## Seminar information archive

Seminar information archive ～02/01｜Today's seminar 02/02 | Future seminars 02/03～

### 2012/12/21

#### GCOE lecture series

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

Evolution equation approach to Fractional Differential Equations (ENGLISH)

**Kazufumi Ito**(North Carolina State Univ.)Evolution equation approach to Fractional Differential Equations (ENGLISH)

[ Abstract ]

A class of fractional differential equations is formulated as an evolution equation on the memory space with non-local boundary condition. Based on such a formulation the mathematical theory of evolution equations is applied to concrete examples of nonlinear fractional PDEs.

A class of fractional differential equations is formulated as an evolution equation on the memory space with non-local boundary condition. Based on such a formulation the mathematical theory of evolution equations is applied to concrete examples of nonlinear fractional PDEs.

### 2012/12/19

#### Operator Algebra Seminars

15:45-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

$W^*$-superrigidity for left-right wreath products (ENGLISH)

A classification of approximate subhomogeneous $C^*$-algebras (ENGLISH)

**Mihaita Berbec**(KU Leuven) 15:45-16:45$W^*$-superrigidity for left-right wreath products (ENGLISH)

**Zhuang Niu**(Univ. Wyoming) 17:00-18:00A classification of approximate subhomogeneous $C^*$-algebras (ENGLISH)

#### Geometry Colloquium

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

Funk metric on Weil-Petersson spaces (JAPANESE)

**FUJIWARA, Koji**(Kyoto University)Funk metric on Weil-Petersson spaces (JAPANESE)

[ Abstract ]

We discuss the Funk function $F(x; y)$ on a Teichmuller space with the Weil-Petersson metric introduced by Yamada, $F(x; y)$ is an asymmetric distance and invariant by the action of the mapping class group. The Funk metric was originally studied for an open convex subset in a Euclidean space by Funk. Its symmetrization is the Hilbert metric.

We discuss the Funk function $F(x; y)$ on a Teichmuller space with the Weil-Petersson metric introduced by Yamada, $F(x; y)$ is an asymmetric distance and invariant by the action of the mapping class group. The Funk metric was originally studied for an open convex subset in a Euclidean space by Funk. Its symmetrization is the Hilbert metric.

#### Number Theory Seminar

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

A generalization of Kato's local epsilon conjecture for

(φ, Γ)-modules over the Robba ring (JAPANESE)

**Kentarou Nakamura**(Hokkaido University)A generalization of Kato's local epsilon conjecture for

(φ, Γ)-modules over the Robba ring (JAPANESE)

[ Abstract ]

In his preprint “Lectures on the approach to Iwasawa theory of Hasse-Weil L-functions via B_dR, Part II ", Kazuya Kato proposed a conjecture called local epsilon conjecture. This conjecture predicts that the determinant of the Galois cohomology of a family of p-adic Galois representations has a canonical base whose specializations at de Rham points can be characterized by using Bloch-Kato exponential, L-factors and Deligne-Langlands epsilon constants of the associated Weil-Deligne representations.

In my talk, I generalize his conjecture for families of (φ, Γ)-modules over the Robba ring, and prove a part of this conjecture in the trianguline case. The two key ingredients are the recent result of Kedlaya-Pottharst-Xiao on the ﬁniteness of cohomologies of (φ, Γ)-modules and my result on Bloch-Kato exponential map for (φ, Γ)-modules.

In his preprint “Lectures on the approach to Iwasawa theory of Hasse-Weil L-functions via B_dR, Part II ", Kazuya Kato proposed a conjecture called local epsilon conjecture. This conjecture predicts that the determinant of the Galois cohomology of a family of p-adic Galois representations has a canonical base whose specializations at de Rham points can be characterized by using Bloch-Kato exponential, L-factors and Deligne-Langlands epsilon constants of the associated Weil-Deligne representations.

In my talk, I generalize his conjecture for families of (φ, Γ)-modules over the Robba ring, and prove a part of this conjecture in the trianguline case. The two key ingredients are the recent result of Kedlaya-Pottharst-Xiao on the ﬁniteness of cohomologies of (φ, Γ)-modules and my result on Bloch-Kato exponential map for (φ, Γ)-modules.

#### GCOE Seminars

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

$W^*$-superrigidity for left-right wreath products (ENGLISH)

**Mihaita Berbec**(KU Leuven)$W^*$-superrigidity for left-right wreath products (ENGLISH)

### 2012/12/18

#### FMSP Lectures

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

On convergence to equilibrium with applications of Lojasiewicz-Simon

inequality (II) (ENGLISH)

**Jie Jiang**(Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences)On convergence to equilibrium with applications of Lojasiewicz-Simon

inequality (II) (ENGLISH)

### 2012/12/17

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

On the existence of strictly effective basis on an arithmetic variety (JAPANESE)

**Hideaki Ikoma**(Kyoto University)On the existence of strictly effective basis on an arithmetic variety (JAPANESE)

[ Abstract ]

I would like to talk about some recent work of mine on the asymptotic behavior of the successive minima associated to a graded arithmetic linear series. A complete arithmetic linear series belonging to a hermitian line bundle on an arithmetic variety is defined as the Z-module of the global sections endowed with the supremum-norm, and the successive minima are invariants that measure the size of the sections with small norms.

If time permits, I would like to also explain some close relationship between the results and the general equi-distribution theory of rational points on an arithmetic variety.

I would like to talk about some recent work of mine on the asymptotic behavior of the successive minima associated to a graded arithmetic linear series. A complete arithmetic linear series belonging to a hermitian line bundle on an arithmetic variety is defined as the Z-module of the global sections endowed with the supremum-norm, and the successive minima are invariants that measure the size of the sections with small norms.

If time permits, I would like to also explain some close relationship between the results and the general equi-distribution theory of rational points on an arithmetic variety.

### 2012/12/15

#### Infinite Analysis Seminar Tokyo

13:30-15:00 Room #117 (Graduate School of Math. Sci. Bldg.)

current and integrability (ENGLISH)

**Vincent Pasquier**(CEA, Saclay, France)current and integrability (ENGLISH)

[ Abstract ]

I will describe some problems related to currents in XXZ chains:

Drude conductivity, Linbladt equation, tasep, matrix ansatz,

in particular the relation of permanent currents with integrability.

If time permits I will also discuss a nonrelated subject:

deformation of fusion rules in minimal models and Macdonald polynomials.

I will describe some problems related to currents in XXZ chains:

Drude conductivity, Linbladt equation, tasep, matrix ansatz,

in particular the relation of permanent currents with integrability.

If time permits I will also discuss a nonrelated subject:

deformation of fusion rules in minimal models and Macdonald polynomials.

### 2012/12/13

#### Algebraic Geometry Seminar

10:40-12:10 Room #118 (Graduate School of Math. Sci. Bldg.)

The asymptotic variety of polynomial maps (ENGLISH)

**Jean-Paul Brasselet**(CNRS (Luminy))The asymptotic variety of polynomial maps (ENGLISH)

[ Abstract ]

The asymptotic variety, or set of non-properness has been intensively studied by Zbigniew Jelonek. In a recent paper, Anna and Guillaume Valette associate to a polynomial map $F: {\\mathbb C}^n \\to {\\mathbb C}^n$ a singular variety $N_F$ and relate properness property of $F$ to the vanishing of some intersection homology groups of $N_F$. I will explain how stratifications of the asymptotic variety of $F$ play an important role in the story and how recently, one of my students, Nguyen Thi Bich Thuy, found a nice way to exhibit such a suitable stratification.

The asymptotic variety, or set of non-properness has been intensively studied by Zbigniew Jelonek. In a recent paper, Anna and Guillaume Valette associate to a polynomial map $F: {\\mathbb C}^n \\to {\\mathbb C}^n$ a singular variety $N_F$ and relate properness property of $F$ to the vanishing of some intersection homology groups of $N_F$. I will explain how stratifications of the asymptotic variety of $F$ play an important role in the story and how recently, one of my students, Nguyen Thi Bich Thuy, found a nice way to exhibit such a suitable stratification.

### 2012/12/12

#### Number Theory Seminar

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

The Tate conjecture for K3 surfaces and holomorphic symplectic varieties over finite fields (ENGLISH)

**François Charles**(CNRS & Université de Rennes 1)The Tate conjecture for K3 surfaces and holomorphic symplectic varieties over finite fields (ENGLISH)

[ Abstract ]

We prove the Tate conjecture for divisors on reductions of holomorphic symplectic varieties over finite fields -- with some restrictions on the characteristic of the base field. We will be concerned mostly with the supersingular case. As a special case, we prove the Tate conjecture, also known as Artin's conjecture in our case, for K3 surfaces over finite fields of characteristic at least 5 and for codimension 2 cycles on cubic fourfolds.

We prove the Tate conjecture for divisors on reductions of holomorphic symplectic varieties over finite fields -- with some restrictions on the characteristic of the base field. We will be concerned mostly with the supersingular case. As a special case, we prove the Tate conjecture, also known as Artin's conjecture in our case, for K3 surfaces over finite fields of characteristic at least 5 and for codimension 2 cycles on cubic fourfolds.

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Large subalgebras of crossed product $C^*$-algebras (ENGLISH)

**N. Christopher Phillips**(Univ. Oregon)Large subalgebras of crossed product $C^*$-algebras (ENGLISH)

#### FMSP Lectures

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Large subalgebras of crossed product C*-algebras (ENGLISH)

**N. Christopher Phillips**(Univ. Oregon)Large subalgebras of crossed product C*-algebras (ENGLISH)

[ Abstract ]

This is work in progress; not everything has been checked.

We define a "large subalgebra" and a "centrally large subalgebra" of a C*-algebra. The motivating example is what we now call the "orbit breaking subalgebra" of the crossed product by a minimal homeomorphism h of a compact metric space X. Let v be the standard unitary in the crossed product C* (Z, X, h). For a closed subset Y of X, we form the subalgebra of C* (Z, X, h) generated by C (X) and all elements f v for f in C (X) such that f vanishes on Y. When each orbit meets Y at most once, this subalgebra is centrally large in the crossed product. Crossed products by smooth free minimal actions of Zd also contain centrally large subalgebras which are simple direct limits, with no dimension growth, of recursive subhomogeneous algebras.

If B is a large subalgebra of A, then the Cuntz semigroups of A and B are the almost the same: if one deletes the classes of nonzero projections, then the inclusion is a bijection on what is left. Also (joint work with Dawn Archey), if B is a centrally large subalgebra of A, and B has stable rank one, then so does A. Moreover, if B is a centrally large subalgebra of A, if B is Z-stable, and if A is nuclear, then A is Z-stable.

This is work in progress; not everything has been checked.

We define a "large subalgebra" and a "centrally large subalgebra" of a C*-algebra. The motivating example is what we now call the "orbit breaking subalgebra" of the crossed product by a minimal homeomorphism h of a compact metric space X. Let v be the standard unitary in the crossed product C* (Z, X, h). For a closed subset Y of X, we form the subalgebra of C* (Z, X, h) generated by C (X) and all elements f v for f in C (X) such that f vanishes on Y. When each orbit meets Y at most once, this subalgebra is centrally large in the crossed product. Crossed products by smooth free minimal actions of Zd also contain centrally large subalgebras which are simple direct limits, with no dimension growth, of recursive subhomogeneous algebras.

If B is a large subalgebra of A, then the Cuntz semigroups of A and B are the almost the same: if one deletes the classes of nonzero projections, then the inclusion is a bijection on what is left. Also (joint work with Dawn Archey), if B is a centrally large subalgebra of A, and B has stable rank one, then so does A. Moreover, if B is a centrally large subalgebra of A, if B is Z-stable, and if A is nuclear, then A is Z-stable.

### 2012/12/11

#### Tuesday Seminar on Topology

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

Homotopy-theoretic methods in the study of spaces of knots and links (ENGLISH)

**Ismar Volic**(Wellesley College)Homotopy-theoretic methods in the study of spaces of knots and links (ENGLISH)

[ Abstract ]

I will survey the ways in which some homotopy-theoretic

methods, manifold calculus of functors main among them, have in recent

years been used for extracting information about the topology of

spaces of knots and links. Cosimplicial spaces and operads will also

be featured. I will end with some recent results about spaces of

homotopy string links and in particular about how one can use functor

calculus in combination with configuration space integrals to extract

information about Milnor invariants.

I will survey the ways in which some homotopy-theoretic

methods, manifold calculus of functors main among them, have in recent

years been used for extracting information about the topology of

spaces of knots and links. Cosimplicial spaces and operads will also

be featured. I will end with some recent results about spaces of

homotopy string links and in particular about how one can use functor

calculus in combination with configuration space integrals to extract

information about Milnor invariants.

#### Tuesday Seminar of Analysis

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

This talk was cancelled! (JAPANESE)

**Rafe Mazzeo**(Stanford University)This talk was cancelled! (JAPANESE)

#### FMSP Lectures

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

On convergence to equilibrium with applications of Lojasiewicz-Simon

inequality (I) (ENGLISH)

**Jie Jiang**(Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences)On convergence to equilibrium with applications of Lojasiewicz-Simon

inequality (I) (ENGLISH)

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants (JAPANESE)

**Tatsuyuki Hikita**(Kyoto University)Affine Springer fibers of type A and combinatorics of diagonal

coinvariants

Affine Springer fibers of type A and combinatorics of diagonal

coinvariants (JAPANESE)

[ Abstract ]

We introduce certain filtrations on the homology of

affine Springer fibers of type A and give combinatorial formulas for the bigraded Frobenius series of the associated graded modules.

The results are essentially given by generalizations of the symmetric function introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov which is conjectured to coincide with the bigraded Frobenius series of the ring of diagonal coinvariants.

We introduce certain filtrations on the homology of

affine Springer fibers of type A and give combinatorial formulas for the bigraded Frobenius series of the associated graded modules.

The results are essentially given by generalizations of the symmetric function introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov which is conjectured to coincide with the bigraded Frobenius series of the ring of diagonal coinvariants.

### 2012/12/10

#### Algebraic Geometry Seminar

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

A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

**Kotaro Kawatani**(Nagoya University)A hyperbolic metric and stability conditions on K3 surfaces with $¥rho=1$ (JAPANESE)

[ Abstract ]

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

We introduce a hyperbolic metric on the (normalized) space of stability conditions on projective K3 surfaces $X$ with Picard rank 1. Furthermore we demonstrate how this hyperbolic metric is helpful for us by discussing two or three topics.

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

A Dirichlet space on ends of tree and Dirichlet forms with a nodewise orthogonal property (JAPANESE)

**Hiroshi Kaneko**(Tokyo University of Science)A Dirichlet space on ends of tree and Dirichlet forms with a nodewise orthogonal property (JAPANESE)

### 2012/12/07

#### Seminar on Probability and Statistics

14:50-16:00 Room #006 (Graduate School of Math. Sci. Bldg.)

Conditional Independence and Linear Algebra (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/12.html

**TANAKA, Kentaro**(Tokyo Institute of Technology)Conditional Independence and Linear Algebra (JAPANESE)

[ Reference URL ]

http://www.sigmath.es.osaka-u.ac.jp/~kamatani/statseminar/2012/12.html

### 2012/12/05

#### PDE Real Analysis Seminar

10:30-11:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Convergence to Equilibrium of Bounded Solutions with Application of Lojasiewicz-Simon's inequality (ENGLISH)

**Jie Jiang**(Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences)Convergence to Equilibrium of Bounded Solutions with Application of Lojasiewicz-Simon's inequality (ENGLISH)

[ Abstract ]

In this talk, we present the application of Lojasiewicz-Simon's inequality to the study on convergence of bounded global solutions to some evolution equations. We take a semi-linear parabolic initial-boundary problem as an example. With the help of Lojasiewicz-Simon's inequality we prove that the bounded global solution will converge to an equilibrium as time goes to infinity provided the nonlinear term is analytic in the unknown function. We also present the application of Lojasiewicz-Simon's inequality to the asymptotic behavior studies on phase-field models with Cattaneo law and chemotaxis models with volume-filling effect.

In this talk, we present the application of Lojasiewicz-Simon's inequality to the study on convergence of bounded global solutions to some evolution equations. We take a semi-linear parabolic initial-boundary problem as an example. With the help of Lojasiewicz-Simon's inequality we prove that the bounded global solution will converge to an equilibrium as time goes to infinity provided the nonlinear term is analytic in the unknown function. We also present the application of Lojasiewicz-Simon's inequality to the asymptotic behavior studies on phase-field models with Cattaneo law and chemotaxis models with volume-filling effect.

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Construction of two dimensional QFT through Longo-Witten endomorphisms (JAPANESE)

**Yoh Tanimoto**(Univ. Goettingen)Construction of two dimensional QFT through Longo-Witten endomorphisms (JAPANESE)

#### Geometry Colloquium

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

Maximal torus actions on complex manifolds (JAPANESE)

**Hiroaki Ishida**(Osaka City University Advanced Mathematical Institute)Maximal torus actions on complex manifolds (JAPANESE)

[ Abstract ]

We say that an effective action of a compact torus $T$ on a connected manifold $M$ is maximal if there is an orbit of dimension $2\\dim T-\\dim M$. In this talk, we give a one-to-one correspondence between the family of connected closed complex manifolds with maximal torus actions and the family of certain combinatorial objects, which is a generalization of the correspondence between complete nonsingular toric varieties and nonsingular complete fans. As an application, we construct a lot of concrete examples of non-K\\"{a}hler manifolds with maximal torus actions.

We say that an effective action of a compact torus $T$ on a connected manifold $M$ is maximal if there is an orbit of dimension $2\\dim T-\\dim M$. In this talk, we give a one-to-one correspondence between the family of connected closed complex manifolds with maximal torus actions and the family of certain combinatorial objects, which is a generalization of the correspondence between complete nonsingular toric varieties and nonsingular complete fans. As an application, we construct a lot of concrete examples of non-K\\"{a}hler manifolds with maximal torus actions.

#### Classical Analysis

16:00-17:30 Room #270 (Graduate School of Math. Sci. Bldg.)

On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)

**Andrei Kapaev**(SISSA, Trieste, Italy)On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)

[ Abstract ]

Equation $P_I^2$ is the second member in the hierarchy of ODEs associated with the classical Painlev\\’e first equation $P_I$ and can be solved via the Riemann-Hilbert (RH) problem approach. It is known also that solutions of equation $P_I^2$ as the functions of $x$ depending on the parameter $t$ can be used to construct a 4-parameter family of isomonodromic solutions to the KdV equation. Given the monodromy data, the set of points $(x,t)$, where the above mentioned RH problem is not solvable, is called the Malgrange divisor. The function $x=a(t)$, which parametrizes locally the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair representation and can be also studied using an RH problem. We discuss the relations between these two kinds of the RH problems and the properties of their $t$-large genus 1 asymptotic solutions.

Equation $P_I^2$ is the second member in the hierarchy of ODEs associated with the classical Painlev\\’e first equation $P_I$ and can be solved via the Riemann-Hilbert (RH) problem approach. It is known also that solutions of equation $P_I^2$ as the functions of $x$ depending on the parameter $t$ can be used to construct a 4-parameter family of isomonodromic solutions to the KdV equation. Given the monodromy data, the set of points $(x,t)$, where the above mentioned RH problem is not solvable, is called the Malgrange divisor. The function $x=a(t)$, which parametrizes locally the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair representation and can be also studied using an RH problem. We discuss the relations between these two kinds of the RH problems and the properties of their $t$-large genus 1 asymptotic solutions.

### 2012/12/04

#### Numerical Analysis Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Tsunami simulation of Hakata Bay using the viscous shallow-water equations (JAPANESE)

http://www.infsup.jp/utnas/

**Hiroshi Kanayama**(Kyushu University)Tsunami simulation of Hakata Bay using the viscous shallow-water equations (JAPANESE)

[ Abstract ]

The tsunami caused by the great East Japan earthquake gave serious damage in the coastal areas of the Tohoku district. Numerical simulation is used for damage prediction as disaster measures to these tsunami hazards. Generally in the numerical simulation about the tsunami propagation to the coast from an open sea, shallow-water equations are used. This research focuses on viscous shallow-water equations and attempts to generate a computational method using finite element techniques based on the previous investigations of Kanayama and Ohtsuka (1978). First, the viscous shallow-water equation system is derived from the Navier-Stokes equations, based on the assumption of hydrostatic pressure in the direction of gravity. Next the numerical scheme is shown. Then, tsunami simulations of Hakata Bay and Tohoku-Oki are shown using the approach. Finally, a stability condition in L2 sense for the numerical scheme of a linearized viscous shallow-water problem is introduced from Kanayama and Ushijima (1988-1989) and its actual effectiveness is discussed from the view point of practical computation. This presentation will be done in Japanese.

[ Reference URL ]The tsunami caused by the great East Japan earthquake gave serious damage in the coastal areas of the Tohoku district. Numerical simulation is used for damage prediction as disaster measures to these tsunami hazards. Generally in the numerical simulation about the tsunami propagation to the coast from an open sea, shallow-water equations are used. This research focuses on viscous shallow-water equations and attempts to generate a computational method using finite element techniques based on the previous investigations of Kanayama and Ohtsuka (1978). First, the viscous shallow-water equation system is derived from the Navier-Stokes equations, based on the assumption of hydrostatic pressure in the direction of gravity. Next the numerical scheme is shown. Then, tsunami simulations of Hakata Bay and Tohoku-Oki are shown using the approach. Finally, a stability condition in L2 sense for the numerical scheme of a linearized viscous shallow-water problem is introduced from Kanayama and Ushijima (1988-1989) and its actual effectiveness is discussed from the view point of practical computation. This presentation will be done in Japanese.

http://www.infsup.jp/utnas/

#### Tuesday Seminar on Topology

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

Conformal field theory for C2-cofinite vertex algebras (JAPANESE)

**Yoshitake Hashimoto**(Tokyo City University)Conformal field theory for C2-cofinite vertex algebras (JAPANESE)

[ Abstract ]

This is a jount work with Akihiro Tsuchiya (Kavli IPMU).

We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces

for a vertex algebra of which the category of modules is not necessarily semi-simple.

We assume the C2-cofiniteness condition for vertex algebras.

We define "tensor product" of two modules over a C2-cofinite vertex algebra.

This is a jount work with Akihiro Tsuchiya (Kavli IPMU).

We consider sheaves of covacua and conformal blocks over parameter spaces of n-pointed Riemann surfaces

for a vertex algebra of which the category of modules is not necessarily semi-simple.

We assume the C2-cofiniteness condition for vertex algebras.

We define "tensor product" of two modules over a C2-cofinite vertex algebra.

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