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Seminar information archive
Seminar information archive ～07/01｜Today's seminar 07/02 | Future seminars 07/03～
2015/07/03
Geometry Colloquium
10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Takayuki OKUDA (HIroshima University)
Proper actions of reductive groups on pseudo-Riemannian symmetric spaces and its compact dual. (日本語)
Takayuki OKUDA (HIroshima University)
Proper actions of reductive groups on pseudo-Riemannian symmetric spaces and its compact dual. (日本語)
[ Abstract ]
Let G be a non-compact semisimple Lie group. We take a pair of symmetric pairs (G,H) and (G,L) such that the diagonal action of G on G/H \times G/L is proper. In this talk, we show that by taking ``the compact dual of triple (G,H,L)'', we obtain a compact symmetric space M = U/K and its reflective submanifolds S_1 and S_2 satisfying that the intersection of S_1 and gS_2 is discrete in M for any g in U. In particular, we give a classification of such triples (G,H,L).
Let G be a non-compact semisimple Lie group. We take a pair of symmetric pairs (G,H) and (G,L) such that the diagonal action of G on G/H \times G/L is proper. In this talk, we show that by taking ``the compact dual of triple (G,H,L)'', we obtain a compact symmetric space M = U/K and its reflective submanifolds S_1 and S_2 satisfying that the intersection of S_1 and gS_2 is discrete in M for any g in U. In particular, we give a classification of such triples (G,H,L).
2015/07/01
Operator Algebra Seminars
16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)
Koichi Shimada (Univ. Tokyo)
Approximate unitary equivalence of finite index endomorphisms of the AFD
factors
Koichi Shimada (Univ. Tokyo)
Approximate unitary equivalence of finite index endomorphisms of the AFD
factors
2015/06/30
Lie Groups and Representation Theory
17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)
Anatoly Vershik (St. Petersburg Department of Steklov Institute of Mathematics)
Random subgroups and representation theory
Anatoly Vershik (St. Petersburg Department of Steklov Institute of Mathematics)
Random subgroups and representation theory
[ Abstract ]
The following problem had been appeared independently in different teams and various reason:
to describe the Borel measures on the lattice of all subgroups of given group, which are invariant with respect to the action of the group by conjugacy. The main interest of course represents nonatomic measures which exist not for any group.
I will explain how these measures connected with characters and representations of the group, and describe the complete list of such measures for infinite symmetric group.
The following problem had been appeared independently in different teams and various reason:
to describe the Borel measures on the lattice of all subgroups of given group, which are invariant with respect to the action of the group by conjugacy. The main interest of course represents nonatomic measures which exist not for any group.
I will explain how these measures connected with characters and representations of the group, and describe the complete list of such measures for infinite symmetric group.
Tuesday Seminar on Topology
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Makoto Sakuma (Hiroshima University)
The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)
Makoto Sakuma (Hiroshima University)
The Cannon-Thurston maps and the canonical decompositions of punctured surface bundles over the circle (JAPANESE)
[ Abstract ]
To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,
there are associated two tessellations of the complex plane:
one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,
and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.
In a joint work with Warren Dicks, I had described the relation between these two tessellations.
This result was recently generalized by Francois Gueritaud to punctured surface bundles
with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.
In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.
To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy,
there are associated two tessellations of the complex plane:
one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra,
and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group.
In a joint work with Warren Dicks, I had described the relation between these two tessellations.
This result was recently generalized by Francois Gueritaud to punctured surface bundles
with pseudo-Anosov monodromy where all singuraities of the invariant foliations are at punctures.
In this talk, I will explain Gueritaud's work and related work by Naoki Sakata.
2015/06/29
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yuta Suzuki (Univ. of Tokyo)
Cohomology Formula for Obstructions to Asymptotic Chow semistability (JAPANESE)
Yuta Suzuki (Univ. of Tokyo)
Cohomology Formula for Obstructions to Asymptotic Chow semistability (JAPANESE)
[ Abstract ]
Odaka and Wang proved the intersection formula for the Donaldson-Futaki invariant. We generalize this result for the higher Futaki invariants which are obstructions to asymptotic Chow semistability.
Odaka and Wang proved the intersection formula for the Donaldson-Futaki invariant. We generalize this result for the higher Futaki invariants which are obstructions to asymptotic Chow semistability.
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Manfred Lehn (Mainz/RIMS)
Twisted cubics and cubic fourfolds (English)
Manfred Lehn (Mainz/RIMS)
Twisted cubics and cubic fourfolds (English)
[ Abstract ]
The moduli scheme of generalised twisted cubics on a smooth
cubic fourfold Y non containing a plane is smooth projective of
dimension 10 and admits a contraction to an 8-dimensional
holomorphic symplectic manifold Z(Y). The latter is shown to be
birational to the Hilbert scheme of four points on a K3 surface if
Y is of Pfaffian type. This is a report on joint work with C. Lehn,
C. Sorger and D. van Straten and with N. Addington.
The moduli scheme of generalised twisted cubics on a smooth
cubic fourfold Y non containing a plane is smooth projective of
dimension 10 and admits a contraction to an 8-dimensional
holomorphic symplectic manifold Z(Y). The latter is shown to be
birational to the Hilbert scheme of four points on a K3 surface if
Y is of Pfaffian type. This is a report on joint work with C. Lehn,
C. Sorger and D. van Straten and with N. Addington.
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Kunio Nishioka (Faculty of Commerce, Chuo University)
Kunio Nishioka (Faculty of Commerce, Chuo University)
Numerical Analysis Seminar
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yoshio Komori (Kyushu Institute of Technology)
Stabilized Runge-Kutta methods for the weak approximation of solutions of stochastic differential equations (日本語)
Yoshio Komori (Kyushu Institute of Technology)
Stabilized Runge-Kutta methods for the weak approximation of solutions of stochastic differential equations (日本語)
[ Abstract ]
We are concerned with numerical methods which give weak approximations for stiff It\^{o} stochastic differential equations (SDEs). Implicit methods are one of good candidates to deal with such SDEs. In fact, a well-designed implicit method has been recently proposed by Abdulle and his colleagues [Abdulle et al. 2013a]. On the other hand, it is well known that the numerical solution of stiff SDEs leads to a stepsize reduction when explicit methods are used. However, there are some classes of explicit methods that are well suited to solving some types of stiff SDEs. One such class is the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods [Abdulle et al. 2013b]. SROCK methods reduce to Runge-Kutta Chebyshev methods when applied to ordinary differential equations (ODEs). Another promising class of methods is the class of explicit methods that reduce to explicit exponential Runge-Kutta (RK) methods [Hochbruck et al. 2005, 2010] when applied to semilinear ODEs.
In this talk, we will propose new exponential RK methods which achieve weak order two for multi-dimensional, non-commutative SDEs with a semilinear drift term. We will analytically investigate their stability properties in mean square, and will check their performance in numerical experiments.
(This is a joint work with D. Cohen and K. Burrage.)
We are concerned with numerical methods which give weak approximations for stiff It\^{o} stochastic differential equations (SDEs). Implicit methods are one of good candidates to deal with such SDEs. In fact, a well-designed implicit method has been recently proposed by Abdulle and his colleagues [Abdulle et al. 2013a]. On the other hand, it is well known that the numerical solution of stiff SDEs leads to a stepsize reduction when explicit methods are used. However, there are some classes of explicit methods that are well suited to solving some types of stiff SDEs. One such class is the class of stochastic orthogonal Runge-Kutta Chebyshev (SROCK) methods [Abdulle et al. 2013b]. SROCK methods reduce to Runge-Kutta Chebyshev methods when applied to ordinary differential equations (ODEs). Another promising class of methods is the class of explicit methods that reduce to explicit exponential Runge-Kutta (RK) methods [Hochbruck et al. 2005, 2010] when applied to semilinear ODEs.
In this talk, we will propose new exponential RK methods which achieve weak order two for multi-dimensional, non-commutative SDEs with a semilinear drift term. We will analytically investigate their stability properties in mean square, and will check their performance in numerical experiments.
(This is a joint work with D. Cohen and K. Burrage.)
Operator Algebra Seminars
16:45-18:15 Room #126 (Graduate School of Math. Sci. Bldg.)
Vaughan F. R. Jones (Vanderbilt University)
Block spin renormalization and R. Thompson's groups F and T
Vaughan F. R. Jones (Vanderbilt University)
Block spin renormalization and R. Thompson's groups F and T
2015/06/26
Colloquium
16:50-17:50 Room #056 (Graduate School of Math. Sci. Bldg.)
Kazushi Ueda (Graduate School of Mathematical Sciences, University of Tokyo)
Dimer models and mirror symmetry (JAPANESE)
Kazushi Ueda (Graduate School of Mathematical Sciences, University of Tokyo)
Dimer models and mirror symmetry (JAPANESE)
2015/06/25
Infinite Analysis Seminar Tokyo
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Akane Nakamura (Tokyo University, Graduate School of Mathematical Sciences)
Autonomous limit of the 4-dimensional Painlev¥.FN"e-type equations and degeneration of curves of genus two (JAPANESE)
Akane Nakamura (Tokyo University, Graduate School of Mathematical Sciences)
Autonomous limit of the 4-dimensional Painlev¥.FN"e-type equations and degeneration of curves of genus two (JAPANESE)
[ Abstract ]
The Painlev¥.FN"e equations have been generalized from various aspects. Recently, the 4-dimensional Painlev¥N"e-type equations were classified by corresponding linear equations(Sakai, Kawakami-N.-Sakai, Kawakami). In this talk, I explain an attempt to characterize the 40 types of integrable systems obtained as the autonomous limit of the 4-dimensional Painlev¥N"e-type equations, by inspecting the degenerations of their spectral curves, which are curves of genus two.
The Painlev¥.FN"e equations have been generalized from various aspects. Recently, the 4-dimensional Painlev¥N"e-type equations were classified by corresponding linear equations(Sakai, Kawakami-N.-Sakai, Kawakami). In this talk, I explain an attempt to characterize the 40 types of integrable systems obtained as the autonomous limit of the 4-dimensional Painlev¥N"e-type equations, by inspecting the degenerations of their spectral curves, which are curves of genus two.
2015/06/24
Operator Algebra Seminars
16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)
Matthew Cha (UC Davis)
Gapped ground state phases, topological order and anyons
Matthew Cha (UC Davis)
Gapped ground state phases, topological order and anyons
2015/06/23
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Takahiro Matsushita (The University of Tokyo)
Box complexes and model structures on the category of graphs (JAPANESE)
Takahiro Matsushita (The University of Tokyo)
Box complexes and model structures on the category of graphs (JAPANESE)
[ Abstract ]
To determine the chromatic numbers of graphs, so-called the graph
coloring problem, is one of the most classical problems in graph theory.
Box complex is a Z_2-space associated to a graph, and it is known that
its equivariant homotopy invariant is related to the chromatic number.
Csorba showed that for each finite Z_2-CW-complex X, there is a graph
whose box complex is Z_2-homotopy equivalent to X. From this result, I
expect that the usual model category of Z_2-topological spaces is
Quillen equivalent to a certain model structure on the category of
graphs, whose weak equivalences are graph homomorphisms inducing Z_2-
homotopy equivalences between their box complexes.
In this talk, we introduce model structures on the category of graphs
whose weak equivalences are described as above. We also compare our
model categories of graphs with the category of Z_2-topological spaces.
To determine the chromatic numbers of graphs, so-called the graph
coloring problem, is one of the most classical problems in graph theory.
Box complex is a Z_2-space associated to a graph, and it is known that
its equivariant homotopy invariant is related to the chromatic number.
Csorba showed that for each finite Z_2-CW-complex X, there is a graph
whose box complex is Z_2-homotopy equivalent to X. From this result, I
expect that the usual model category of Z_2-topological spaces is
Quillen equivalent to a certain model structure on the category of
graphs, whose weak equivalences are graph homomorphisms inducing Z_2-
homotopy equivalences between their box complexes.
In this talk, we introduce model structures on the category of graphs
whose weak equivalences are described as above. We also compare our
model categories of graphs with the category of Z_2-topological spaces.
2015/06/22
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Martí Lahoz (Institut de Mathématiques de Jussieu )
Rational cohomology tori
(English)
http://webusers.imj-prg.fr/~marti.lahoz/
Martí Lahoz (Institut de Mathématiques de Jussieu )
Rational cohomology tori
(English)
[ Abstract ]
Complex tori can be topologically characterised among compact Kähler
manifolds by their integral cohomology ring. I will discuss the
structure of compact Kähler manifolds whose rational cohomology ring is
isomorphic to the rational cohomology ring of a torus and give some
examples. This is joint work with Olivier Debarre and Zhi Jiang.
[ Reference URL ]Complex tori can be topologically characterised among compact Kähler
manifolds by their integral cohomology ring. I will discuss the
structure of compact Kähler manifolds whose rational cohomology ring is
isomorphic to the rational cohomology ring of a torus and give some
examples. This is joint work with Olivier Debarre and Zhi Jiang.
http://webusers.imj-prg.fr/~marti.lahoz/
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Susumu Tanabé (Université Galatasaray)
Amoebas and Horn hypergeometric functions
Susumu Tanabé (Université Galatasaray)
Amoebas and Horn hypergeometric functions
[ Abstract ]
Since 10 years, the utility of the Horn hypergeometric functions in Algebraic Geometry has been recognized in a small circle of specialists. The main reason for this interest lies in the fact that every period integral of an affine non-degenerate complete intersection variety can be described as a Horn hypergeometric function (HGF). Therefore the monodromy of the middle dimensional homology can be calculated as the monodromy of an Horn HGF’s.
There is a slight difference between the Gel’fand-Kapranov-Zelevinski HGF’s and the Horn HGF’s. The latter may contain so called “persistent polynomial solutions” that cannot be mapped to GKZ HGF’s via a natural isomorphism between two spaces of HGF’s. In this talk, I will review basic facts on the Horn HGF’s. As a main tool to study the topology of the discriminant loci together with the
analytic aspects of the story, amoebas – image by the log map of the discriminant- will be highlighted.
As an application of this theory the following theorem can be established. For a bivariate Horn HGF system, its monodromy invariant space is always one dimensional if and only if its Ore-Sato polygon is either a zonotope or a Minkowski sum of a triangle and some segments.
This is a collaboration with Timur Sadykov.
Since 10 years, the utility of the Horn hypergeometric functions in Algebraic Geometry has been recognized in a small circle of specialists. The main reason for this interest lies in the fact that every period integral of an affine non-degenerate complete intersection variety can be described as a Horn hypergeometric function (HGF). Therefore the monodromy of the middle dimensional homology can be calculated as the monodromy of an Horn HGF’s.
There is a slight difference between the Gel’fand-Kapranov-Zelevinski HGF’s and the Horn HGF’s. The latter may contain so called “persistent polynomial solutions” that cannot be mapped to GKZ HGF’s via a natural isomorphism between two spaces of HGF’s. In this talk, I will review basic facts on the Horn HGF’s. As a main tool to study the topology of the discriminant loci together with the
analytic aspects of the story, amoebas – image by the log map of the discriminant- will be highlighted.
As an application of this theory the following theorem can be established. For a bivariate Horn HGF system, its monodromy invariant space is always one dimensional if and only if its Ore-Sato polygon is either a zonotope or a Minkowski sum of a triangle and some segments.
This is a collaboration with Timur Sadykov.
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Chikara Nakamura (Research Institute for Mathematical Sciences, Kyoto University)
Lamplighter random walks on fractals
Chikara Nakamura (Research Institute for Mathematical Sciences, Kyoto University)
Lamplighter random walks on fractals
2015/06/17
Operator Algebra Seminars
16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)
Yoh Tanimoto (Univ. Tokyo)
Self-adjointness of bound state operators in integrable quantum field theory
Yoh Tanimoto (Univ. Tokyo)
Self-adjointness of bound state operators in integrable quantum field theory
Number Theory Seminar
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Norifumi Seki (University of Tokyo)
Hodge-Tate weights of p-adic Galois representations and Banach representations of GL_2(Q_p)
(Japanese)
Norifumi Seki (University of Tokyo)
Hodge-Tate weights of p-adic Galois representations and Banach representations of GL_2(Q_p)
(Japanese)
Mathematical Biology Seminar
14:55-16:40 Room #128演習室 (Graduate School of Math. Sci. Bldg.)
Yusuke Kakizoe (Graduate school of systems life sciences, Kyushu University)
A conservation law and time-delay for viral infection dynamics (JAPANESE)
Yusuke Kakizoe (Graduate school of systems life sciences, Kyushu University)
A conservation law and time-delay for viral infection dynamics (JAPANESE)
2015/06/16
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Masaharu Ishikawa (Tohoku University)
Stable maps and branched shadows of 3-manifolds (JAPANESE)
Masaharu Ishikawa (Tohoku University)
Stable maps and branched shadows of 3-manifolds (JAPANESE)
[ Abstract ]
We study what kind of stable map to the real plane a 3-manifold has. It
is known by O. Saeki that there exists a stable map without certain
singular fibers if and only if the 3-manifold is a graph manifold. According to
F. Costantino and D. Thurston, we identify the Stein factorization of a
stable map with a shadow of the 3-manifold under some modification,
where the above singular fibers correspond to the vertices of the shadow. We
define the notion of stable map complexity by counting the number of
such singular fibers and prove that this equals the branched shadow
complexity. With this equality, we give an estimation of the Gromov norm of the
3-manifold by the stable map complexity. This is a joint work with Yuya Koda.
We study what kind of stable map to the real plane a 3-manifold has. It
is known by O. Saeki that there exists a stable map without certain
singular fibers if and only if the 3-manifold is a graph manifold. According to
F. Costantino and D. Thurston, we identify the Stein factorization of a
stable map with a shadow of the 3-manifold under some modification,
where the above singular fibers correspond to the vertices of the shadow. We
define the notion of stable map complexity by counting the number of
such singular fibers and prove that this equals the branched shadow
complexity. With this equality, we give an estimation of the Gromov norm of the
3-manifold by the stable map complexity. This is a joint work with Yuya Koda.
2015/06/15
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Christopher Hacon (University of Utah/RIMS)
Boundedness of the KSBA functor of
SLC models (English)
http://www.math.utah.edu/~hacon/
Christopher Hacon (University of Utah/RIMS)
Boundedness of the KSBA functor of
SLC models (English)
[ Abstract ]
Let $X$ be a canonically polarized smooth $n$-dimensional projective variety over $\mathbb C$ (so that $\omega _X$ is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of $X$ in projective space. It then follows easily that if we fix certain invariants of $X$, then $X$ belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized $n$-dimensional projectivevarieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.This is joint work with C. Xu and J. McKernan
[ Reference URL ]Let $X$ be a canonically polarized smooth $n$-dimensional projective variety over $\mathbb C$ (so that $\omega _X$ is ample), then it is well-known that a fixed multiple of the canonical line bundle defines an embedding of $X$ in projective space. It then follows easily that if we fix certain invariants of $X$, then $X$ belongs to finitely many deformation types. Since canonical models are rarely smooth, it is important to generalize this result to canonically polarized $n$-dimensional projectivevarieties with canonical singularities. Moreover, since these varieties specialize to non-normal varieties it is also important to generalize this result to semi-log canonical pairs. In this talk we will explain a strong version of the above result that applies to semi-log canonical pairs.This is joint work with C. Xu and J. McKernan
http://www.math.utah.edu/~hacon/
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Saotome Takanari
The Lyapunov-Schmidt reduction for the CR Yamabe equation on the Heisenberg group (Japanese)
Saotome Takanari
The Lyapunov-Schmidt reduction for the CR Yamabe equation on the Heisenberg group (Japanese)
[ Abstract ]
We will study CR Yamabe equation for a CR structure on the Heisenberg group which is deformed from the standard structure. By using Lyapunov-Schmidt reduction, it is shown that the perturbation of the standard CR Yamabe solution is a solution to the deformed CR Yamabe equation, under certain conditions of the deformation.
We will study CR Yamabe equation for a CR structure on the Heisenberg group which is deformed from the standard structure. By using Lyapunov-Schmidt reduction, it is shown that the perturbation of the standard CR Yamabe solution is a solution to the deformed CR Yamabe equation, under certain conditions of the deformation.
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Hiroshi Takahashi (College of Science and Technology, Nihon University)
Hiroshi Takahashi (College of Science and Technology, Nihon University)
Numerical Analysis Seminar
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Yuto Miyatake (Nagoya University)
Parallel energy-preserving methods for Hamiltonian systems (日本語)
Yuto Miyatake (Nagoya University)
Parallel energy-preserving methods for Hamiltonian systems (日本語)
2015/06/12
Geometry Colloquium
10:00-11:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Kota Hattori (Keio University)
The nonuniqueness of tangent cone at infinity of Ricci-flat manifolds (Japanese)
Kota Hattori (Keio University)
The nonuniqueness of tangent cone at infinity of Ricci-flat manifolds (Japanese)
[ Abstract ]
For a complete Riemannian manifold (M,g), the Gromov-Hausdorff limit of (M, r^2g) as r to 0 is called the tangent cone at infinity. By the Gromov's Compactness Theorem, there exists tangent cone at infinity for every complete Riemannian manifolds with nonnegative Ricci curvatures. Moreover, if it is Ricci-flat, with Euclidean volume growth and having at least one tangent cone at infinity with a smooth cross section, then it is uniquely determined by the result of Colding and Minicozzi. In this talk I will explain that the assumption of the volume growth is essential for their uniqueness theorem.
For a complete Riemannian manifold (M,g), the Gromov-Hausdorff limit of (M, r^2g) as r to 0 is called the tangent cone at infinity. By the Gromov's Compactness Theorem, there exists tangent cone at infinity for every complete Riemannian manifolds with nonnegative Ricci curvatures. Moreover, if it is Ricci-flat, with Euclidean volume growth and having at least one tangent cone at infinity with a smooth cross section, then it is uniquely determined by the result of Colding and Minicozzi. In this talk I will explain that the assumption of the volume growth is essential for their uniqueness theorem.
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