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Seminar information archive
Seminar information archive ～04/24｜Today's seminar 04/25 | Future seminars 04/26～
Mathematical Biology Seminar
14:55-16:40 Room #128演習室 (Graduate School of Math. Sci. Bldg.)
Shigehide Iwata (The graduate school of marine science and technology, Tokyo University of Marine Science and Technology)
Population dynamics of fish stock with migration and its management strategy
Shigehide Iwata (The graduate school of marine science and technology, Tokyo University of Marine Science and Technology)
Population dynamics of fish stock with migration and its management strategy
2015/06/01
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Daizo Ishikawa (Waseda University)
Rank 2 weak Fano bundles on cubic 3-folds (日本語)
Daizo Ishikawa (Waseda University)
Rank 2 weak Fano bundles on cubic 3-folds (日本語)
[ Abstract ]
A vector bundle on a projective variety is called weak Fano if its
projectivization is a weak Fano manifold. This is a generalization of
Fano bundles.
In this talk, we will obtain a classification of rank 2 weak Fano
bundles on a nonsingular cubic hypersurface in a projective 4-space.
Specifically, we will show that there exist rank 2 indecomposable weak
Fano bundles on it.
A vector bundle on a projective variety is called weak Fano if its
projectivization is a weak Fano manifold. This is a generalization of
Fano bundles.
In this talk, we will obtain a classification of rank 2 weak Fano
bundles on a nonsingular cubic hypersurface in a projective 4-space.
Specifically, we will show that there exist rank 2 indecomposable weak
Fano bundles on it.
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Masato Hoshino (Graduate School of Mathematical Sciences, The University of Tokyo)
Masato Hoshino (Graduate School of Mathematical Sciences, The University of Tokyo)
2015/05/28
Infinite Analysis Seminar Tokyo
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Yuki Arano (Graduate School of Mathematical Sciences, the University of Tokyo)
Unitary spherical representations of Drinfeld doubles (JAPANESE)
Yuki Arano (Graduate School of Mathematical Sciences, the University of Tokyo)
Unitary spherical representations of Drinfeld doubles (JAPANESE)
[ Abstract ]
It is known that the Drinfeld double of the quantized
enveloping algebra of a semisimple Lie algebra looks similar to the
quantized enveloping algebra of the complexification of the Lie algebra.
In this talk, we investigate the unitary representation theory of such
Drinfeld double via its analogy to that of the complex Lie group.
We also talk on an application to operator algebras.
It is known that the Drinfeld double of the quantized
enveloping algebra of a semisimple Lie algebra looks similar to the
quantized enveloping algebra of the complexification of the Lie algebra.
In this talk, we investigate the unitary representation theory of such
Drinfeld double via its analogy to that of the complex Lie group.
We also talk on an application to operator algebras.
2015/05/27
Operator Algebra Seminars
16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)
John F. R. Duncan (Case Western Reserve Univ.)
Vertex operator algebras in umbral Moonshine
John F. R. Duncan (Case Western Reserve Univ.)
Vertex operator algebras in umbral Moonshine
Number Theory Seminar
17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Ippei Nagamachi (University of Tokyo)
On a good reduction criterion for polycurves with sections (Japanese)
Ippei Nagamachi (University of Tokyo)
On a good reduction criterion for polycurves with sections (Japanese)
2015/05/26
Lie Groups and Representation Theory
17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)
Takeyoshi Kogiso (Josai University)
Local functional equations of Clifford quartic forms and homaloidal EKP-polynomials
Takeyoshi Kogiso (Josai University)
Local functional equations of Clifford quartic forms and homaloidal EKP-polynomials
[ Abstract ]
It is known that one can associate local functional equation to the irreducible relative invariant of an irreducible regular prehomogeneous vector spaces. We construct Clifford quartic forms that cannot obtained from prehomogeneous vector spaces, but, for which one can associate local functional equations. The characterization of polynomials which satisfy local functional equations is an interesting problem. In relation to this characterization problem (in a more general form), Etingof, Kazhdan and Polishchuk raised a conjecture. We make a counter example of this conjecture from Clifford quartic forms. (This is based on the joint work with F.Sato)
It is known that one can associate local functional equation to the irreducible relative invariant of an irreducible regular prehomogeneous vector spaces. We construct Clifford quartic forms that cannot obtained from prehomogeneous vector spaces, but, for which one can associate local functional equations. The characterization of polynomials which satisfy local functional equations is an interesting problem. In relation to this characterization problem (in a more general form), Etingof, Kazhdan and Polishchuk raised a conjecture. We make a counter example of this conjecture from Clifford quartic forms. (This is based on the joint work with F.Sato)
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Ken'ichi Kuga (Chiba University)
Introduction to formalization of topology using a proof assistant. (JAPANESE)
Ken'ichi Kuga (Chiba University)
Introduction to formalization of topology using a proof assistant. (JAPANESE)
[ Abstract ]
Although the program of formalization goes back to David
Hilbert, it is only recently that we can actually formalize
substantial theorems in modern mathematics. It is made possible by the
development of certain type theory and a computer software called a
proof assistant. We begin this talk by showing our formalization of
some basic geometric topology using a proof assistant COQ. Then we
introduce homotopy type theory (HoTT) of Voevodsky et al., which
interprets type theory from abstract homotopy theoretic perspective.
HoTT proposes "univalent" foundation of mathematics which is
particularly suited for computer formalization.
Although the program of formalization goes back to David
Hilbert, it is only recently that we can actually formalize
substantial theorems in modern mathematics. It is made possible by the
development of certain type theory and a computer software called a
proof assistant. We begin this talk by showing our formalization of
some basic geometric topology using a proof assistant COQ. Then we
introduce homotopy type theory (HoTT) of Voevodsky et al., which
interprets type theory from abstract homotopy theoretic perspective.
HoTT proposes "univalent" foundation of mathematics which is
particularly suited for computer formalization.
2015/05/25
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Tomoyuki Hisamoto (Nagoya Univ.)
On uniform K-stability (Japanese)
Tomoyuki Hisamoto (Nagoya Univ.)
On uniform K-stability (Japanese)
[ Abstract ]
It is a joint work with Sébastien Boucksom and Mattias Jonsson. We first introduce functionals on the space of test configurations, as non-Archimedean analogues of classical functionals on the space of Kähler metrics. Then, uniform K-stability is defined as a counterpart of K-energy's coercivity condition. Finally, reproving and strengthening Y. Odaka's results, we study uniform K-stability of Kähler-Einstein manifolds.
It is a joint work with Sébastien Boucksom and Mattias Jonsson. We first introduce functionals on the space of test configurations, as non-Archimedean analogues of classical functionals on the space of Kähler metrics. Then, uniform K-stability is defined as a counterpart of K-energy's coercivity condition. Finally, reproving and strengthening Y. Odaka's results, we study uniform K-stability of Kähler-Einstein manifolds.
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Yuya Matsumoto (University of Tokyo)
Good reduction of K3 surfaces (日本語 or English)
https://www.ms.u-tokyo.ac.jp/~ymatsu/index_j.html
Yuya Matsumoto (University of Tokyo)
Good reduction of K3 surfaces (日本語 or English)
[ Abstract ]
We consider degeneration of K3 surfaces over a 1-dimensional base scheme
of mixed characteristic (e.g. Spec of the p-adic integers).
Under the assumption of potential semistable reduction, we first prove
that a trivial monodromy action on the l-adic etale cohomology group
implies potential good reduction, where potential means that we allow a
finite base extension.
Moreover we show that a finite etale base change suffices.
The proof for the first part involves a mixed characteristic
3-dimensional MMP (Kawamata) and the classification of semistable
degeneration of K3 surfaces (Kulikov, Persson--Pinkham, Nakkajima).
For the second part, we consider flops and descent arguments. This is a joint work with Christian Liedtke.
[ Reference URL ]We consider degeneration of K3 surfaces over a 1-dimensional base scheme
of mixed characteristic (e.g. Spec of the p-adic integers).
Under the assumption of potential semistable reduction, we first prove
that a trivial monodromy action on the l-adic etale cohomology group
implies potential good reduction, where potential means that we allow a
finite base extension.
Moreover we show that a finite etale base change suffices.
The proof for the first part involves a mixed characteristic
3-dimensional MMP (Kawamata) and the classification of semistable
degeneration of K3 surfaces (Kulikov, Persson--Pinkham, Nakkajima).
For the second part, we consider flops and descent arguments. This is a joint work with Christian Liedtke.
https://www.ms.u-tokyo.ac.jp/~ymatsu/index_j.html
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Yu Kitabeppu (Graduate School of Sciences, Kyoto University)
A finite diameter theorem on RCD spaces
Yu Kitabeppu (Graduate School of Sciences, Kyoto University)
A finite diameter theorem on RCD spaces
2015/05/21
Lectures
16:00-17:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Gunnar Carlsson (Stanford University, Ayasdi INC)
The Shape of Data
(ENGLISH)
http://faculty.ms.u-tokyo.ac.jp/Carlsson.html
Gunnar Carlsson (Stanford University, Ayasdi INC)
The Shape of Data
(ENGLISH)
[ Abstract ]
There is a tremendous amount of attention being paid to the notion of
"Big Data". In many situations, however, the problem is not so much the
size of the data but rather its complexity. This observation shows that
it is now important to find methods for representing complex data in a
compressed and understandable fashion. Representing data by shapes
turns out to be useful in many situations, and therefore topology, the
mathematical sub discipline which studies shape, becomes quite
relevant. There is now a collection of methods based on topology for
analyzing complex data, and in this talk we will discuss these methods,
with numerous examples.
[ Reference URL ]There is a tremendous amount of attention being paid to the notion of
"Big Data". In many situations, however, the problem is not so much the
size of the data but rather its complexity. This observation shows that
it is now important to find methods for representing complex data in a
compressed and understandable fashion. Representing data by shapes
turns out to be useful in many situations, and therefore topology, the
mathematical sub discipline which studies shape, becomes quite
relevant. There is now a collection of methods based on topology for
analyzing complex data, and in this talk we will discuss these methods,
with numerous examples.
http://faculty.ms.u-tokyo.ac.jp/Carlsson.html
2015/05/20
Number Theory Seminar
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Shou-Wu Zhang (Princeton University)
Colmez' conjecture in average (English)
Shou-Wu Zhang (Princeton University)
Colmez' conjecture in average (English)
[ Abstract ]
This is a report on a joint work with Xinyi Yuan on a conjectured formula of Colmez about the Faltings heights of CM abelian varieties. I will sketch a deduction of this formula in average of CM types from our early work on Gross-Zagier formula. When combined with a recent work of Tsimerman, this result implies the Andre-Oort conjecture for the moduli of abelian varieties.
Our method is different than a recently announced proof of a weaker form of the average formula by Andreatta, Howard, Goren, and Madapusi Pera: we use neither high dimensional Shimura varieties nor Borcherds' liftings.
This is a report on a joint work with Xinyi Yuan on a conjectured formula of Colmez about the Faltings heights of CM abelian varieties. I will sketch a deduction of this formula in average of CM types from our early work on Gross-Zagier formula. When combined with a recent work of Tsimerman, this result implies the Andre-Oort conjecture for the moduli of abelian varieties.
Our method is different than a recently announced proof of a weaker form of the average formula by Andreatta, Howard, Goren, and Madapusi Pera: we use neither high dimensional Shimura varieties nor Borcherds' liftings.
Operator Algebra Seminars
16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)
Yusuke Isono (RIMS, Kyoto Univ.)
Unique prime factorization and bicentralizer problem for a class of type III factors
Yusuke Isono (RIMS, Kyoto Univ.)
Unique prime factorization and bicentralizer problem for a class of type III factors
2015/05/19
PDE Real Analysis Seminar
10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Sumio Yamada (Gakushuin University)
Convex bodies and geometry of some associated Minkowski functionals (日本語)
Sumio Yamada (Gakushuin University)
Convex bodies and geometry of some associated Minkowski functionals (日本語)
[ Abstract ]
In this talk, we will investigate the construction of so-called Hilbert metric, as well as Funk metric, defined on convex set from a new variational viewpoint. The local and global aspects of the geometry of the resulting Minkowski functionals will be contrasted. As an application, some remarks on the Perron-Frobenius theorem will be made. Part of the project is a joint work with Athanase Papadopoulos (Strasbourg).
In this talk, we will investigate the construction of so-called Hilbert metric, as well as Funk metric, defined on convex set from a new variational viewpoint. The local and global aspects of the geometry of the resulting Minkowski functionals will be contrasted. As an application, some remarks on the Perron-Frobenius theorem will be made. Part of the project is a joint work with Athanase Papadopoulos (Strasbourg).
Lie Groups and Representation Theory
17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)
Anton Evseev (University of Birmingham)
RoCK blocks, wreath products and KLR algebras (English)
Anton Evseev (University of Birmingham)
RoCK blocks, wreath products and KLR algebras (English)
[ Abstract ]
The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses an isomorphism between a Hecke algebra at a root of unity and a cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the Hecke algebra and the ideas behind a construction of R-matrices for modules over KLR algebras due to Kang-Kashiwara-Kim.
The so-called RoCK (or Rouquier) blocks play an important role in representation theory of symmetric groups over a finite field of characteristic $p$, as well as of Hecke algebras at roots of unity. Turner has conjectured that a certain idempotent truncation of a RoCK block is Morita equivalent to the principal block $B_0$ of the wreath product $S_p\wr S_d$ of symmetric groups, where $d$ is the "weight" of the block. The talk will outline a proof of this conjecture, which generalizes a result of Chuang-Kessar proved for $d < p$. The proof uses an isomorphism between a Hecke algebra at a root of unity and a cyclotomic Khovanov-Lauda-Rouquier algebra, the resulting grading on the Hecke algebra and the ideas behind a construction of R-matrices for modules over KLR algebras due to Kang-Kashiwara-Kim.
Tuesday Seminar on Topology
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Akishi Kato (The University of Tokyo)
Quiver mutation loops and partition q-series (JAPANESE)
Akishi Kato (The University of Tokyo)
Quiver mutation loops and partition q-series (JAPANESE)
[ Abstract ]
Quivers and their mutations are ubiquitous in mathematics and
mathematical physics; they play a key role in cluster algebras,
wall-crossing phenomena, gluing of ideal tetrahedra, etc.
Recently, we introduced a partition q-series for a quiver mutation loop
(a loop in a quiver exchange graph) using the idea of state sum of statistical
mechanics. The partition q-series enjoy some nice properties such
as pentagon move invariance. We also discuss their relation with combinatorial
Donaldson-Thomas invariants, as well as fermionic character formulas of
certain conformal field theories.
This is a joint work with Yuji Terashima.
Quivers and their mutations are ubiquitous in mathematics and
mathematical physics; they play a key role in cluster algebras,
wall-crossing phenomena, gluing of ideal tetrahedra, etc.
Recently, we introduced a partition q-series for a quiver mutation loop
(a loop in a quiver exchange graph) using the idea of state sum of statistical
mechanics. The partition q-series enjoy some nice properties such
as pentagon move invariance. We also discuss their relation with combinatorial
Donaldson-Thomas invariants, as well as fermionic character formulas of
certain conformal field theories.
This is a joint work with Yuji Terashima.
2015/05/18
Seminar on Geometric Complex Analysis
10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Masanori Adachi (Tokyo Univ. of Science)
On a global estimate of the Diederich–Fornaess index of Levi-flat real hypersurfaces (Japanese)
Masanori Adachi (Tokyo Univ. of Science)
On a global estimate of the Diederich–Fornaess index of Levi-flat real hypersurfaces (Japanese)
[ Abstract ]
We give yet another proof for a global estimate of the Diederich-Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. Although the Diederich-Fornaess index is originally defined for relatively compact domains in complex manifolds, our formulation reveals that it makes sense for abstract Levi-flat CR manifolds.
We give yet another proof for a global estimate of the Diederich-Fornaess index of relatively compact domains with Levi-flat boundary, namely, the index must be smaller than or equal to the reciprocal of the dimension of the ambient space. Although the Diederich-Fornaess index is originally defined for relatively compact domains in complex manifolds, our formulation reveals that it makes sense for abstract Levi-flat CR manifolds.
Algebraic Geometry Seminar
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Will Donovan (IPMU)
Twists and braids for general 3-fold flops (English)
http://db.ipmu.jp/member/personal/4007en.html
Will Donovan (IPMU)
Twists and braids for general 3-fold flops (English)
[ Abstract ]
When a 3-fold contains a floppable rational curve, a theorem of Bridgeland provides a derived equivalence between the 3-fold and its flop. I will discuss recent joint work with Michael Wemyss, showing that these flop functors satisfy Coxeter-type braid relations. Using this result, we construct an action of a braid-type group on the derived category of the 3-fold. This group arises from the topology of a certain simplicial hyperplane arrangement, determined by the local geometry of the curve. I will give examples and explain key elements in the construction, including the noncommutative deformations of curves introduced in our previous work.
[ Reference URL ]When a 3-fold contains a floppable rational curve, a theorem of Bridgeland provides a derived equivalence between the 3-fold and its flop. I will discuss recent joint work with Michael Wemyss, showing that these flop functors satisfy Coxeter-type braid relations. Using this result, we construct an action of a braid-type group on the derived category of the 3-fold. This group arises from the topology of a certain simplicial hyperplane arrangement, determined by the local geometry of the curve. I will give examples and explain key elements in the construction, including the noncommutative deformations of curves introduced in our previous work.
http://db.ipmu.jp/member/personal/4007en.html
Numerical Analysis Seminar
16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)
Katsuhisa Ozaki (Shibaura Institute of Technology)
Accurate matrix multiplication by error-free transformation (日本語)
Katsuhisa Ozaki (Shibaura Institute of Technology)
Accurate matrix multiplication by error-free transformation (日本語)
Tokyo Probability Seminar
16:50-18:20 Room #128 (Graduate School of Math. Sci. Bldg.)
Lu Xu (Graduate School of Mathematical Sciences, The University of Tokyo)
Central limit theorem for stochastic heat equations in random environments
Lu Xu (Graduate School of Mathematical Sciences, The University of Tokyo)
Central limit theorem for stochastic heat equations in random environments
2015/05/14
Applied Analysis
16:00-17:30 Room #128 (Graduate School of Math. Sci. Bldg.)
Masahito Ohta (Tokyo University of Science)
Strong instability of standing waves for some nonlinear Schr\"odinger equations (Japanese)
Masahito Ohta (Tokyo University of Science)
Strong instability of standing waves for some nonlinear Schr\"odinger equations (Japanese)
2015/05/13
Operator Algebra Seminars
16:45-18:15 Room #122 (Graduate School of Math. Sci. Bldg.)
Yosuke Kubota (Univ. Tokyo)
Controlled topological phases and the bulk-edge correspondence for
topological insulators (English)
Yosuke Kubota (Univ. Tokyo)
Controlled topological phases and the bulk-edge correspondence for
topological insulators (English)
2015/05/12
Tuesday Seminar of Analysis
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Keisuke Takasao (Graduate School of Mathematical Sciences, the University of Tokyo)
Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow (Japanese)
Keisuke Takasao (Graduate School of Mathematical Sciences, the University of Tokyo)
Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow (Japanese)
[ Abstract ]
In this talk we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.
In this talk we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.
Tuesday Seminar on Topology
17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Masayuki Asaoka (Kyoto University)
Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)
Masayuki Asaoka (Kyoto University)
Growth rate of the number of periodic points for generic dynamical systems (JAPANESE)
[ Abstract ]
For any hyperbolic dynamical system, the number of periodic
points grows at most exponentially and the growth rate
reflects statistic property of the system. For dynamics far
from hyperbolicity, the situation is different. In 1999,
Kaloshin proved genericity of super-exponential growth in the
region where dense set of dynamical systems exhibits homoclinic
tangency (so called the Newhouse region).
How does the number of periodic points grow for generic
partially hyperbolic dynamical systems? Such systems are known
to be far from homoclinic tangency. Is the growth at most
exponential like hyperbolic system, or super-exponential by
a mechanism different from homoclinic tangency?
The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved
super-exponential growth of the number of periodic points for
generic one-dimensional iterated function systems under some
reasonable conditions. Such systems are models of dynamics
of partially hyperbolic systems in neutral direction. So, we
expect genericity of super-exponential growth in a region of
partially hyperbolic systems.
In this talk, we start with a brief history of the problem on
growth rate of the number of periodic point and discuss two
mechanisms which lead to genericity of super-exponential growth,
Kaloshin's one and ours.
For any hyperbolic dynamical system, the number of periodic
points grows at most exponentially and the growth rate
reflects statistic property of the system. For dynamics far
from hyperbolicity, the situation is different. In 1999,
Kaloshin proved genericity of super-exponential growth in the
region where dense set of dynamical systems exhibits homoclinic
tangency (so called the Newhouse region).
How does the number of periodic points grow for generic
partially hyperbolic dynamical systems? Such systems are known
to be far from homoclinic tangency. Is the growth at most
exponential like hyperbolic system, or super-exponential by
a mechanism different from homoclinic tangency?
The speaker, Katsutoshi Shinohara, and Dimitry Turaev proved
super-exponential growth of the number of periodic points for
generic one-dimensional iterated function systems under some
reasonable conditions. Such systems are models of dynamics
of partially hyperbolic systems in neutral direction. So, we
expect genericity of super-exponential growth in a region of
partially hyperbolic systems.
In this talk, we start with a brief history of the problem on
growth rate of the number of periodic point and discuss two
mechanisms which lead to genericity of super-exponential growth,
Kaloshin's one and ours.
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