a conjecture that helps us to prove polynomial growth of a certain class of equations including $Q_V$ and its non-autonomous generalization. In addition, we also study growth of degrees of several non-integrable equations. Exponential growth of degrees of these equations is also proved subject to a conjecture. Our technique is to determine the ambient degree growth of the equations and a conjectured growth of their common factors at each vertex, allowing the true degree growth to be found. Moreover, our results can also be used for mappings obtained as periodic reductions of integrable lattice equations. We also study signatures of growth of degrees of lattice equations over finite fields.

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