## Seminar information archive

Seminar information archive ～02/18｜Today's seminar 02/19 | Future seminars 02/20～

### 2011/05/10

#### Numerical Analysis Seminar

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

Conservative finite difference method for the three body problem (JAPANESE)

[ Reference URL ]

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

**Yukitaka Minesaki**(Tokushima Bunri University)Conservative finite difference method for the three body problem (JAPANESE)

[ Reference URL ]

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

#### Tuesday Seminar on Topology

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

Isotated points in the space of group left orderings (JAPANESE)

**Tetsuya Ito**(The University of Tokyo)Isotated points in the space of group left orderings (JAPANESE)

[ Abstract ]

The set of all left orderings of a group G admits a natural

topology. In general the space of left orderings is homeomorphic to the

union of Cantor set and finitely many isolated points. In this talk I

will give a new method to construct left orderings corresponding to

isolated points, and will explain how such isolated orderings reflect

the structures of groups.

The set of all left orderings of a group G admits a natural

topology. In general the space of left orderings is homeomorphic to the

union of Cantor set and finitely many isolated points. In this talk I

will give a new method to construct left orderings corresponding to

isolated points, and will explain how such isolated orderings reflect

the structures of groups.

### 2011/05/09

#### Algebraic Geometry Seminar

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

Fourier--Mukai partners of elliptic ruled surfaces (JAPANESE)

**Hokuto Uehara**(Tokyo Metropolitan University)Fourier--Mukai partners of elliptic ruled surfaces (JAPANESE)

[ Abstract ]

Atiyah classifies vector bundles on elliptic curves E over an algebraically closed field of any characteristic. On the other hand, a rank 2 vector bundle on E defines a surface S with P^1-bundle structure on E.

We study when S has an elliptic fibration according to the Atiyah's classification. As its application, we determines the set of Fourier--Mukai partners of elliptic ruled surfaces over the complex number field.

Atiyah classifies vector bundles on elliptic curves E over an algebraically closed field of any characteristic. On the other hand, a rank 2 vector bundle on E defines a surface S with P^1-bundle structure on E.

We study when S has an elliptic fibration according to the Atiyah's classification. As its application, we determines the set of Fourier--Mukai partners of elliptic ruled surfaces over the complex number field.

#### Seminar on Geometric Complex Analysis

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

Order of meromorphic maps and rationality of the image space (JAPANESE)

**Junjiro Noguchi**(University of Tokyo)Order of meromorphic maps and rationality of the image space (JAPANESE)

### 2011/05/02

#### Algebraic Geometry Seminar

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

Projective varieties admitting an embedding with Gauss map of rank zero (JAPANESE)

**Katsuhisa Furukawa**(Waseda University)Projective varieties admitting an embedding with Gauss map of rank zero (JAPANESE)

[ Abstract ]

I will talk about the study of Gauss map in positivity characteristic which is a joint work with S. Fukasawa and H. Kaji. I will also talk about my resent research of this topic.

We call that a projective variety $X$ satisfies (GMRZ) if there exists an embedding $¥iota: X ¥hookrightarrow ¥mathbb{P}^M$ whose Gauss map $X ¥dashrightarrow G(¥dim(X), ¥mathbb{P}^M)$ is of rank zero at a general point.

We study the case where $X$ has a rational curve $C$. Then, as a fundamental theorem, it follows that the property (GMRZ) makes the splitting type of the normal bundle $N_{C/X}$ very special. We also have a characterization of the Fermat cubic hypersurface in characteristic two in terms of (GMRZ). In this talk, I will also explain the relation of blow-ups and the property (GMRZ).

I will talk about the study of Gauss map in positivity characteristic which is a joint work with S. Fukasawa and H. Kaji. I will also talk about my resent research of this topic.

We call that a projective variety $X$ satisfies (GMRZ) if there exists an embedding $¥iota: X ¥hookrightarrow ¥mathbb{P}^M$ whose Gauss map $X ¥dashrightarrow G(¥dim(X), ¥mathbb{P}^M)$ is of rank zero at a general point.

We study the case where $X$ has a rational curve $C$. Then, as a fundamental theorem, it follows that the property (GMRZ) makes the splitting type of the normal bundle $N_{C/X}$ very special. We also have a characterization of the Fermat cubic hypersurface in characteristic two in terms of (GMRZ). In this talk, I will also explain the relation of blow-ups and the property (GMRZ).

### 2011/04/27

#### Number Theory Seminar

16:30-17:30 Room #056 (Graduate School of Math. Sci. Bldg.)

An analogue of Sturm's theorem for Hilbert modular forms (JAPANESE)

**Yuuki Takai**(University of Tokyo)An analogue of Sturm's theorem for Hilbert modular forms (JAPANESE)

### 2011/04/26

#### Numerical Analysis Seminar

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

Remarks on the discontinuous Galerkin finite element method (JAPANESE)

[ Reference URL ]

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

**Fumio Kikuchi**(Hitotsubashi University)Remarks on the discontinuous Galerkin finite element method (JAPANESE)

[ Reference URL ]

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

#### Lie Groups and Representation Theory

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

Topological Blow-up (JAPANESE)

**Taro YOSHINO**(the University of Tokyo)Topological Blow-up (JAPANESE)

[ Abstract ]

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Rougly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Rougly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

#### Tuesday Seminar on Topology

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

Topological Blow-up (JAPANESE)

**Taro Yoshino**(The University of Tokyo)Topological Blow-up (JAPANESE)

[ Abstract ]

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Roughly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

Suppose that a Lie group $G$ acts on a manifold

$M$. The quotient space $X:=G\\backslash M$ is locally compact,

but not Hausdorff in general. Our aim is to understand

such a non-Hausdorff space $X$.

The space $X$ has the crack $S$. Roughly speaking, $S$ is

the causal subset of non-Hausdorffness of $X$, and especially

$X\\setminus S$ is Hausdorff.

We introduce the concept of `topological blow-up' as a `repair'

of the crack. The `repaired' space $\\tilde{X}$ is

locally compact and Hausdorff space containing $X\\setminus S$

as its open subset. Moreover, the original space $X$ can be

recovered from the pair of $(\\tilde{X}, S)$.

#### Tuesday Seminar of Analysis

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

On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)

**Kiyoomi KATAOKA**(Graduate School of Mathematical Sciences, the University of Tokyo)On the system of fifth-order differential equations which describes surfaces containing six continuous families of circles (JAPANESE)

### 2011/04/25

#### Algebraic Geometry Seminar

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

Mirror symmetry and projective geometry of Reye congruences (JAPANESE)

**Hiromichi Takagi**(University of Tokyo)Mirror symmetry and projective geometry of Reye congruences (JAPANESE)

[ Abstract ]

This is a joint work with Shinobu Hosono.

It is well-known that the projective dual of the second Veronese variety v_2(P^n) is the symmetric determinantal hypersurface H. However, in the context of homological projective duality after Kuznetsov, it is natural to consider that the Chow^2 P^n and H are dual (note that Chow^2 P^n is the secant variety of v_2(P^n)).

Though we did not yet formulate what this duality exactly means in full generality, we show some results in this context for the values n¥leq 4.

For example, let n=4. We consider Chow^2 P^4 in P(S^2 V) and H in P(S^2 V^*), where V is the vector space such that P^4 =P(V). Take a general 4-plane P in

P(S^2 V^*) and let P' be the orthogonal space to P in P(S^2 V). Then X:=Chow^2 P^4 ¥cap P' is a smooth Calabi-Yau 3-fold, and there exists a natural double cover Y -> H¥cap P with a smooth Calabi-Yau 3-fold Y. It is easy to check

that X and Y are not birational each other.

Our main result asserts the derived equivalence of X and Y. This derived equivalence is given by the Fourier Mukai functor D(X)-> D(Y) whose kernel is the ideal sheaf in X×Y of a flat family of curves on Y parameterized by X.

Curves on Y in this family have degree 5 and arithmetic genus 3, and these have a nice interpretation by a BPS number of Y. The proof of the derived equivalence is slightly involved so I explain a similar result in the case where n=3. In this case, we obtain a fully faithful functor from D(X)-> D(Y), where X is a so called the Reye congruence Enriques surface and Y is the 'big resolution' of the Artin-Mumford quartic double solid.

This is a joint work with Shinobu Hosono.

It is well-known that the projective dual of the second Veronese variety v_2(P^n) is the symmetric determinantal hypersurface H. However, in the context of homological projective duality after Kuznetsov, it is natural to consider that the Chow^2 P^n and H are dual (note that Chow^2 P^n is the secant variety of v_2(P^n)).

Though we did not yet formulate what this duality exactly means in full generality, we show some results in this context for the values n¥leq 4.

For example, let n=4. We consider Chow^2 P^4 in P(S^2 V) and H in P(S^2 V^*), where V is the vector space such that P^4 =P(V). Take a general 4-plane P in

P(S^2 V^*) and let P' be the orthogonal space to P in P(S^2 V). Then X:=Chow^2 P^4 ¥cap P' is a smooth Calabi-Yau 3-fold, and there exists a natural double cover Y -> H¥cap P with a smooth Calabi-Yau 3-fold Y. It is easy to check

that X and Y are not birational each other.

Our main result asserts the derived equivalence of X and Y. This derived equivalence is given by the Fourier Mukai functor D(X)-> D(Y) whose kernel is the ideal sheaf in X×Y of a flat family of curves on Y parameterized by X.

Curves on Y in this family have degree 5 and arithmetic genus 3, and these have a nice interpretation by a BPS number of Y. The proof of the derived equivalence is slightly involved so I explain a similar result in the case where n=3. In this case, we obtain a fully faithful functor from D(X)-> D(Y), where X is a so called the Reye congruence Enriques surface and Y is the 'big resolution' of the Artin-Mumford quartic double solid.

#### Seminar on Geometric Complex Analysis

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

On the classification of CR mappings between generalized pseudoellipsoids (JAPANESE)

**Atsushi Hayashimoto**(Nagano National College of Technology)On the classification of CR mappings between generalized pseudoellipsoids (JAPANESE)

### 2011/04/20

#### PDE Real Analysis Seminar

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

Stochastic power law fluids (JAPANESE)

http://www.math.kyoto-u.ac.jp/~nobuo/

**Yoshida, Nobuo**(Department of Mathematics, Kyoto University)Stochastic power law fluids (JAPANESE)

[ Abstract ]

This talk is based in part on a joint work with Yutaka Terasawa.

We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force.

Here, the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force.

We first investigate the existence and the uniqueness of weak solutions to this SPDE.

We next turn to the special case: $p \\in [1 + {d \\over 2},{2d\\overd-2})$,

where $d$ is the dimension of the space. We prove there that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.

[ Reference URL ]This talk is based in part on a joint work with Yutaka Terasawa.

We consider a SPDE (stochastic partial differential equation) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force.

Here, the extra stress tensor of the fluid is given by a polynomial of degree $p-1$ of the rate of strain tensor, while the colored noise is considered as a random force.

We first investigate the existence and the uniqueness of weak solutions to this SPDE.

We next turn to the special case: $p \\in [1 + {d \\over 2},{2d\\overd-2})$,

where $d$ is the dimension of the space. We prove there that the Galerkin scheme approximates the velocity field in a strong sense. As a consequence, we establish the energy equality for the velocity field.

http://www.math.kyoto-u.ac.jp/~nobuo/

### 2011/04/18

#### Algebraic Geometry Seminar

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

Ideal-adic semi-continuity problem for minimal log discrepancies (JAPANESE)

**Masayuki Kawakita**(Research Institute for Mathematical Sciences, Kyoto University)Ideal-adic semi-continuity problem for minimal log discrepancies (JAPANESE)

[ Abstract ]

De Fernex, Ein and Mustaţă, after Kollár, proved the ideal-adic semi-continuity of log canonicity to obtain Shokurov's ACC conjecture for log canonical thresholds on l.c.i. varieties. I discuss its generalisation to minimal log discrepancies, proposed by Mustaţă.

De Fernex, Ein and Mustaţă, after Kollár, proved the ideal-adic semi-continuity of log canonicity to obtain Shokurov's ACC conjecture for log canonical thresholds on l.c.i. varieties. I discuss its generalisation to minimal log discrepancies, proposed by Mustaţă.

#### Seminar on Geometric Complex Analysis

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

Asymptotic cohomology vanishing and a converse of the Andreotti-Grauert theorem on surface (JAPANESE)

**Shinichi Matsumura**(University of Tokyo)Asymptotic cohomology vanishing and a converse of the Andreotti-Grauert theorem on surface (JAPANESE)

### 2011/04/14

#### Operator Algebra Seminars

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

Amenable actions and crossed products of $C^*$-algebras (JAPANESE)

**Masayoshi Matsumura**(Univ. Tokyo)Amenable actions and crossed products of $C^*$-algebras (JAPANESE)

#### Applied Analysis

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

Homoclinic and heteroclinic orbits for a semilinear parabolic equation (ENGLISH)

**Marek FILA**(Comenius University (Slovakia))Homoclinic and heteroclinic orbits for a semilinear parabolic equation (ENGLISH)

[ Abstract ]

We study the existence of connecting orbits for the Fujita equation

u_t=\\Delta u+u^p

with a critical or supercritical exponent $p$. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.

We study the existence of connecting orbits for the Fujita equation

u_t=\\Delta u+u^p

with a critical or supercritical exponent $p$. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and the existence of a homoclinic orbit with respect to zero. This is a joint work with Eiji Yanagida.

### 2011/04/13

#### Functional Analysis Seminar

15:00-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Spectral theory for functions of self-adjoint operators (ENGLISH)

**Alexander Pushnitski**(King's College, London)Spectral theory for functions of self-adjoint operators (ENGLISH)

[ Abstract ]

Let A, B be self-adjoint operators such that the standard assumptions of smooth scattering theory for the pair A, B are satisfied. The spectral theory of the operators of the type f(A)-f(B) will be discussed, with a particular attention to the case of discontinuous functions f. It turns out that the spectrum of f(A)-f(B) can often be explicitly described in terms of the spectrum of the scattering matrix for the pair A,B. This is joint work with D.Yafaev.

Let A, B be self-adjoint operators such that the standard assumptions of smooth scattering theory for the pair A, B are satisfied. The spectral theory of the operators of the type f(A)-f(B) will be discussed, with a particular attention to the case of discontinuous functions f. It turns out that the spectrum of f(A)-f(B) can often be explicitly described in terms of the spectrum of the scattering matrix for the pair A,B. This is joint work with D.Yafaev.

### 2011/04/12

#### Tuesday Seminar on Topology

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

On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere (JAPANESE)

**Susumu Hirose**(Tokyo University of Science)On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere (JAPANESE)

[ Abstract ]

For a closed orientable surface standardly embedded in the 4-sphere,

it was known that a diffeomorphism over this surface is extendable to

the 4-sphere if and only if this diffeomorphism preserves

the Rokhlin quadratic form of this surafce.

In this talk, we will explain an approach to the same kind of problem for

closed non-orientable surfaces.

For a closed orientable surface standardly embedded in the 4-sphere,

it was known that a diffeomorphism over this surface is extendable to

the 4-sphere if and only if this diffeomorphism preserves

the Rokhlin quadratic form of this surafce.

In this talk, we will explain an approach to the same kind of problem for

closed non-orientable surfaces.

### 2011/04/11

#### Seminar on Geometric Complex Analysis

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

Algebraic analysis of resolvents and an exact algorithm for computing Spectral decomposition matrices (JAPANESE)

**Shinichi Tajima**(University of Tsukuba)Algebraic analysis of resolvents and an exact algorithm for computing Spectral decomposition matrices (JAPANESE)

### 2011/03/31

#### Lectures

13:00-14:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Dynamical localization for unitary Anderson models (JAPANESE)

**Alain Joye**(Univ. Grenoble)Dynamical localization for unitary Anderson models (JAPANESE)

#### Lectures

14:30-15:30 Room #118 (Graduate School of Math. Sci. Bldg.)

Stable limits for biased random walks on random trees (JAPANESE)

**Gerard Ben Arous**(Courant Institute, New York Univ.)Stable limits for biased random walks on random trees (JAPANESE)

[ Abstract ]

It is well know that transport in random media can be hampered by dead-end regions and that the velocity can even vanish for strong drifts. We study this phenomenon in great detail for random trees. That is, we study the behavior of biased random walks on supercritical random trees with leaves, in the sub-ballistic regime. When the drift is strong enough it is well known that trapping in the dead-ends of the tree, causes the velocity to vanish. We study the behavior of the walk in this regime, and in particular find the exponents for the mean displacement and the time to reach a given large distance. We also establish a scaling limit result in the case where the drift are random and a non-lattice condition is satisfied. (Joint work with Alexander Fribergh, Alan Hammond, Nina Gantert)

It is well know that transport in random media can be hampered by dead-end regions and that the velocity can even vanish for strong drifts. We study this phenomenon in great detail for random trees. That is, we study the behavior of biased random walks on supercritical random trees with leaves, in the sub-ballistic regime. When the drift is strong enough it is well known that trapping in the dead-ends of the tree, causes the velocity to vanish. We study the behavior of the walk in this regime, and in particular find the exponents for the mean displacement and the time to reach a given large distance. We also establish a scaling limit result in the case where the drift are random and a non-lattice condition is satisfied. (Joint work with Alexander Fribergh, Alan Hammond, Nina Gantert)

### 2011/03/22

#### Lectures

14:00-15:00 Room #128 (Graduate School of Math. Sci. Bldg.)

Potts models and Bethe states on sparse random graphs (JAPANESE)

**Amir Dembo**(Stanford Univ.)Potts models and Bethe states on sparse random graphs (JAPANESE)

[ Abstract ]

Theoretical models of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying mathematical structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph. Focusing on ferromagnetic Potts measures on random finite graphs that converge locally to trees we validate the `cavity' prediction for the limiting free energy per spin and show that local marginals are approximated well by the belief propagation algorithm. This is a concrete example of the more general approximation by Bethe measures, namely, the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on an appropriate infinite random tree (this talk is based on a joint work with Andrea Montanari and Nike Sun).

Theoretical models of disordered materials lead to challenging mathematical problems with applications to random combinatorial problems and coding theory. The underlying mathematical structure is that of many discrete variables that are strongly interacting according to a mean field model determined by a random sparse graph. Focusing on ferromagnetic Potts measures on random finite graphs that converge locally to trees we validate the `cavity' prediction for the limiting free energy per spin and show that local marginals are approximated well by the belief propagation algorithm. This is a concrete example of the more general approximation by Bethe measures, namely, the local convergence of the Boltzmann distribution on the original graph to the Boltzmann distribution on an appropriate infinite random tree (this talk is based on a joint work with Andrea Montanari and Nike Sun).

### 2011/03/08

#### GCOE Seminars

15:00-16:30 Room #128 (Graduate School of Math. Sci. Bldg.)

Diagonal singularities of the scattering matrix and the inverse problem at a fixed energy (ENGLISH)

**Dimitri Yafaev**(Univ. Rennes 1)Diagonal singularities of the scattering matrix and the inverse problem at a fixed energy (ENGLISH)

### 2011/03/04

#### GCOE Seminars

17:00-18:00 Room #370 (Graduate School of Math. Sci. Bldg.)

Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets (ENGLISH)

**Oleg Emanouilov**(Colorado State University)Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets (ENGLISH)

[ Abstract ]

We discuss the inverse boundary value problem of determining the conductivity in two dimensions from the pair of all input Dirichlet data supported on an open subset S1 and all the corresponding Neumann data measured on an open subset S2.

We prove the global uniqueness under some additional geometric condition, in the case where the intersection of S_1 and S_2 has no interior points, and we prove also the uniqueness for a similar inverse problem for the stationary Schr"odinger equation.

The key of the proof isthe construction of appropriate complex geometrical optics solutions using Carleman estimates with a singular weight.

We discuss the inverse boundary value problem of determining the conductivity in two dimensions from the pair of all input Dirichlet data supported on an open subset S1 and all the corresponding Neumann data measured on an open subset S2.

We prove the global uniqueness under some additional geometric condition, in the case where the intersection of S_1 and S_2 has no interior points, and we prove also the uniqueness for a similar inverse problem for the stationary Schr"odinger equation.

The key of the proof isthe construction of appropriate complex geometrical optics solutions using Carleman estimates with a singular weight.

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