## Seminar information archive

Seminar information archive ～09/10｜Today's seminar 09/11 | Future seminars 09/12～

### 2021/11/26

#### Colloquium

15:30-16:30 Online

Registration is closed (12:00, November 26).

Ricci flow on Fano manifolds (ENGLISH)

Registration is closed (12:00, November 26).

**Gang Tian**(BICMR, Peking University)Ricci flow on Fano manifolds (ENGLISH)

[ Abstract ]

Ricci flow was introduced by Hamilton in early 80s. It preserves the Kahlerian structure and has found many applications in Kahler geometry. In this expository talk, I will focus on Ricci flow on Fano manifolds. I will first survey some results in recent years, then I will discuss my joint work with Li and Zhu. I will also discuss the connection between the long time behavior of Ricci flow and some algebraic geometric problems for Fano manifolds.

Ricci flow was introduced by Hamilton in early 80s. It preserves the Kahlerian structure and has found many applications in Kahler geometry. In this expository talk, I will focus on Ricci flow on Fano manifolds. I will first survey some results in recent years, then I will discuss my joint work with Li and Zhu. I will also discuss the connection between the long time behavior of Ricci flow and some algebraic geometric problems for Fano manifolds.

#### Mathematical Biology Seminar

15:00-16:30 Online

Derivation of structured population models of cellular proliferation on an

energy landscape

[ Reference URL ]

オンライン参加希望の方は，inaba@ms.u-tokyo.ac.jp までご連絡ください．

**Shinji NAKAOKA**(Faculty of Advanced Life Science, Hokkaido University)Derivation of structured population models of cellular proliferation on an

energy landscape

[ Reference URL ]

オンライン参加希望の方は，inaba@ms.u-tokyo.ac.jp までご連絡ください．

### 2021/11/25

#### Applied Analysis

#### Information Mathematics Seminar

16:50-18:35 Online

Classification and Clustering in the Machine Learning (Japanese)

https://docs.google.com/forms/d/1I3XD63V937BT_IoqRWBVN67goQAtbkSoIKs-6hfLUAM

**Hiroshi Fujiwara**(BroadBand Tower, Inc.)Classification and Clustering in the Machine Learning (Japanese)

[ Abstract ]

Explanation on classification and clustering in the Machine Learning

[ Reference URL ]Explanation on classification and clustering in the Machine Learning

https://docs.google.com/forms/d/1I3XD63V937BT_IoqRWBVN67goQAtbkSoIKs-6hfLUAM

### 2021/11/24

#### Number Theory Seminar

17:00-18:00 Online

On the formal degree conjecture for non-singular supercuspidal representations (Japanese)

**Kazuma Ohara**(University of Tokyo)On the formal degree conjecture for non-singular supercuspidal representations (Japanese)

[ Abstract ]

We prove the formal degree conjecture for non-singular supercuspidal representations based on Schwein's work proving the formal degree conjecture for regular supercuspidal representations. The main difference between our work and Schwein's work is that in non-singular case, the Deligne--Lusztig representations can be reducible, and the $S$-groups are not necessarily abelian. Therefore, we have to compare the dimensions of irreducible constituents of the Deligne--Lusztig representations and the dimensions of irreducible representations of $S$-groups.

We prove the formal degree conjecture for non-singular supercuspidal representations based on Schwein's work proving the formal degree conjecture for regular supercuspidal representations. The main difference between our work and Schwein's work is that in non-singular case, the Deligne--Lusztig representations can be reducible, and the $S$-groups are not necessarily abelian. Therefore, we have to compare the dimensions of irreducible constituents of the Deligne--Lusztig representations and the dimensions of irreducible representations of $S$-groups.

#### Seminar on Mathematics for various disciplines

10:30-11:30 Room #Zoomによるオンライン開催 (Graduate School of Math. Sci. Bldg.)

乱流熱輸送現象の最適制御と複雑伝熱面の形状最適化 (日本語)

**Yosuke Hasegawa**(Institute of Industrial Science, the University of Tokyo)乱流熱輸送現象の最適制御と複雑伝熱面の形状最適化 (日本語)

### 2021/11/23

#### Lie Groups and Representation Theory

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

A Cartan decomposition for a reductive real spherical subgroup

(Japanese)

**Yuichiro Tanaka**(The University of Tokyo)A Cartan decomposition for a reductive real spherical subgroup

(Japanese)

[ Abstract ]

A closed subgroup H of a real reductive Lie group G is real spherical if a minimal parabolic subgroup of G has an open orbit on G/H.

In this talk I would like to show a proof of a Cartan decomposition G=KAH when H is reductive.

This is a conjecture of T. Kobayashi, introduced in the 3rd Number Theory Summer School in 1995.

A closed subgroup H of a real reductive Lie group G is real spherical if a minimal parabolic subgroup of G has an open orbit on G/H.

In this talk I would like to show a proof of a Cartan decomposition G=KAH when H is reductive.

This is a conjecture of T. Kobayashi, introduced in the 3rd Number Theory Summer School in 1995.

### 2021/11/19

#### Tokyo-Nagoya Algebra Seminar

17:00-18:30 Online

Please see the URL below for details on the online seminar.

有限群のブロック上の$\tau$-傾理論 (Japanese) (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

Please see the URL below for details on the online seminar.

**Yuta Kozakai**(Tokyo University of Science)有限群のブロック上の$\tau$-傾理論 (Japanese) (Japanese)

[ Reference URL ]

http://www.math.nagoya-u.ac.jp/~aaron.chan/TNAseminar.html

### 2021/11/18

#### Information Mathematics Seminar

16:50-18:35 Online

Reinforcement learning and Regression algorithm to support AI (Japanese)

https://docs.google.com/forms/d/1I3XD63V937BT_IoqRWBVN67goQAtbkSoIKs-6hfLUAM

**Hiroshi Fujiwara**(BroadBand Tower, Inc.)Reinforcement learning and Regression algorithm to support AI (Japanese)

[ Abstract ]

Explanation on reinforcement learning and regression algorithm to support AI

[ Reference URL ]Explanation on reinforcement learning and regression algorithm to support AI

https://docs.google.com/forms/d/1I3XD63V937BT_IoqRWBVN67goQAtbkSoIKs-6hfLUAM

### 2021/11/17

#### Seminar on Probability and Statistics

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

On the local times of noise reinforced Bessel processes

https://docs.google.com/forms/d/e/1FAIpQLSeuK9AOw6QUqvUge9ukw__v04j5jpfogzrGxlPLpEgNhW09kg/viewform

**Jean Bertoin**(Institut of Mathematics, University of Zurich (UZH))On the local times of noise reinforced Bessel processes

[ Abstract ]

Asia-Pacific Seminar in Probability and Statistics (APSPS)

https://sites.google.com/view/apsps/home

Bessel processes form a one-parameter family of self-similar diffusion on $[0,\infty)$ with the same Hurst exponent 1/2 as Brownian motion. Loosely speaking, in this setting, linear noise reinforcement with reinforcement parameter $p$ consists of repeating (if $p>0$) or counterbalancing (if $p<0$)infinitesimal increments of the process, uniformly at random and at a fixed rate as time passes. In this talk, we will investigate the effect of noise reinforcement on the local time at level $0$, that is, informally, the time that the process spends at $0$. A connection with increasing self-similar Markov processes will play a key role.

[ Reference URL ]Asia-Pacific Seminar in Probability and Statistics (APSPS)

https://sites.google.com/view/apsps/home

Bessel processes form a one-parameter family of self-similar diffusion on $[0,\infty)$ with the same Hurst exponent 1/2 as Brownian motion. Loosely speaking, in this setting, linear noise reinforcement with reinforcement parameter $p$ consists of repeating (if $p>0$) or counterbalancing (if $p<0$)infinitesimal increments of the process, uniformly at random and at a fixed rate as time passes. In this talk, we will investigate the effect of noise reinforcement on the local time at level $0$, that is, informally, the time that the process spends at $0$. A connection with increasing self-similar Markov processes will play a key role.

https://docs.google.com/forms/d/e/1FAIpQLSeuK9AOw6QUqvUge9ukw__v04j5jpfogzrGxlPLpEgNhW09kg/viewform

### 2021/11/16

#### Tuesday Seminar of Analysis

16:00-17:30 Online

TBA (Japanese)

[ Reference URL ]

https://forms.gle/6ZCp8hQxKA3vq3DB9

**KUBO Hideo**(Hokkaido University)TBA (Japanese)

[ Reference URL ]

https://forms.gle/6ZCp8hQxKA3vq3DB9

#### Tuesday Seminar on Topology

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Skein and cluster algebras of marked surfaces without punctures for sl(3) (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Wataru Yuasa**(RIMS, Kyoto University)Skein and cluster algebras of marked surfaces without punctures for sl(3) (JAPANESE)

[ Abstract ]

We consider a skein algebra consisting of sl(3)-webs with the boundary skein relations for a marked surface without punctures. We construct a quantum cluster algebra coming from the moduli space of decorated SL(3)-local systems of the surface inside the skew-field of fractions of the skein algebra. In this talk, we introduce the sticking trick and the cutting trick for sl(3)-webs. The sticking trick expands the boundary-localized skein algebra into the cluster algebra. The cutting trick gives Laurent expressions of "elevation-preserving" webs with positive coefficients in certain clusters. We can also apply these tricks in the case of sp(4). This talk is based on joint works with Tsukasa Ishibashi.

[ Reference URL ]We consider a skein algebra consisting of sl(3)-webs with the boundary skein relations for a marked surface without punctures. We construct a quantum cluster algebra coming from the moduli space of decorated SL(3)-local systems of the surface inside the skew-field of fractions of the skein algebra. In this talk, we introduce the sticking trick and the cutting trick for sl(3)-webs. The sticking trick expands the boundary-localized skein algebra into the cluster algebra. The cutting trick gives Laurent expressions of "elevation-preserving" webs with positive coefficients in certain clusters. We can also apply these tricks in the case of sp(4). This talk is based on joint works with Tsukasa Ishibashi.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

#### Operator Algebra Seminars

16:45-18:15 Online

On regular $*$-algebras of bounded linear operators: A new approach towards a theory of noncommutative Boolean algebras

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

**Michiya Mori**(RIKEN)On regular $*$-algebras of bounded linear operators: A new approach towards a theory of noncommutative Boolean algebras

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

#### Lie Groups and Representation Theory

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

Computation of weighted Bergman norms on block diagonal matrices in bounded symmetric domains (Japanese)

**Ryosuke Nakahama**(Kyushu University)Computation of weighted Bergman norms on block diagonal matrices in bounded symmetric domains (Japanese)

[ Abstract ]

Let $G/K\simeq D\subset\mathfrak{p}^+$ be a Hermitian symmetric space realized as a bounded symmetric domain, and we consider the weighted Bergman space $\mathcal{H}_\lambda(D)$ on $D$.

Then the norm on each $K$-type in $\mathcal{H}_\lambda(D)$ is explicitly computed by Faraut--Kor\'anyi (1990).

In this talk, we consider the cases $\mathfrak{p}^+=\operatorname{Sym}(r,\mathbb{C})$, $M(r,\mathbb{C})$, $\operatorname{Alt}(2r,\mathbb{C})$, fix $r=r'+r''$, and decompose $\mathfrak{p}^+$ into $2\times 2$ block matrices.

Then the speaker presents the results on explicit computation of the norm of $\mathcal{H}_\lambda(D)$ on each $K'$-type in the space of polynomials on the block diagonal matrices $\mathfrak{p}^+_{11}\oplus\mathfrak{p}^+_{22}$.

Also, as an application, the speaker presents the results on Plancherel-type formulas on the branching laws for symmetric pairs $(Sp(r,\mathbb{R}),U(r',r''))$, $(U(r,r),U(r',r'')\times U(r'',r'))$, $(SO^*(4r),U(2r',2r''))$.

Let $G/K\simeq D\subset\mathfrak{p}^+$ be a Hermitian symmetric space realized as a bounded symmetric domain, and we consider the weighted Bergman space $\mathcal{H}_\lambda(D)$ on $D$.

Then the norm on each $K$-type in $\mathcal{H}_\lambda(D)$ is explicitly computed by Faraut--Kor\'anyi (1990).

In this talk, we consider the cases $\mathfrak{p}^+=\operatorname{Sym}(r,\mathbb{C})$, $M(r,\mathbb{C})$, $\operatorname{Alt}(2r,\mathbb{C})$, fix $r=r'+r''$, and decompose $\mathfrak{p}^+$ into $2\times 2$ block matrices.

Then the speaker presents the results on explicit computation of the norm of $\mathcal{H}_\lambda(D)$ on each $K'$-type in the space of polynomials on the block diagonal matrices $\mathfrak{p}^+_{11}\oplus\mathfrak{p}^+_{22}$.

Also, as an application, the speaker presents the results on Plancherel-type formulas on the branching laws for symmetric pairs $(Sp(r,\mathbb{R}),U(r',r''))$, $(U(r,r),U(r',r'')\times U(r'',r'))$, $(SO^*(4r),U(2r',2r''))$.

### 2021/11/15

#### Seminar on Geometric Complex Analysis

10:30-12:00 Online

Computing logarithmic vector fields along an isolated singularity and Bruce-Roberts Milnor ideals (Japanese)

https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

**Katsusuke Nabeshima**(Tokyo University of Science)Computing logarithmic vector fields along an isolated singularity and Bruce-Roberts Milnor ideals (Japanese)

[ Abstract ]

The concept of logarithmic vector fields along a hypersurface, introduced by K. Saito (1980), is of considerable importance in singularity theory.

Logarithmic vector fields have been extensively studied and utilized by several researchers. A. G. Aleksandrov (1986) and J. Wahl (1983) considered quasihomogeneous complete intersection cases and gave independently, among other things, a closed formula of generators of logarithmic vector fields. However, there is no closed formula for generators of logarithmic vector fields, even for semi-quasihomogeneous hypersurface isolated singularity cases. Many problems related with logarithmic vector fields remain still unsolved, especially for non-quasihomogeneous cases.

Bruce-Roberts Milnor number was introduced in 1988 by J. W. Bruce and R. M. Roberts as a generalization of the Milnor number, a multiplicity of an isolated critical point of a holomorphic function germ. This number is defined for a critical point of a holomorphic function on a singular variety in terms of logarithmic vector fields. Recently, Bruce-Robert Milnor numbers are investigated by several researchers. However, many problems related with Bruce-Roberts Milnor numbers remain unsolved.

In this talk, we consider logarithmic vector fields along a hypersurface with an isolated singularity. We present methods to study complex analytic properties of logarithmic vector fields and illustrate an algorithm for computing logarithmic vector fields. As an application of logarithmic vector fields, we consider Bruce-Roberts Milnor numbers in the context of symbolic computation.

[ Reference URL ]The concept of logarithmic vector fields along a hypersurface, introduced by K. Saito (1980), is of considerable importance in singularity theory.

Logarithmic vector fields have been extensively studied and utilized by several researchers. A. G. Aleksandrov (1986) and J. Wahl (1983) considered quasihomogeneous complete intersection cases and gave independently, among other things, a closed formula of generators of logarithmic vector fields. However, there is no closed formula for generators of logarithmic vector fields, even for semi-quasihomogeneous hypersurface isolated singularity cases. Many problems related with logarithmic vector fields remain still unsolved, especially for non-quasihomogeneous cases.

Bruce-Roberts Milnor number was introduced in 1988 by J. W. Bruce and R. M. Roberts as a generalization of the Milnor number, a multiplicity of an isolated critical point of a holomorphic function germ. This number is defined for a critical point of a holomorphic function on a singular variety in terms of logarithmic vector fields. Recently, Bruce-Robert Milnor numbers are investigated by several researchers. However, many problems related with Bruce-Roberts Milnor numbers remain unsolved.

In this talk, we consider logarithmic vector fields along a hypersurface with an isolated singularity. We present methods to study complex analytic properties of logarithmic vector fields and illustrate an algorithm for computing logarithmic vector fields. As an application of logarithmic vector fields, we consider Bruce-Roberts Milnor numbers in the context of symbolic computation.

https://u-tokyo-ac-jp.zoom.us/meeting/register/tJ0vcu2rrDIqG9Rv5AT0Mpi37urIkJ1IRldB

### 2021/11/11

#### Information Mathematics Seminar

16:50-18:35 Online

Supervised/Unsupervised Learning and Reinforcement learning for Deep Learning (Japanese)

https://docs.google.com/forms/d/1I3XD63V937BT_IoqRWBVN67goQAtbkSoIKs-6hfLUAM

**Hiroshi Fujiwara**(BroadBand Tower, Inc.)Supervised/Unsupervised Learning and Reinforcement learning for Deep Learning (Japanese)

[ Abstract ]

Explanation on supervised/unsupervised learning and reinforcement learning for deep learning

[ Reference URL ]Explanation on supervised/unsupervised learning and reinforcement learning for deep learning

https://docs.google.com/forms/d/1I3XD63V937BT_IoqRWBVN67goQAtbkSoIKs-6hfLUAM

### 2021/11/09

#### Tuesday Seminar on Topology

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

The spaces of non-descendible quasimorphisms and bounded characteristic classes (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Shuhei Maruyama**(Nagoya University)The spaces of non-descendible quasimorphisms and bounded characteristic classes (JAPANESE)

[ Abstract ]

A quasimorphism is a real-valued function on a group which is a homomorphism up to bounded error. In this talk, we discuss the (non-)descendibility of quasimorphisms. In particular, we consider the space of non-descendible quasimorphisms on universal covering groups and explain its relation to the space of bounded characteristic classes of foliated bundles. This talk is based on a joint work with Morimichi Kawasaki.

[ Reference URL ]A quasimorphism is a real-valued function on a group which is a homomorphism up to bounded error. In this talk, we discuss the (non-)descendibility of quasimorphisms. In particular, we consider the space of non-descendible quasimorphisms on universal covering groups and explain its relation to the space of bounded characteristic classes of foliated bundles. This talk is based on a joint work with Morimichi Kawasaki.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

#### Operator Algebra Seminars

16:45-18:15 Online

Construction of Haag-Kastler nets for factorizing S-matrices with bound states

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

**Yoh Tanimoto**(University of Rome, Tor Vergata)Construction of Haag-Kastler nets for factorizing S-matrices with bound states

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

#### Lie Groups and Representation Theory

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

On the support of Plancherel measures and the image of moment map

(Japanese)

**Yoshiki Oshima**(Osaka University)On the support of Plancherel measures and the image of moment map

(Japanese)

[ Abstract ]

We see a relationship between the support of Plancherel measures for homogeneous spaces and the image of moment maps from cotangent bundles based on a joint work with Benjamin Harris.

Moreover, we discuss related problems and conjectures about inductions and restrictions for representations of Lie groups in general settings.

We see a relationship between the support of Plancherel measures for homogeneous spaces and the image of moment maps from cotangent bundles based on a joint work with Benjamin Harris.

Moreover, we discuss related problems and conjectures about inductions and restrictions for representations of Lie groups in general settings.

### 2021/11/04

#### Information Mathematics Seminar

16:50-18:35 Online

Internet business establishment/GPU application/Quantum computer design (Japanese)

https://docs.google.com/forms/d/1I3XD63V937BT_IoqRWBVN67goQAtbkSoIKs-6hfLUAM

**Hiroshi Fujiwara**(BroadBand Tower, Inc.)Internet business establishment/GPU application/Quantum computer design (Japanese)

[ Abstract ]

Explanation on Internet business establishment, GPU application and quantum computer design

[ Reference URL ]Explanation on Internet business establishment, GPU application and quantum computer design

https://docs.google.com/forms/d/1I3XD63V937BT_IoqRWBVN67goQAtbkSoIKs-6hfLUAM

### 2021/11/02

#### Operator Algebra Seminars

16:45-18:15 Online

Magnetic properties of ground states in many-electron systems

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

**Tarahiro Miyao**(Hokkaido Univ.)Magnetic properties of ground states in many-electron systems

[ Reference URL ]

https://www.ms.u-tokyo.ac.jp/~yasuyuki/tokyo-seminar.htm

#### Tuesday Seminar on Topology

17:00-18:00 Online

Pre-registration required. See our seminar webpage.

Meta-nilpotent knot invariants and symplectic automorphism groups of free nilpotent groups (JAPANESE)

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

Pre-registration required. See our seminar webpage.

**Takefumi Nosaka**(Tokyo Institute of Technolog)Meta-nilpotent knot invariants and symplectic automorphism groups of free nilpotent groups (JAPANESE)

[ Abstract ]

There are many developments of fibered knots and homology cylinders from topological and algebraic viewpoints. In a converse sense, we discuss meta-nilpotent localization of knot groups,

which can deal with any knot like fibered knots. The monodoromy can be regarded as a symplectic automorphism of free nilpotent group, and the conjugacy classes in the outer automorphism groups produce knot invariants in terms of Johnson homomorphisms. In this talk, I show the construction of the monodoromies, and some results on the knot invariants. I also talk approaches from Fox pairings.

[ Reference URL ]There are many developments of fibered knots and homology cylinders from topological and algebraic viewpoints. In a converse sense, we discuss meta-nilpotent localization of knot groups,

which can deal with any knot like fibered knots. The monodoromy can be regarded as a symplectic automorphism of free nilpotent group, and the conjugacy classes in the outer automorphism groups produce knot invariants in terms of Johnson homomorphisms. In this talk, I show the construction of the monodoromies, and some results on the knot invariants. I also talk approaches from Fox pairings.

https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html

#### Lie Groups and Representation Theory

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

Difference between the distribution and hyperfunction solution spaces of an irregular holonomic D-module (Japanese)

**Taito TAUCHI**(Kyushu University)Difference between the distribution and hyperfunction solution spaces of an irregular holonomic D-module (Japanese)

[ Abstract ]

Let M be a holonomic D-module. Then the distribution and hyperfunction solution spaces of M coincide if M is regular. However, there is a difference between them in general if M is irregular. In this talk, we explain this phenomena taking a Whittaker functional of the principal series representation of SL(2, R) as a concrete example.

Let M be a holonomic D-module. Then the distribution and hyperfunction solution spaces of M coincide if M is regular. However, there is a difference between them in general if M is irregular. In this talk, we explain this phenomena taking a Whittaker functional of the principal series representation of SL(2, R) as a concrete example.

### 2021/10/29

#### Colloquium

15:30-16:30 Online

Registration was closed

Well-posedness of friction- or Signorini-type boundary value problems in the non-stationary case (JAPANESE)

Registration was closed

**Takahito Kashiwabara**(Graduate School of Mathematical Sciences, University of Tokyo)Well-posedness of friction- or Signorini-type boundary value problems in the non-stationary case (JAPANESE)

### 2021/10/28

#### Applied Analysis

16:00-17:00 Online

Quasiconformal and Sobolev mappings on metric measure

https://forms.gle/QATECqmwmWGvXoU56

**Xiaodan Zhou**(OIST)Quasiconformal and Sobolev mappings on metric measure

[ Abstract ]

The study of quasiconformal mappings has been an important and active topic since its introduction in the 1930s and the theory has been widely applied to different fields including differential geometry, harmonic analysis, PDEs, etc. In the Euclidean space, it is a fundamental result that three definitions (metric, geometric and analytic) of quasiconformality are equivalent. The theory of quasiconformal mappings has been extended to metric measure spaces by Heinonen and Koskela in the 1990s and their work laid the foundation of analysis on metric spaces. In general, the equivalence of the three characterizations will no longer hold without appropriate assumptions on the spaces and mappings. It is a question of general interest to find minimal assumptions on the metric spaces and on the mapping to guarantee the metric definition implies the analytic characterization or geometric characterization. In this talk, we will give an brief review of the above mentioned classical theory and present some recent results we achieved in obtaining the analytic property, in particular, the Sobolev regularity of a metric quasiconformal mapping with relaxed spaces and mapping conditions. Unexpectedly, we can apply this to prove results that are new even in the classical Euclidean setting. This is joint work with Panu Lahti (Chinese Academy of Sciences).

[ Reference URL ]The study of quasiconformal mappings has been an important and active topic since its introduction in the 1930s and the theory has been widely applied to different fields including differential geometry, harmonic analysis, PDEs, etc. In the Euclidean space, it is a fundamental result that three definitions (metric, geometric and analytic) of quasiconformality are equivalent. The theory of quasiconformal mappings has been extended to metric measure spaces by Heinonen and Koskela in the 1990s and their work laid the foundation of analysis on metric spaces. In general, the equivalence of the three characterizations will no longer hold without appropriate assumptions on the spaces and mappings. It is a question of general interest to find minimal assumptions on the metric spaces and on the mapping to guarantee the metric definition implies the analytic characterization or geometric characterization. In this talk, we will give an brief review of the above mentioned classical theory and present some recent results we achieved in obtaining the analytic property, in particular, the Sobolev regularity of a metric quasiconformal mapping with relaxed spaces and mapping conditions. Unexpectedly, we can apply this to prove results that are new even in the classical Euclidean setting. This is joint work with Panu Lahti (Chinese Academy of Sciences).

https://forms.gle/QATECqmwmWGvXoU56

< Previous 123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188 Next >