਍ഀഀ ਍㰀洀攀琀愀 栀琀琀瀀ⴀ攀焀甀椀瘀㴀䌀漀渀琀攀渀琀ⴀ吀礀瀀攀 挀漀渀琀攀渀琀㴀∀琀攀砀琀⼀栀琀洀氀㬀 挀栀愀爀猀攀琀㴀甀渀椀挀漀搀攀∀㸀ഀഀ ਍㰀洀攀琀愀 渀愀洀攀㴀䜀攀渀攀爀愀琀漀爀 挀漀渀琀攀渀琀㴀∀䴀椀挀爀漀猀漀昀琀 圀漀爀搀 ㄀㔀∀㸀ഀഀ ਍㰀氀椀渀欀 爀攀氀㴀䘀椀氀攀ⴀ䰀椀猀琀 栀爀攀昀㴀∀猀攀洀椀渀愀爀开猀椀琀攀开昀椀氀攀猀⼀昀椀氀攀氀椀猀琀⸀砀洀氀∀㸀ഀഀ Conformal Dynamics and Groups ਍㰀℀ⴀⴀ嬀椀昀 最琀攀 洀猀漀 㤀崀㸀㰀砀洀氀㸀ഀഀ ਍  㰀漀㨀䄀甀琀栀漀爀㸀娀栀椀焀椀愀渀最 䰀椀㰀⼀漀㨀䄀甀琀栀漀爀㸀ഀഀ 李智强_数学学院 ਍  㰀漀㨀刀攀瘀椀猀椀漀渀㸀㌀㌀㰀⼀漀㨀刀攀瘀椀猀椀漀渀㸀ഀഀ 429 ਍  㰀漀㨀䌀爀攀愀琀攀搀㸀㈀　㈀㌀ⴀ㄀㈀ⴀ㈀㐀吀　㄀㨀㌀㈀㨀　　娀㰀⼀漀㨀䌀爀攀愀琀攀搀㸀ഀഀ 2025-02-14T15:26:00Z ਍  㰀漀㨀倀愀最攀猀㸀㄀㰀⼀漀㨀倀愀最攀猀㸀ഀഀ 10035 ਍  㰀漀㨀䌀栀愀爀愀挀琀攀爀猀㸀㔀㜀㈀　㈀㰀⼀漀㨀䌀栀愀爀愀挀琀攀爀猀㸀ഀഀ 476 ਍  㰀漀㨀倀愀爀愀最爀愀瀀栀猀㸀㄀㌀㐀㰀⼀漀㨀倀愀爀愀最爀愀瀀栀猀㸀ഀഀ 67103 ਍  㰀漀㨀嘀攀爀猀椀漀渀㸀㄀㘀⸀　　㰀⼀漀㨀嘀攀爀猀椀漀渀㸀ഀഀ ਍ 㰀漀㨀伀昀昀椀挀攀䐀漀挀甀洀攀渀琀匀攀琀琀椀渀最猀㸀ഀഀ ਍ 㰀⼀漀㨀伀昀昀椀挀攀䐀漀挀甀洀攀渀琀匀攀琀琀椀渀最猀㸀ഀഀ ਍㰀氀椀渀欀 爀攀氀㴀琀栀攀洀攀䐀愀琀愀 栀爀攀昀㴀∀猀攀洀椀渀愀爀开猀椀琀攀开昀椀氀攀猀⼀琀栀攀洀攀搀愀琀愀⸀琀栀洀砀∀㸀ഀഀ ਍㰀℀ⴀⴀ嬀椀昀 最琀攀 洀猀漀 㤀崀㸀㰀砀洀氀㸀ഀഀ ਍  㰀眀㨀娀漀漀洀㸀㄀㈀　㰀⼀眀㨀娀漀漀洀㸀ഀഀ Clean ਍  㰀眀㨀䜀爀愀洀洀愀爀匀琀愀琀攀㸀䌀氀攀愀渀㰀⼀眀㨀䜀爀愀洀洀愀爀匀琀愀琀攀㸀ഀഀ false ਍  㰀眀㨀吀爀愀挀欀䘀漀爀洀愀琀琀椀渀最⼀㸀ഀഀ ਍  㰀眀㨀匀愀瘀攀䤀昀堀䴀䰀䤀渀瘀愀氀椀搀㸀昀愀氀猀攀㰀⼀眀㨀匀愀瘀攀䤀昀堀䴀䰀䤀渀瘀愀氀椀搀㸀ഀഀ false ਍  㰀眀㨀䄀氀眀愀礀猀匀栀漀眀倀氀愀挀攀栀漀氀搀攀爀吀攀砀琀㸀昀愀氀猀攀㰀⼀眀㨀䄀氀眀愀礀猀匀栀漀眀倀氀愀挀攀栀漀氀搀攀爀吀攀砀琀㸀ഀഀ ਍  㰀眀㨀䰀椀搀吀栀攀洀攀伀琀栀攀爀㸀䔀一ⴀ唀匀㰀⼀眀㨀䰀椀搀吀栀攀洀攀伀琀栀攀爀㸀ഀഀ ZH-CN ਍  㰀眀㨀䰀椀搀吀栀攀洀攀䌀漀洀瀀氀攀砀匀挀爀椀瀀琀㸀堀ⴀ一伀一䔀㰀⼀眀㨀䰀椀搀吀栀攀洀攀䌀漀洀瀀氀攀砀匀挀爀椀瀀琀㸀ഀഀ ਍   㰀眀㨀䈀爀攀愀欀圀爀愀瀀瀀攀搀吀愀戀氀攀猀⼀㸀ഀഀ ਍   㰀眀㨀唀猀攀䘀䔀䰀愀礀漀甀琀⼀㸀ഀഀ ਍  㰀眀㨀䐀漀挀甀洀攀渀琀嘀愀爀椀愀戀氀攀猀㸀ഀഀ H4sIAAAAAAAEAKtWckksSQxILCpxzi/NK1GyMqwFAAEhoTITAAAA ਍   㰀眀㨀开开䜀爀愀洀洀愀爀氀礀开㐀㈀开开开㄀㸀䠀㐀猀䤀䄀䄀䄀䄀䄀䄀䄀䔀䄀䬀琀圀挀猀氀倀㤀欀砀刀猀氀䤀礀一䐀夀㈀䴀稀䜀㈀猀䐀匀砀䴀䐀䄀㄀一䐀唀㄀一䐀娀唀　氀䔀䬀吀椀　甀稀猀稀倀䄀礀欀眀爀䄀唀䄀㠀䜀挀儀昀礀眀䄀䄀䄀䄀㴀㰀⼀眀㨀开开䜀爀愀洀洀愀爀氀礀开㐀㈀开开开㄀㸀ഀഀ ਍  㰀眀㨀䈀爀漀眀猀攀爀䰀攀瘀攀氀㸀䴀椀挀爀漀猀漀昀琀䤀渀琀攀爀渀攀琀䔀砀瀀氀漀爀攀爀㐀㰀⼀眀㨀䈀爀漀眀猀攀爀䰀攀瘀攀氀㸀ഀഀ ਍   㰀洀㨀洀愀琀栀䘀漀渀琀 洀㨀瘀愀氀㴀∀䌀愀洀戀爀椀愀 䴀愀琀栀∀⼀㸀ഀഀ ਍   㰀洀㨀戀爀欀䈀椀渀匀甀戀 洀㨀瘀愀氀㴀∀☀⌀㐀㔀㬀ⴀ∀⼀㸀ഀഀ ਍   㰀洀㨀搀椀猀瀀䐀攀昀⼀㸀ഀഀ ਍   㰀洀㨀爀䴀愀爀最椀渀 洀㨀瘀愀氀㴀∀　∀⼀㸀ഀഀ ਍   㰀洀㨀眀爀愀瀀䤀渀搀攀渀琀 洀㨀瘀愀氀㴀∀㄀㐀㐀　∀⼀㸀ഀഀ ਍   㰀洀㨀渀愀爀礀䰀椀洀 洀㨀瘀愀氀㴀∀甀渀搀伀瘀爀∀⼀㸀ഀഀ ਍㰀⼀砀洀氀㸀㰀℀嬀攀渀搀椀昀崀ⴀⴀ㸀㰀℀ⴀⴀ嬀椀昀 最琀攀 洀猀漀 㤀崀㸀㰀砀洀氀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀　∀ 儀䘀漀爀洀愀琀㴀∀琀爀甀攀∀ 一愀洀攀㴀∀一漀爀洀愀氀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㤀∀ 儀䘀漀爀洀愀琀㴀∀琀爀甀攀∀ 一愀洀攀㴀∀栀攀愀搀椀渀最 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" QFormat="true" Name="heading 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" QFormat="true" Name="heading 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" QFormat="true" Name="heading 6"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" QFormat="true" Name="heading 7"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" QFormat="true" Name="heading 8"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" QFormat="true" Name="heading 9"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 6"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 7"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 8"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="index 9"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" Name="toc 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" Name="toc 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" Name="toc 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" Name="toc 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" Name="toc 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ 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਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List Bullet 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List Bullet 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List Bullet 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List Bullet 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List Number 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List Number 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List Number 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="List Number 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㄀　∀ 儀䘀漀爀洀愀琀㴀∀琀爀甀攀∀ 一愀洀攀㴀∀吀椀琀氀攀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Salutation"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Date"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Body Text First Indent"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Body Text First Indent 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Note Heading"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Body Text 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Body Text 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Body Text Indent 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Body Text Indent 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Block Text"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Hyperlink"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="FollowedHyperlink"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㈀㈀∀ 儀䘀漀爀洀愀琀㴀∀琀爀甀攀∀ 一愀洀攀㴀∀匀琀爀漀渀最∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Document Map"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Plain Text"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="E-mail Signature"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Top of Form"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Bottom of Form"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Normal (Web)"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Acronym"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Address"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Cite"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Code"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Definition"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Keyboard"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Preformatted"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Sample"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Typewriter"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="HTML Variable"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Normal Table"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="annotation subject"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="No List"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Outline List 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Outline List 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Outline List 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Simple 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Simple 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Simple 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Classic 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Classic 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Classic 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Classic 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Colorful 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Colorful 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Colorful 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Columns 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Columns 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Columns 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Columns 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Columns 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Grid 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Grid 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Grid 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Grid 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Grid 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Grid 6"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Grid 7"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Grid 8"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table List 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table List 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table List 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table List 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table List 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table List 6"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table List 7"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table List 8"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table 3D effects 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table 3D effects 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table 3D effects 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Contemporary"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Elegant"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Professional"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Subtle 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Subtle 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Web 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Web 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Table Web 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Balloon Text"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㤀∀ 一愀洀攀㴀∀吀愀戀氀攀 䜀爀椀搀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㄀∀ 儀䘀漀爀洀愀琀㴀∀琀爀甀攀∀ 一愀洀攀㴀∀一漀 匀瀀愀挀椀渀最∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㄀∀ 一愀洀攀㴀∀䰀椀最栀琀 䰀椀猀琀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㌀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 匀栀愀搀椀渀最 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㔀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䰀椀猀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㜀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㤀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㄀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 匀栀愀搀椀渀最∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㌀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 䜀爀椀搀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㄀∀ 一愀洀攀㴀∀䰀椀最栀琀 䰀椀猀琀 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㌀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 匀栀愀搀椀渀最 ㄀ 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㔀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䰀椀猀琀 ㄀ 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㐀∀ 儀䘀漀爀洀愀琀㴀∀琀爀甀攀∀ഀഀ Name="List Paragraph"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㈀㤀∀ 儀䘀漀爀洀愀琀㴀∀琀爀甀攀∀ 一愀洀攀㴀∀儀甀漀琀攀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㜀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㄀ 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㤀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㌀ 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㄀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 匀栀愀搀椀渀最 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㌀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 䜀爀椀搀 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㄀∀ 一愀洀攀㴀∀䰀椀最栀琀 䰀椀猀琀 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㌀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 匀栀愀搀椀渀最 ㄀ 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㔀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䰀椀猀琀 ㄀ 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㜀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㄀ 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㤀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㌀ 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㄀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 匀栀愀搀椀渀最 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㌀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 䜀爀椀搀 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㄀∀ 一愀洀攀㴀∀䰀椀最栀琀 䰀椀猀琀 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㌀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 匀栀愀搀椀渀最 ㄀ 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㔀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䰀椀猀琀 ㄀ 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㜀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㄀ 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㤀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㌀ 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㄀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 匀栀愀搀椀渀最 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㌀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 䜀爀椀搀 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㄀∀ 一愀洀攀㴀∀䰀椀最栀琀 䰀椀猀琀 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㌀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 匀栀愀搀椀渀最 ㄀ 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㔀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䰀椀猀琀 ㄀ 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㜀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㄀ 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㤀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㌀ 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㄀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 匀栀愀搀椀渀最 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㌀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 䜀爀椀搀 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㄀∀ 一愀洀攀㴀∀䰀椀最栀琀 䰀椀猀琀 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㌀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 匀栀愀搀椀渀最 ㄀ 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㔀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䰀椀猀琀 ㄀ 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㜀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㄀ 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㤀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㌀ 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㄀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 匀栀愀搀椀渀最 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㌀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 䜀爀椀搀 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㄀∀ 一愀洀攀㴀∀䰀椀最栀琀 䰀椀猀琀 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㌀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 匀栀愀搀椀渀最 ㄀ 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㔀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䰀椀猀琀 ㄀ 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㜀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㄀ 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㘀㤀∀ 一愀洀攀㴀∀䴀攀搀椀甀洀 䜀爀椀搀 ㌀ 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㄀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 匀栀愀搀椀渀最 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㜀㌀∀ 一愀洀攀㴀∀䌀漀氀漀爀昀甀氀 䜀爀椀搀 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㜀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" Name="Bibliography"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㌀㤀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ഀഀ UnhideWhenUsed="true" QFormat="true" Name="TOC Heading"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㄀∀ 一愀洀攀㴀∀倀氀愀椀渀 吀愀戀氀攀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㌀∀ 一愀洀攀㴀∀倀氀愀椀渀 吀愀戀氀攀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㔀∀ 一愀洀攀㴀∀倀氀愀椀渀 吀愀戀氀攀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㘀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 ㄀ 䰀椀最栀琀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㠀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀　∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 㔀 䐀愀爀欀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㈀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 㜀 䌀漀氀漀爀昀甀氀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㠀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 ㌀ 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀　∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 㔀 䐀愀爀欀 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㠀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 ㌀ 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀　∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 㔀 䐀愀爀欀 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㠀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 ㌀ 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀　∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 㔀 䐀愀爀欀 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㠀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 ㌀ 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀　∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 㔀 䐀愀爀欀 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㠀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 ㌀ 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀　∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 㔀 䐀愀爀欀 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㠀∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 ㌀ 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀　∀ 一愀洀攀㴀∀䜀爀椀搀 吀愀戀氀攀 㔀 䐀愀爀欀 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㜀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㤀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㄀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 㘀 䌀漀氀漀爀昀甀氀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㘀∀ഀഀ Name="List Table 1 Light Accent 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㜀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 ㈀ 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㤀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 㐀 䄀挀挀攀渀琀 ㄀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㄀∀ഀഀ Name="List Table 6 Colorful Accent 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㈀∀ഀഀ Name="List Table 7 Colorful Accent 1"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㘀∀ഀഀ Name="List Table 1 Light Accent 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㜀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 ㈀ 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㤀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 㐀 䄀挀挀攀渀琀 ㈀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㄀∀ഀഀ Name="List Table 6 Colorful Accent 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㈀∀ഀഀ Name="List Table 7 Colorful Accent 2"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㘀∀ഀഀ Name="List Table 1 Light Accent 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㜀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 ㈀ 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㤀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 㐀 䄀挀挀攀渀琀 ㌀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㄀∀ഀഀ Name="List Table 6 Colorful Accent 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㈀∀ഀഀ Name="List Table 7 Colorful Accent 3"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㘀∀ഀഀ Name="List Table 1 Light Accent 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㜀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 ㈀ 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㤀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 㐀 䄀挀挀攀渀琀 㐀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㄀∀ഀഀ Name="List Table 6 Colorful Accent 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㈀∀ഀഀ Name="List Table 7 Colorful Accent 4"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㘀∀ഀഀ Name="List Table 1 Light Accent 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㜀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 ㈀ 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㤀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 㐀 䄀挀挀攀渀琀 㔀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㄀∀ഀഀ Name="List Table 6 Colorful Accent 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㈀∀ഀഀ Name="List Table 7 Colorful Accent 5"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㘀∀ഀഀ Name="List Table 1 Light Accent 6"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㜀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 ㈀ 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㐀㤀∀ 一愀洀攀㴀∀䰀椀猀琀 吀愀戀氀攀 㐀 䄀挀挀攀渀琀 㘀∀⼀㸀ഀഀ ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㄀∀ഀഀ Name="List Table 6 Colorful Accent 6"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 倀爀椀漀爀椀琀礀㴀∀㔀㈀∀ഀഀ Name="List Table 7 Colorful Accent 6"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Mention"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Smart Hyperlink"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Hashtag"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Unresolved Mention"/> ਍  㰀眀㨀䰀猀搀䔀砀挀攀瀀琀椀漀渀 䰀漀挀欀攀搀㴀∀昀愀氀猀攀∀ 匀攀洀椀䠀椀搀搀攀渀㴀∀琀爀甀攀∀ 唀渀栀椀搀攀圀栀攀渀唀猀攀搀㴀∀琀爀甀攀∀ഀഀ Name="Smart Link"/> ਍ 㰀⼀眀㨀䰀愀琀攀渀琀匀琀礀氀攀猀㸀ഀഀ ਍㰀猀琀礀氀攀㸀ഀഀ ਍㰀⼀猀琀礀氀攀㸀ഀഀ ਍㰀洀攀琀愀 栀琀琀瀀ⴀ攀焀甀椀瘀㴀䌀漀渀琀攀渀琀ⴀ匀琀礀氀攀ⴀ吀礀瀀攀 挀漀渀琀攀渀琀㴀∀琀攀砀琀⼀挀猀猀∀㸀ഀഀ ਍㰀℀ⴀⴀ嬀椀昀 最琀攀 洀猀漀 㤀崀㸀㰀砀洀氀㸀ഀഀ ਍㰀⼀砀洀氀㸀㰀℀嬀攀渀搀椀昀崀ⴀⴀ㸀㰀℀ⴀⴀ嬀椀昀 最琀攀 洀猀漀 㤀崀㸀㰀砀洀氀㸀ഀഀ ਍  㰀漀㨀椀搀洀愀瀀 瘀㨀攀砀琀㴀∀攀搀椀琀∀ 搀愀琀愀㴀∀㈀∀⼀㸀ഀഀ ਍㰀⼀栀攀愀搀㸀ഀഀ ਍㰀戀漀搀礀 氀愀渀最㴀娀䠀ⴀ䌀一 氀椀渀欀㴀戀氀甀攀 瘀氀椀渀欀㴀瀀甀爀瀀氀攀 猀琀礀氀攀㴀✀琀愀戀ⴀ椀渀琀攀爀瘀愀氀㨀㈀㄀⸀　瀀琀㬀眀漀爀搀ⴀ眀爀愀瀀㨀ഀഀ break-word'> ਍ഀഀ
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Conformal ਍䐀礀渀愀洀椀挀猀 愀渀搀 䜀爀漀甀瀀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㈀㘀⸀　瀀琀㬀ഀഀ font-family:"Arial",sans-serif;color:black'>

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 Junyi Xie and

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਍㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㄀㘀⸀　瀀琀㬀昀漀渀琀ⴀ昀愀洀椀氀礀㨀ഀഀ "Arial",sans-serif;color:black'>Speakers:崔贵珍

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Time: 3:00-4:00pm (Beijing

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Title: Rational maps with ਍挀漀渀猀琀愀渀琀 吀栀甀爀猀琀漀渀 洀愀瀀猀⨀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀戀猀琀爀愀挀琀㨀 䄀 爀愀琀椀漀渀愀氀 洀愀瀀 眀椀琀栀ഀഀ marked points induces a holomorphic map between Teichmüller ਍猀瀀愀挀攀猀 漀昀 瀀甀渀挀琀甀爀攀搀 猀瀀栀攀爀攀猀Ⰰ 眀栀椀挀栀 椀猀 挀愀氀氀攀搀 琀栀攀 吀栀甀爀猀琀漀渀 瀀甀氀氀戀愀挀欀 洀愀瀀⸀ 伀渀攀ഀഀ natural problem is to classify the marked rational maps with constant Thurston ਍洀愀瀀猀⸀ 䤀渀 琀栀椀猀 琀愀氀欀Ⰰ 䤀 眀椀氀氀 椀渀琀爀漀搀甀挀攀 琀栀攀 戀愀挀欀最爀漀甀渀搀 愀戀漀甀琀 琀栀椀猀 瀀爀漀戀氀攀洀 愀琀ഀഀ first. Then present a new class of rational maps with constant Thurston ਍瀀甀氀氀戀愀挀欀 洀愀瀀猀⸀ 吀栀椀猀 爀攀猀甀氀琀 挀愀渀 戀攀 猀琀愀琀攀搀 椀渀 攀氀攀洀攀渀琀愀爀礀 最攀漀洀攀琀爀礀⸀ 䤀渀 琀栀攀 氀愀猀琀 䤀ഀഀ will present some relations about this problem with monodromy ਍最爀漀甀瀀猀 漀昀 爀愀琀椀漀渀愀氀 洀愀瀀猀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀一漀瘀⸀ ㈀㜀 ㈀　㈀　 ⴀ 㰀戀㸀刀漀戀攀爀琀 吀愀渀最㰀⼀戀㸀ഀഀ (Xi’an Jiaotong-Liverpool University)

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Time: 10:00-11:00am (Beijing

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Title: Affine diffeomorphism ਍最爀漀甀瀀猀 愀爀攀 甀渀搀椀猀琀漀爀琀攀搀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀戀猀琀爀愀挀琀㨀 匀琀甀搀礀椀渀最 琀栀攀 搀椀猀琀漀爀琀椀漀渀ഀഀ properties of naturally occurring subgroups of the mapping class group plays an ਍椀洀瀀漀爀琀愀渀琀 瀀愀爀琀 椀渀 甀渀搀攀爀猀琀愀渀搀椀渀最 椀琀猀 氀愀爀最攀ⴀ猀挀愀氀攀 最攀漀洀攀琀爀礀⸀ 䤀渀 琀栀椀猀 琀愀氀欀Ⰰ 䤀 眀椀氀氀ഀഀ focus on the affine diffeomorphism group Aff(S,q). This is the group of self-diffeomorphisms with ਍挀漀渀猀琀愀渀琀 搀椀昀昀攀爀攀渀琀椀愀氀 愀眀愀礀 昀爀漀洀 琀栀攀 猀椀渀最甀氀愀爀椀琀椀攀猀⸀ 吀栀椀猀 最爀漀甀瀀 愀氀猀漀 挀漀椀渀挀椀搀攀猀ഀഀ with the stabilizer of the associated Teichmüller ਍搀椀猀挀 甀渀搀攀爀 琀栀攀 愀挀琀椀漀渀 漀昀 琀栀攀 洀愀瀀瀀椀渀最 挀氀愀猀猀 最爀漀甀瀀 漀渀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀吀攀椀挀栀洀ﰀ氀氀攀爀㰀⼀猀瀀愀渀㸀ഀഀ space. Our main result is that any finitely generated subgroup of Aff(

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Dec. 4 2020 - Mario Bonk ਍⠀唀䌀䰀䄀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 ㄀㄀㨀　　愀洀ⴀ㄀㈀㨀　　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom (ID: 649 4519 4579, ਍倀愀猀猀眀漀爀搀㨀 ㄀㐀㌀㌀㠀㘀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀琀氀攀㨀 匀甀氀氀椀瘀愀渀✀猀 搀椀挀琀椀漀渀愀爀礀 愀渀搀ഀഀ beyond*****

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Abstract: The classical theory of ਍栀漀氀漀洀漀爀瀀栀椀挀 搀礀渀愀洀椀挀猀 椀渀挀氀甀搀攀猀 琀栀攀 琀栀攀漀爀礀 漀昀 䬀氀攀椀渀椀愀渀 最爀漀甀瀀猀 愀渀搀 琀栀攀 椀琀攀爀愀琀椀漀渀ഀഀ theory of rational maps. In the 1980s Sullivan pointed out several analogies ਍戀攀琀眀攀攀渀 琀栀攀猀攀 琀眀漀 猀甀戀樀攀挀琀猀⸀ 䘀漀爀 攀砀愀洀瀀氀攀Ⰰ 琀栀攀 挀漀渀挀攀瀀琀 漀昀 愀 䨀甀氀椀愀 猀攀琀 漀昀 愀ഀഀ rational map corresponds to the concept of a limit set of a Kleinian group. ਍䔀瘀攀渀 椀昀 漀渀攀 最漀攀猀 戀攀礀漀渀搀 栀漀氀漀洀漀爀瀀栀椀挀 搀礀渀愀洀椀挀猀Ⰰ 愀渀 ᰀ攠砀琀攀渀搀攀搀ᴀ†匀甀氀氀椀瘀愀渀 搀椀挀琀椀漀渀愀爀礀ഀഀ can be very helpful as a guide for investigations. In my talk I will 高延Time)    Location: Zoom (ID: 690 5359 1120, ਍倀愀猀猀眀漀爀搀㨀 ㈀㤀㐀㈀㄀㐀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀琀氀攀㨀 䈀漀甀渀搀攀搀渀攀猀猀 愀渀搀ഀഀ compactification of hyperbolic components for Newton mapsIE(immediately escaping); ਍昀甀爀琀栀攀爀洀漀爀攀Ⰰ 琀栀攀 䜀䤀吀ⴀ挀漀洀瀀愀挀琀椀昀椀挀愀琀椀漀渀 漀昀 攀愀挀栀 琀礀瀀攀ⴀ䤀䔀 栀礀瀀攀爀戀漀氀椀挀 挀漀洀瀀漀渀攀渀琀 愀琀ഀഀ infinity boundary is either an analytic closed disk or one point. The proof is ਍戀愀猀攀搀 漀渀 愀 挀漀渀瘀攀爀最攀渀挀攀 琀栀攀漀爀攀洀 漀昀 椀渀琀攀爀渀愀氀 爀愀礀猀 眀攀 攀猀琀愀戀氀椀猀栀 昀漀爀 搀攀最攀渀攀爀愀琀攀ഀഀ Newton sequences. This is a joint work with Hongming ਍一椀攀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀愀渀⸀ ㄀㰀戀㸀 㰀⼀戀㸀㈀　㈀㄀ ⴀ 䈀爀攀愀欀 昀漀爀 一攀眀ഀഀ Year

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Jan. 8 2021 -  

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Abstract: In this talk, ਍眀攀☀渀戀猀瀀㬀搀椀猀挀甀猀猀☀渀戀猀瀀㬀琀栀攀 挀漀渀昀漀爀洀愀氀 搀椀洀攀渀猀椀漀渀 漀昀 猀漀洀攀☀渀戀猀瀀㬀猀琀漀挀栀愀猀琀椀挀 漀戀樀攀挀琀猀⸀ഀഀ The conformal dimension of a metric space is the infimum of the Hausdorff ਍搀椀洀攀渀猀椀漀渀 漀昀 愀氀氀 椀琀猀 焀甀愀猀椀猀礀洀洀攀琀爀椀挀 椀洀愀最攀猀⸀☀渀戀猀瀀㬀圀攀☀渀戀猀瀀㬀挀愀氀氀 愀 洀攀琀爀椀挀 猀瀀愀挀攀 洀椀渀椀洀愀氀ഀഀ if its conformal dimension equals its Hausdorff dimension. We begin ਍眀椀琀栀 愀 挀漀渀猀琀爀甀挀琀椀漀渀 漀昀 愀 最爀愀瀀栀 漀昀 愀 爀愀渀搀漀洀 昀甀渀挀琀椀漀渀 眀栀椀挀栀 椀猀 㰀猀瀀愀渀ഀഀ class=SpellE>a.s. minimal. Inspired by this, we apply the same ਍琀攀挀栀渀椀焀甀攀猀☀渀戀猀瀀㬀琀漀 琀栀攀 猀琀甀搀礀☀渀戀猀瀀㬀漀昀 ㄀ⴀ搀椀洀攀渀猀椀漀渀愀氀 䈀爀漀眀渀椀愀渀 最爀愀瀀栀猀⸀ 吀栀攀 洀愀椀渀ഀഀ tool is the Fuglede modulus. This is a joint work with Ilia Binder ਍愀渀搀 䠀爀愀渀琀 䠀愀欀漀戀礀愀渀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀愀渀⸀ ㄀㌀㰀戀㸀 㰀⼀戀㸀㈀　㈀㄀ ⴀ㰀戀㸀 吀椀洀 㰀猀瀀愀渀ഀഀ class=SpellE>Mesikepp (University of Washington)

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Time: 11:00am-12:00pm (Beijing  Title:How

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Abstract: How do you ਍挀漀渀昀漀爀洀愀氀氀礀 眀攀氀搀㼀 吀栀愀琀 椀猀Ⰰ 最椀瘀攀渀 愀 眀攀氀搀椀渀最 栀漀洀攀漀洀漀爀瀀栀椀猀洀 ␀尀㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀瘀愀爀瀀栀椀㰀⼀猀瀀愀渀㸀㨀尀㰀猀瀀愀渀ഀഀ class=SpellE>mathbb{S}^1 \rightarrow \weldings to driving functions. In the ਍瀀爀漀挀攀猀猀Ⰰ 眀攀 愀氀猀漀 最椀瘀攀 猀漀洀攀 爀攀猀甀氀琀猀 漀渀 洀椀渀椀洀愀氀ⴀ攀渀攀爀最礀 挀甀爀瘀攀猀 愀渀搀Ⰰ 椀昀 琀椀洀攀ഀഀ permits, draw connections to universal Teichmuller ਍猀瀀愀挀攀⸀ 吀栀攀爀攀 眀椀氀氀 戀攀 洀愀渀礀 瀀椀挀琀甀爀攀猀 愀渀搀 琀栀攀 琀愀氀欀 猀栀漀甀氀搀 戀攀 眀椀搀攀氀礀 愀挀挀攀猀猀椀戀氀攀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䐀䔀 猀琀礀氀攀㴀✀洀猀漀ⴀ愀渀猀椀ⴀ氀愀渀最甀愀最攀㨀䐀䔀✀㸀䴀愀爀⸀ ㈀㐀㰀戀㸀ഀഀ 2021 – Zhe Sun (IHES)--Shen related Kuntson-Tao ਍椀渀攀焀甀愀氀椀琀椀攀猀 琀漀 琀栀攀 瀀漀猀椀琀椀瘀椀琀礀 漀昀 䄀⸀ 伀渀 琀栀攀 猀甀爀昀愀挀攀Ⰰ 眀攀 椀搀攀渀琀椀昀礀 琀栀攀 猀瀀愀挀攀 漀昀ഀഀ 3-webs up to homotopy with certain lattice of A ਍洀愀瀀瀀椀渀最 挀氀愀猀猀 最爀漀甀瀀 攀焀甀椀瘀愀爀椀愀渀琀氀礀⸀ 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀䄀猀 愀 挀漀渀猀攀焀甀攀渀挀攀㰀⼀猀瀀愀渀㸀Ⰰഀഀ as predicted by Fock--Goncharov duality conjecture, these tropical points ਍瀀愀爀愀洀攀琀攀爀椀稀攀 愀 氀椀渀攀愀爀 戀愀猀椀猀 漀昀 琀栀攀 爀攀最甀氀愀爀 昀甀渀挀琀椀漀渀 爀椀渀最 漀昀 琀栀攀 搀甀愀氀 猀瀀愀挀攀ഀഀ explicitly. This is a joint work with Daniel Douglas.

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Mar. 30 2021 - Suzhen ਍䠀愀渀 ⠀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㔀㸀㰀戀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ昀愀洀椀氀礀㨀謀卛㭏ഀഀ mso-ascii-font-family:"Songti SC";mso-hansi-font-family:"Songti SC"'>韩素珍Time)    Location: Room 1114, Science ਍䈀甀椀氀搀椀渀最 ㄀Ⰰ 倀攀欀椀渀最 唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀琀氀攀㨀 吀栀攀 瀀爀漀瀀攀爀 愀挀琀椀漀渀猀 漀昀ഀഀ relatively hyperbolic groups on finite products of quasi-treesBestvina, K. Bromberg and K. ਍䘀甀樀椀眀愀爀愀Ⰰ 眀栀漀 愀氀猀漀 攀猀琀愀戀氀椀猀栀攀搀 猀甀挀栀 瀀爀漀瀀攀爀琀礀 昀漀爀 爀攀猀椀搀甀愀氀氀礀 昀椀渀椀琀攀 栀礀瀀攀爀戀漀氀椀挀ഀഀ groups and mapping class groups. In a joint work with H.T. Nguyen and 骆宇盛

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Time: 9:00-10:00am (Beijing  In particular, we will explain how the analogue of ਍吀栀甀爀猀琀漀渀ᤀ猠 挀漀洀瀀愀挀琀渀攀猀猀 琀栀攀漀爀攀洀 昀漀爀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀愀挀礀氀椀渀搀爀椀挀愀氀㰀⼀猀瀀愀渀㸀ഀഀ hyperbolic 3-manifold holds for critically fixed anti-rational maps.

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We will also brief discuss some ਍漀瀀攀渀 焀甀攀猀琀椀漀渀猀 洀漀琀椀瘀愀琀攀搀 戀礀 琀栀攀 挀漀爀爀攀猀瀀漀渀搀攀渀挀攀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀瀀爀椀氀 ㈀㄀㰀戀㸀 㰀⼀戀㸀㈀　㈀㄀ ጀ㰠戀㸀 䴀愀 䈀椀愀漀 ⠀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=s5> 

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Abstract: Let S =

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May 12 2021 – Lizhi Chen ਍⠀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㔀㸀㰀戀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ昀愀洀椀氀礀㨀謀卛㭏洀猀漀ⴀ愀猀挀椀椀ⴀ昀漀渀琀ⴀ昀愀洀椀氀礀㨀ഀഀ "Songti SC";mso-hansi-font-family:"Songti SC"'>陈立志Time)    Location: Zoom (ID: 895 8085 2996, Password: 339132)

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Title:  Homology and  homotopy complexity of manifolds in terms of Gromov’s ਍猀礀猀琀漀氀椀挀 椀渀攀焀甀愀氀椀琀礀⸀ 吀栀攀 漀瀀琀椀洀愀氀 挀漀渀猀琀愀渀琀 椀渀 猀礀猀琀漀氀椀挀 椀渀攀焀甀愀氀椀琀礀 椀猀 甀猀甀愀氀氀礀ഀഀ called systolic volume. A central theorem in systolic geometry relates systolic ਍瘀漀氀甀洀攀 琀漀 猀椀洀瀀氀椀挀椀愀氀 瘀漀氀甀洀攀⸀ 匀椀渀挀攀 昀漀爀 栀礀瀀攀爀戀漀氀椀挀 洀愀渀椀昀漀氀搀猀 琀栀攀爀攀 攀砀椀猀琀ഀഀ proportionality principle, this theorem builds a bridge between systolic ਍最攀漀洀攀琀爀礀 愀渀搀 栀礀瀀攀爀戀漀氀椀挀 最攀漀洀攀琀爀礀⸀ 䤀渀 琀栀攀 琀愀氀欀Ⰰ 眀攀 眀椀氀氀 瀀爀攀猀攀渀琀 猀漀洀攀ഀഀ applications of this theorem to the problem of homology and homotopy ਍挀漀洀瀀氀攀砀椀琀礀 漀昀 洀愀渀椀昀漀氀搀猀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䴀愀礀 ㄀㜀㰀戀㸀 㰀⼀戀㸀㈀　㈀㄀ ጀ㰠戀㸀 刀漀氀愀渀搀ഀഀ Roeder (Indiana University – Purdue University Indianapolis)

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Time: 9:00-10:00am (Beijing  

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Abstract: We study the dynamics of ਍琀栀攀 最爀漀甀瀀 漀昀 栀漀氀漀洀漀爀瀀栀椀挀 愀甀琀漀洀漀爀瀀栀椀猀洀猀 漀昀 琀栀攀 愀昀昀椀渀攀 挀甀戀椀挀 猀甀爀昀愀挀攀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀匀开笀㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀䄀Ⰰ䈀㰀⼀猀瀀愀渀㸀Ⰰ䌀Ⰰ䐀紀ഀഀ = {(x,y,z)  \in C^3   ਍㨀☀渀戀猀瀀㬀☀渀戀猀瀀㬀☀渀戀猀瀀㬀 砀帀㈀ ⬀ 礀帀㈀ ⬀ 稀帀㈀ ⬀㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀砀礀稀㰀⼀猀瀀愀渀㸀 㴀 䄀砀 ⬀ 㰀猀瀀愀渀ഀഀ class=SpellE>By+Cz+D},

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where A,B,C, ਍愀渀搀 䐀 愀爀攀 挀漀洀瀀氀攀砀 瀀愀爀愀洀攀琀攀爀猀⸀☀渀戀猀瀀㬀 䤀琀 愀爀椀猀攀猀 渀愀琀甀爀愀氀氀礀 椀渀 琀栀攀 搀礀渀愀洀椀挀猀 漀渀ഀഀ character varieties and it also describes the monodromy ਍漀昀 琀栀攀 昀愀洀漀甀猀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀倀愀椀渀氀攀瘀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㜀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀 㘀ഀഀ differential equation.  We explore the Fatou/Julia dichotomy for this ਍最爀漀甀瀀 愀挀琀椀漀渀 ⠀搀攀昀椀渀攀搀 愀渀愀氀漀最漀甀猀氀礀 琀漀 琀栀攀 甀猀甀愀氀 搀攀昀椀渀椀琀椀漀渀 昀漀爀 椀琀攀爀愀琀椀漀渀 漀昀 愀ഀഀ single rational map) and also the Locally Discrete / ਍一漀渀ⴀ䐀椀猀挀爀攀琀攀 搀椀挀栀漀琀漀洀礀 ⠀愀 渀漀渀ⴀ氀椀渀攀愀爀 瘀攀爀猀椀漀渀 昀爀漀洀 琀栀攀 挀氀愀猀猀椀挀愀氀ഀഀ discrete/non-discrete dichotomy for Lie groups). The interplay between these ਍琀眀漀 搀椀挀栀漀琀漀洀椀攀猀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀愀氀氀漀眀㰀⼀猀瀀愀渀㸀 甀猀 琀漀 瀀爀漀瘀攀 猀攀瘀攀爀愀氀 爀攀猀甀氀琀猀ഀഀ about the topological dynamics of this group.  Moreover, we show the ਍挀漀攀砀椀猀琀攀渀挀攀 漀昀 渀漀渀ⴀ攀洀瀀琀礀 䘀愀琀漀甀 猀攀琀猀 愀渀搀 䨀甀氀椀愀 猀攀琀猀 眀椀琀栀 渀漀渀ⴀ琀爀椀瘀椀愀氀 椀渀琀攀爀椀漀爀ഀഀ for a large set of parameters.

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This is joint work with Julio ਍刀攀戀攀氀漀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䴀愀礀 ㈀㘀㰀戀㸀 㰀⼀戀㸀㈀　㈀㄀ ጀ㰠戀㸀 䘀攀椀 夀愀渀最 ⠀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=s5> of f centered at z0. The ਍琀漀瀀漀氀漀最礀 漀昀 琀栀攀 戀漀甀渀搀愀爀椀攀猀 漀昀 匀椀攀最攀氀 搀椀猀欀猀 栀愀猀 戀攀攀渀 猀琀甀搀椀攀搀 攀砀琀攀渀猀椀瘀攀氀礀 椀渀 瀀愀猀琀ഀഀ 3 decades. This was motivated by the prediction of Douady ਍愀渀搀 匀甀氀氀椀瘀愀渀 琀栀愀琀 琀栀攀 匀椀攀最攀氀 搀椀猀欀 漀昀 攀瘀攀爀礀 渀漀渀ⴀ氀椀渀攀愀爀 爀愀琀椀漀渀愀氀 洀愀瀀 椀猀 愀 䨀漀爀搀愀渀ഀഀ domain.

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For the topology of whole Julia ਍猀攀琀猀 漀昀 栀漀氀漀洀漀爀瀀栀椀挀 洀愀瀀猀 眀椀琀栀 匀椀攀最攀氀 搀椀猀欀猀Ⰰ 琀栀攀 爀攀猀甀氀琀猀 愀瀀瀀攀愀爀 氀攀猀猀⸀ 倀攀琀攀爀猀攀渀ഀഀ proved that the quadratic Julia sets with bounded type Siegel disks are locally ਍挀漀渀渀攀挀琀攀搀⸀ 䰀愀琀攀爀 夀愀洀瀀漀氀猀欀礀 瀀爀漀瘀攀搀 琀栀攀 猀愀洀攀 爀攀猀甀氀琀 戀礀 愀渀 愀氀琀攀爀渀愀琀椀瘀攀 洀攀琀栀漀搀ഀഀ based on the existence of complex a prior bound of unicritical ਍挀椀爀挀氀攀 洀愀瀀猀⸀ 䄀 戀椀最 瀀爀漀最爀攀猀猀 眀愀猀 洀愀搀攀 戀礀 倀攀琀攀爀猀攀渀 愀渀搀 娀愀欀攀爀椀 椀渀 ㈀　　㐀⸀ 吀栀攀礀ഀഀ proved that for almost all rotation numbers, the quadratic Julia sets with ਍匀椀攀最攀氀 搀椀猀欀猀 愀爀攀 氀漀挀愀氀氀礀 挀漀渀渀攀挀琀攀搀⸀ 刀攀挀攀渀琀氀礀 䨀⸀ 夀愀渀最 瀀爀漀瘀攀搀 愀 猀琀爀椀欀椀渀最 爀攀猀甀氀琀ഀഀ that the Julia set of any polynomial (assumed to be connected) is locally ਍挀漀渀渀攀挀琀攀搀 愀琀 琀栀攀 戀漀甀渀搀愀爀礀 瀀漀椀渀琀猀 漀昀 琀栀攀椀爀 戀漀甀渀搀攀搀 琀礀瀀攀 匀椀攀最攀氀 搀椀猀欀猀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀猀 愀 最攀渀攀爀愀氀椀稀愀琀椀漀渀 漀昀 倀攀琀攀爀猀攀渀✀猀ഀഀ result, we prove that the Julia sets of a number of ਍爀愀琀椀漀渀愀氀 洀愀瀀猀 愀渀搀 琀爀愀渀猀挀攀渀搀攀渀琀愀氀 攀渀琀椀爀攀 昀甀渀挀琀椀漀渀猀 眀椀琀栀 戀漀甀渀搀攀搀 琀礀瀀攀 匀椀攀最攀氀ഀഀ disks are locally connected. This is based on establishing an expanding ਍瀀爀漀瀀攀爀琀礀 漀昀 愀 氀漀渀最 椀琀攀爀愀琀椀漀渀 漀昀 愀 挀氀愀猀猀 漀昀 焀甀愀猀椀ⴀ䈀氀愀猀挀栀欀攀 洀漀搀攀氀猀 渀攀愀爀 琀栀攀 甀渀椀琀ഀഀ circle.

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June 9 2021 – Yulan Qing ਍⠀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㔀㸀㰀戀㸀㰀猀瀀愀渀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ昀愀洀椀氀礀㨀謀卛㭏洀猀漀ⴀ愀猀挀椀椀ⴀ昀漀渀琀ⴀ昀愀洀椀氀礀㨀ഀഀ "Songti SC";mso-hansi-font-family:"Songti SC"'>卿于兰Time)    Location: Zoom (ID: 857 3683 8743, Password: 566197)

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Title:  The

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Abstract: In this talk, we ਍椀渀琀爀漀搀甀挀攀 琀栀攀 昀爀愀洀攀眀漀爀欀 漀昀 琀栀攀 挀漀愀爀猀攀 最攀漀洀攀琀爀礀 漀昀 渀漀渀ⴀ氀漀挀愀氀氀礀 挀漀洀瀀愀挀琀 最爀漀甀瀀猀 椀渀ഀഀ the setting of big mapping class groups, as studied by Rosendal. We will ਍搀椀猀挀甀猀猀 琀栀攀 挀栀愀爀愀挀琀攀爀椀稀愀琀椀漀渀 爀攀猀甀氀琀猀 漀昀 䴀愀渀渀ⴀ刀愀昀椀 愀渀搀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䠀漀爀戀攀稀㰀⼀猀瀀愀渀㸀ⴀ儀椀渀最ⴀ刀愀昀椀ഀഀ that illustrate big mapping groups' rich geometric and algebraic structures. We ਍眀椀氀氀 漀甀琀氀椀渀攀 琀栀攀 瀀爀漀漀昀猀 椀渀 琀栀攀猀攀 爀攀猀甀氀琀猀 愀渀搀 琀栀攀椀爀 椀洀瀀氀椀挀愀琀椀漀渀猀⸀ 䤀昀 琀椀洀攀ഀഀ permits, we will discuss some open problems in this area.(Nanjing University)

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Time: 3:00-4:00pm (Beijing

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Title:  On the local ਍挀漀渀渀攀挀琀椀瘀椀琀礀 漀昀 琀栀攀 䨀甀氀椀愀 猀攀琀猀 漀昀 爀愀琀椀漀渀愀氀 洀愀瀀猀 眀椀琀栀 戀漀甀渀搀攀搀 琀礀瀀攀 匀椀攀最攀氀ഀഀ disks***

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Abstract: In the talk, I will ਍猀欀攀琀挀栀 琀栀攀 椀搀攀愀 漀昀 琀栀攀 瀀爀漀漀昀 漀昀 琀栀攀 昀漀氀氀漀眀椀渀最 爀攀猀甀氀琀⸀☀渀戀猀瀀㬀 匀甀瀀瀀漀猀攀 昀 椀猀 愀ഀഀ rational map with bounded type Siegel disks such that any infinite critical ਍漀爀戀椀琀 椀渀琀攀爀猀攀挀琀猀 攀椀琀栀攀爀 琀栀攀 挀氀漀猀甀爀攀 漀昀 猀漀洀攀 戀漀甀渀搀攀搀 琀礀瀀攀 匀椀攀最攀氀 搀椀猀欀 漀爀 琀栀攀ഀഀ basin of some attracting periodic point.  Suppose additionally the Julia ਍猀攀琀 㰀椀㸀䨀⠀昀⤀㰀⼀椀㸀 椀猀 挀漀渀渀攀挀琀攀搀⸀ 吀栀攀渀 㰀椀㸀䨀⠀昀⤀㰀⼀椀㸀 椀猀 氀漀挀愀氀氀礀 挀漀渀渀攀挀琀攀搀⸀☀渀戀猀瀀㬀 吀栀椀猀ഀഀ was previously proved under the assumption that the boundaries of attracting ਍戀愀猀琀椀漀渀猀 搀漀 渀漀琀 椀渀琀攀爀猀攀挀琀 琀栀攀 戀漀甀渀搀愀爀礀 漀昀 愀渀礀 匀椀攀最攀氀 搀椀猀欀⸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=apple-converted-space> 

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September 24 2021 – 陈剑宇Time)    Location: Zoom (ID: 886 6229 2968, ਍倀愀猀猀眀漀爀搀㨀 㜀㌀㌀㈀㈀㤀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀琀氀攀㨀☀渀戀猀瀀㬀 䤀渀搀甀挀椀渀最 猀挀栀攀洀攀猀ഀഀ with finite weighted complexity

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Abstract: We consider a measurable ਍洀愀瀀 漀昀 愀 挀漀洀瀀愀挀琀 洀攀琀爀椀挀 猀瀀愀挀攀 眀栀椀挀栀 愀搀洀椀琀猀 愀渀 椀渀搀甀挀椀渀最 猀挀栀攀洀攀⸀ 唀渀搀攀爀 琀栀攀 昀椀渀椀琀攀ഀഀ weighted complexity condition, we establish a thermodynamic formalism for a ਍瀀愀爀愀洀攀琀攀爀 昀愀洀椀氀礀 漀昀 瀀漀琀攀渀琀椀愀氀猀 ␀尀㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀瘀愀爀瀀栀椀⬀琀㰀⼀猀瀀愀渀㸀尀瀀猀椀␀ 椀渀 愀渀ഀഀ interval containing $t=0$.  Furthermore, if there is a generating ਍瀀愀爀琀椀琀椀漀渀 挀漀洀瀀愀琀椀戀氀攀 琀漀 琀栀攀 椀渀搀甀挀椀渀最 猀挀栀攀洀攀Ⰰ 眀攀 猀栀漀眀 琀栀愀琀 愀氀氀 攀爀最漀搀椀挀 椀渀瘀愀爀椀愀渀琀ഀഀ measures with sufficiently large pressure are liftable. Our results are ਍愀瀀瀀氀椀挀愀戀氀攀 琀漀 琀栀攀 匀椀渀愀椀 搀椀猀瀀攀爀猀椀渀最 戀椀氀氀椀愀爀搀猀 眀椀琀栀 昀椀渀椀琀攀 栀漀爀椀稀漀渀Ⰰ 琀栀愀琀 椀猀Ⰰ 眀攀ഀഀ establish the equilibrium measures for the family of geometric potentials in a ਍猀氀椀最栀琀氀礀 爀攀猀琀爀椀挀琀攀搀 挀氀愀猀猀⸀ 吀栀椀猀 椀猀 愀 樀漀椀渀琀 眀漀爀欀 眀椀琀栀 䘀愀渀最 圀愀渀最 愀渀搀 䠀漀渀最ⴀ䬀甀渀ഀഀ Zhang.

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November 9 2021 – 余斌Time)    Location: Zoom (ID:879 8424 0017, Password: 928052)

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Title:  R-covered Anosov ਍昀氀漀眀猀 愀渀搀 昀爀攀攀氀礀 栀漀洀漀琀漀瀀椀挀 瀀攀爀椀漀搀椀挀 漀爀戀椀琀猀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀戀㸀㰀猀瀀愀渀ഀഀ lang=EN-US>

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Abstract: In the last 30 years, by ਍甀猀椀渀最 昀漀氀椀愀琀椀漀渀 琀栀攀漀爀礀 愀渀搀 最爀漀甀瀀 愀挀琀椀漀渀Ⰰ 䈀愀爀戀漀琀 愀渀搀 䘀攀渀氀攀礀 搀攀瘀攀氀漀瀀攀搀 愀 瀀漀眀攀爀昀甀氀ഀഀ tool to qualitatively understand 3-dimensioanl Anosov flows.  In this ਍猀攀洀椀渀愀爀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀琀愀氀欀Ⰰ☀渀戀猀瀀㬀 眀攀㰀⼀猀瀀愀渀㸀 眀椀氀氀 椀渀琀爀漀搀甀挀攀 猀漀洀攀 戀愀猀椀挀 椀搀攀愀猀ഀഀ due to Fenley in this direction.  In particular, we will explain a ਍猀椀最渀椀昀椀挀愀渀琀 瀀爀漀瀀攀爀琀礀 搀椀猀挀漀瘀攀爀攀搀 戀礀 䘀攀渀氀攀礀㨀 攀瘀攀爀礀 昀爀攀攀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀栀漀洀漀琀漀瀀礀㰀⼀猀瀀愀渀㸀ഀഀ class of an  R-covered Anosov flow on a ਍栀礀瀀攀爀戀漀氀椀挀 ㌀ⴀ洀愀渀椀昀漀氀搀 挀漀渀琀愀椀渀猀 椀渀昀椀渀椀琀攀氀礀 洀愀渀礀 瀀攀爀椀漀搀椀挀 漀爀戀椀琀猀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䐀攀挀攀洀戀攀爀 ㌀ ㈀　㈀㄀ ጀ㰠戀㸀 㰀猀瀀愀渀ഀഀ class=SpellE>Inhyeok Choi (Seoul National University)

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Time: 3:00-4:00pm (Beijing

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Abstract: Random walks on groups ਍愀挀琀椀渀最 漀渀 渀漀渀ⴀ瀀漀猀椀琀椀瘀攀氀礀ⴀ挀甀爀瘀攀搀 猀瀀愀挀攀猀 栀愀瘀攀 戀攀攀渀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀猀琀甀搀椀攀搀 椀渀ഀഀ depth. Various limiting behaviors of these random walks, including the ਍瀀漀猀椀琀椀瘀攀 攀猀挀愀瀀攀 爀愀琀攀Ⰰ 氀愀眀猀 漀昀 氀愀爀最攀 渀甀洀戀攀爀猀 愀渀搀 挀攀渀琀爀愀氀 氀椀洀椀琀 琀栀攀漀爀攀洀猀Ⰰ 眀攀爀攀ഀഀ known under some geometric assumptions on the space and the group action. ਍䠀漀眀攀瘀攀爀Ⰰ 琀栀攀 愀挀琀椀漀渀猀 漀昀 洀愀瀀瀀椀渀最 挀氀愀猀猀 最爀漀甀瀀猀 漀渀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀吀攀椀挀栀洀ﰀ氀氀攀爀㰀⼀猀瀀愀渀㸀ഀഀ spaces and curve complexes do not satisfy these assumptions, the former one ਍戀攀椀渀最 渀漀琀 䜀爀漀洀漀瘀 栀礀瀀攀爀戀漀氀椀挀 愀渀搀 琀栀攀 氀愀琀琀攀爀 漀渀攀 戀攀椀渀最 渀漀琀 氀漀挀愀氀氀礀 挀漀洀瀀愀挀琀⸀ഀഀ Nonetheless, recent developments imply that many limiting behaviors of random ਍眀愀氀欀猀 漀渀 洀愀瀀瀀椀渀最 挀氀愀猀猀 最爀漀甀瀀猀 昀漀氀氀漀眀 搀椀爀攀挀琀氀礀 昀爀漀洀 琀栀攀 ⠀瀀愀爀琀椀愀氀⤀ 栀礀瀀攀爀戀漀氀椀挀椀琀礀ഀഀ of Teichmüller spaces and curve complexes. In this ਍琀愀氀欀Ⰰ 䤀 眀椀氀氀 昀漀挀甀猀 漀渀 琀栀攀 瀀爀椀渀挀椀瀀氀攀 戀攀栀椀渀搀 琀栀攀猀攀 爀攀猀甀氀琀猀 愀渀搀 琀眀漀 瀀漀猀猀椀戀氀攀ഀഀ applications: the regularity of the harmonic measure and the counting problem ਍椀渀 洀愀瀀瀀椀渀最 挀氀愀猀猀 最爀漀甀瀀猀⸀ 倀愀爀琀椀愀氀氀礀 樀漀椀渀琀 眀椀琀栀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䠀礀甀渀最爀礀甀氀㰀⼀猀瀀愀渀㸀ഀഀ Baik and Dongryul M. Kim.

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December 24 2021 –

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Abstract: This talk is mainly about ਍琀栀攀 搀礀渀愀洀椀挀猀 漀昀 爀愀琀椀漀渀愀氀 洀愀瀀猀 漀渀 琀栀攀 刀椀攀洀愀渀渀 猀瀀栀攀爀攀⸀ 圀攀 眀椀氀氀 猀栀漀眀 琀栀愀琀 攀瘀攀爀礀ഀഀ postcritically finite rational map with non-empty Fatou set can be decomposed ਍椀渀琀漀 戀甀戀戀氀攀 爀愀琀椀漀渀愀氀 洀愀瀀猀 愀渀搀 匀椀攀爀瀀椀渀猀欀椀 爀愀琀椀漀渀愀氀 洀愀瀀猀⸀ 䈀愀猀攀搀 漀渀 琀栀椀猀 琀栀攀漀爀礀Ⰰഀഀ an invariant and finite connected graph can be constructed in the Julia set. ਍吀栀椀猀 椀猀 愀 樀漀椀渀琀 眀漀爀欀 眀椀琀栀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䜀甀椀稀栀攀渀㰀⼀猀瀀愀渀㸀 䌀甀椀 愀渀搀 夀愀渀 䜀愀漀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䴀愀爀挀栀 ㄀㈀ ㈀　㈀㈀ ጀ㰠戀㸀 㰀猀瀀愀渀ഀഀ class=SpellE>Zhuchao Ji (冀诸超

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Time: 2:00-3:00pm (Beijing Jingchunyuan 78, BICMR, Peking University

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Title:  On wandering ਍搀漀洀愀椀渀猀 椀渀 栀椀最栀攀爀 搀椀洀攀渀猀椀漀渀猀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀戀猀琀爀愀挀琀㨀☀渀戀猀瀀㬀 㰀愀ഀഀ href="../../../../../../../../GoogleDrive/From%20360/Academics/Homepage/public_html/seminars/conformal_dynamics_and_groups/ZhuchaoJiAbstract.pdf">Mesikepp (University of Washington)

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Time: 10:00-11:00am (Beijing

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Abstract: The Loewner energy is a ਍挀漀渀昀漀爀洀愀氀氀礀ⴀ椀渀瘀愀爀椀愀渀琀 昀甀渀挀琀椀漀渀愀氀 漀昀 挀甀爀瘀攀猀 椀渀 琀栀攀 甀瀀瀀攀爀 栀愀氀昀 瀀氀愀渀攀 䠀 琀栀愀琀 栀愀猀ഀഀ connections to hyperbolic geometry, SLE theory, Teichmuller ਍琀栀攀漀爀礀 愀渀搀 最攀漀洀攀琀爀椀挀 昀甀渀挀琀椀漀渀 琀栀攀漀爀礀⸀☀渀戀猀瀀㬀 䤀渀 琀栀椀猀 琀愀氀欀 眀攀 猀琀甀搀礀ഀഀ "one-point" minimizers of Loewner energy, asking what curves minimize ਍琀栀攀 攀渀攀爀最礀 愀洀漀渀最 愀氀氀 眀栀椀挀栀 瀀愀猀猀 琀栀爀漀甀最栀 愀 最椀瘀攀渀 瀀漀椀渀琀 椀渀 䠀Ⰰ 愀渀搀 眀栀愀琀 挀甀爀瘀攀猀ഀഀ minimize the energy among all which weld given points x<0<y.  The ਍昀漀爀洀攀爀 焀甀攀猀琀椀漀渀 眀愀猀 瀀愀爀琀椀愀氀氀礀 猀琀甀搀椀攀搀 戀礀 夀椀氀椀渀 圀愀渀最Ⰰ 眀栀漀 甀猀攀搀 匀䰀䔀 琀攀挀栀渀椀焀甀攀猀 琀漀ഀഀ calculate the minimal energy and show it is uniquely attained.  We revisit ਍琀栀攀 焀甀攀猀琀椀漀渀 甀猀椀渀最 愀 瀀甀爀攀氀礀 搀攀琀攀爀洀椀渀椀猀琀椀挀 洀攀琀栀漀搀漀氀漀最礀Ⰰ 栀漀眀攀瘀攀爀Ⰰ 愀渀搀 爀攀ⴀ搀攀爀椀瘀攀ഀഀ the energy formula and also obtain further results, ਍猀甀挀栀 愀猀 愀渀 攀砀瀀氀椀挀椀琀 挀漀洀瀀甀琀愀琀椀漀渀 漀昀 琀栀攀 搀爀椀瘀椀渀最 昀甀渀挀琀椀漀渀⸀ 伀甀爀 愀瀀瀀爀漀愀挀栀 愀氀猀漀ഀഀ yields existence and uniqueness of minimizers for the welding question, as well ਍愀猀 愀渀 攀砀瀀氀椀挀椀琀 攀渀攀爀最礀 昀漀爀洀甀氀愀 愀渀搀 攀砀瀀氀椀挀椀琀 搀爀椀瘀椀渀最 昀甀渀挀琀椀漀渀⸀☀渀戀猀瀀㬀 圀攀 搀椀猀挀甀猀猀ഀഀ parallel properties of both families, such as "universality" (all curves ਍椀渀 攀椀琀栀攀爀 昀愀洀椀氀礀 挀愀渀 戀攀 最攀渀攀爀愀琀攀搀 戀礀 愀 猀椀渀最氀攀 搀爀椀瘀椀渀最 昀甀渀挀琀椀漀渀⤀Ⰰ 攀渀攀爀最礀 甀猀愀最攀Ⰰഀഀ and connections to SLE with forcing. 

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May 26 2022 – Time)    Location: Zoom (ID:860 9860 6310, Password: 399654)

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Title: The non-positive ਍挀甀爀瘀愀琀甀爀攀 最攀漀洀攀琀爀礀 漀昀 猀漀洀攀 昀甀渀搀愀洀攀渀琀愀氀 最爀漀甀瀀猀 漀昀 挀漀洀瀀氀攀砀 栀礀瀀攀爀瀀氀愀渀攀 愀爀爀愀渀最攀洀攀渀琀ഀഀ complements

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Abstract: A complex hyperplane ਍挀漀洀瀀氀攀洀攀渀琀 椀猀 愀 琀漀瀀漀氀漀最椀挀愀氀 猀瀀愀挀攀 漀戀琀愀椀渀攀搀 戀礀 爀攀洀漀瘀椀渀最 愀 挀漀氀氀攀挀琀椀漀渀 漀昀 挀漀洀瀀氀攀砀ഀഀ codimension one affine hyperplanes from Osajda.

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June 21 2022 – Fabrizio Bianchi ()

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Time: 4:00-5:00pm (Beijing

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Title: A spectral gap for the ਍琀爀愀渀猀昀攀爀 漀瀀攀爀愀琀漀爀 漀渀 挀漀洀瀀氀攀砀 瀀爀漀樀攀挀琀椀瘀攀 猀瀀愀挀攀猀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀戀㸀㰀猀瀀愀渀ഀഀ lang=EN-US>

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Abstract: We study the transfer ਍⠀倀攀爀爀漀渀ⴀ䘀爀漀戀攀渀椀甀猀⤀ 漀瀀攀爀愀琀漀爀 漀渀 倀欀⠀䌀⤀ 椀渀搀甀挀攀搀 戀礀 愀 最攀渀攀爀椀挀 栀漀氀漀洀漀爀瀀栀椀挀ഀഀ endomorphism and a suitable continuous weight. We prove the existence of a ਍甀渀椀焀甀攀 攀焀甀椀氀椀戀爀椀甀洀 猀琀愀琀攀 愀渀搀 眀攀 椀渀琀爀漀搀甀挀攀 瘀愀爀椀漀甀猀 渀攀眀 椀渀瘀愀爀椀愀渀琀 昀甀渀挀琀椀漀渀愀氀 猀瀀愀挀攀猀Ⰰഀഀ including a dynamical Sobolev space, on which the action of ਍昀㰀⼀猀瀀愀渀㸀 愀搀洀椀琀猀 愀 猀瀀攀挀琀爀愀氀 最愀瀀⸀ 吀栀椀猀 椀猀 漀渀攀 漀昀 琀栀攀 洀漀猀琀 搀攀猀椀爀攀搀 瀀爀漀瀀攀爀琀椀攀猀 椀渀ഀഀ dynamics. It allows us to obtain a list of statistical properties for the ਍攀焀甀椀氀椀戀爀椀甀洀 猀琀愀琀攀猀 猀甀挀栀 愀猀 琀栀攀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀攀焀甀椀搀椀猀琀爀椀戀甀琀椀漀渀㰀⼀猀瀀愀渀㸀 漀昀ഀഀ points, speed of convergence, K-mixing, mixing of all orders, exponential ਍洀椀砀椀渀最Ⰰ 挀攀渀琀爀愀氀 氀椀洀椀琀 琀栀攀漀爀攀洀Ⰰ 䈀攀爀爀礀ⴀ㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䔀猀猀攀攀渀㰀⼀猀瀀愀渀㸀 琀栀攀漀爀攀洀Ⰰഀഀ local central limit theorem, almost sure invariant principle, law of iterated ਍氀漀最愀爀椀琀栀洀猀Ⰰ 愀氀洀漀猀琀 猀甀爀攀 挀攀渀琀爀愀氀 氀椀洀椀琀 琀栀攀漀爀攀洀 愀渀搀 琀栀攀 氀愀爀最攀 搀攀瘀椀愀琀椀漀渀ഀഀ principle. Most of the results are new even in dimension 1 and in the case of ਍挀漀渀猀琀愀渀琀 眀攀椀最栀琀 昀甀渀挀琀椀漀渀Ⰰ 椀⸀攀⸀Ⰰ 昀漀爀 琀栀攀 漀瀀攀爀愀琀漀爀 昀开⨀⸀ 伀甀爀 挀漀渀猀琀爀甀挀琀椀漀渀 漀昀 琀栀攀ഀഀ invariant functional spaces uses ideas from pluripotential theory and interpolation ਍戀攀琀眀攀攀渀 䈀愀渀愀挀栀 猀瀀愀挀攀猀⸀ 吀栀椀猀 椀猀 愀 樀漀椀渀琀 眀漀爀欀 眀椀琀栀 吀椀攀渀ⴀ䌀甀漀渀最 䐀椀渀栀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀匀攀瀀琀攀洀戀攀爀 ㈀　 ㈀　㈀㈀ ጀ†㰀猀瀀愀渀ഀഀ class=SpellE>Carlangelo Liverani (University of Rome Tor  

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Time: 4:00-5:00pm (Beijing Jingchunyuan 78, BICMR, Peking University; Zoom (ID: 812 ਍　㤀㜀㜀 㐀㠀㈀㠀Ⰰ 倀愀猀猀眀漀爀搀㨀 㠀㄀㌀　　㐀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black; ਍昀漀渀琀ⴀ眀攀椀最栀琀㨀渀漀爀洀愀氀✀㸀吀椀琀氀攀㨀 匀琀愀琀椀猀琀椀挀愀氀 瀀爀漀瀀攀爀琀椀攀猀 漀昀 栀礀瀀攀爀戀漀氀椀挀 戀椀氀氀椀愀爀搀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=s10>

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Abstract: I will discuss some ਍爀攀挀攀渀琀 愀渀搀 氀攀猀猀 爀攀挀攀渀琀 爀攀猀甀氀琀猀 挀漀渀挀攀爀渀椀渀最 琀栀攀 猀琀愀琀椀猀琀椀挀愀氀 瀀爀漀瀀攀爀琀椀攀猀 漀昀ഀഀ hyperbolic billiards, with particular emphasis on properties motivated by ਍昀甀渀搀愀洀攀渀琀愀氀 焀甀攀猀琀椀漀渀猀 椀渀 渀漀渀ⴀ攀焀甀椀氀椀戀爀椀甀洀 猀琀愀琀椀猀琀椀挀愀氀 洀攀挀栀愀渀椀挀猀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀伀挀琀漀戀攀爀 㜀 ㈀　㈀㈀ ጀ†㰀猀瀀愀渀ഀഀ class=SpellE>Wenbo Li (李文博)Time)    Location: Room 77201,

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quasisymmetries. In an effort ਍琀漀㰀⼀猀瀀愀渀㸀 愀渀猀眀攀爀 琀栀攀猀攀 焀甀攀猀琀椀漀渀猀Ⰰ 洀愀渀礀 瀀攀漀瀀氀攀 栀愀瘀攀 猀琀甀搀椀攀搀 琀栀攀洀 昀爀漀洀 搀椀昀昀攀爀攀渀琀ഀഀ approaches. We will go through the work by Bonk and Kleiner. In the end, we ਍瀀爀漀瘀椀搀攀 漀甀爀 琀爀礀椀渀最 漀渀 琀栀攀猀攀 挀漀渀樀攀挀琀甀爀攀猀⸀ 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀䤀渀 瀀愀爀琀椀挀甀氀愀爀Ⰰ 眀攀㰀⼀猀瀀愀渀㸀ഀഀ have constructed a special case of metric Sierpinski carpet, dyadic slit ਍挀愀爀瀀攀琀猀Ⰰ 愀渀搀 挀漀洀瀀氀攀琀攀氀礀 挀栀愀爀愀挀琀攀爀椀稀攀 椀琀猀 瀀氀愀渀愀爀 焀甀愀猀椀猀礀洀洀攀琀爀椀挀 攀洀戀攀搀搀愀戀椀氀椀琀礀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀伀挀琀漀戀攀爀 ㄀㐀 ㈀　㈀㈀ ጀ†㰀猀瀀愀渀ഀഀ class=SpellE>Xiaolei Wu (伍晓磊) Time)    Location: Tencent (ID: 522-848-241, ਍倀愀猀猀眀漀爀搀㨀 ㈀㈀㄀　㄀㐀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍䘀椀渀椀琀攀渀攀猀猀 瀀爀漀瀀攀爀琀椀攀猀 漀昀 愀猀礀洀瀀琀漀琀椀挀 洀愀瀀瀀椀渀最 挀氀愀猀猀 最爀漀甀瀀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>

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Abstract: Asymptotic mapping class ਍最爀漀甀瀀猀 眀愀猀 椀渀琀爀漀搀甀挀攀搀 戀礀 䘀甀渀愀爀 愀渀搀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䬀愀瀀漀甀搀樀椀愀渀㰀⼀猀瀀愀渀㸀⸀ 吀栀攀礀ഀഀ can be treated as a generalization of the Thompson groups in the mapping class ਍最爀漀甀瀀 猀攀琀琀椀渀最⸀ 圀攀 眀椀氀氀 搀椀猀挀甀猀猀 栀漀眀 琀栀攀猀攀 最爀漀甀瀀猀 愀爀攀 挀漀渀猀琀爀甀挀琀攀搀 愀渀搀 猀栀漀眀 琀栀愀琀ഀഀ they are in fact of type F_infinity. The proof boils ਍搀漀眀渀 琀漀 瀀爀漀瘀攀 挀攀爀琀愀椀渀 猀甀戀猀甀爀昀愀挀攀 挀漀洀瀀氀攀砀攀猀 愀爀攀 栀椀最栀氀礀 挀漀渀渀攀挀琀攀搀⸀ 吀栀椀猀 椀猀 戀愀猀攀搀ഀഀ on joint work with Javier Aramayona, Kai-Uwe Bux, ਍䨀漀渀愀猀 䘀氀攀挀栀猀椀最 愀渀搀 一愀渀猀攀渀 倀攀琀爀漀猀礀愀渀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㠀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀伀挀琀漀戀攀爀 ㈀㄀ ㈀　㈀㈀ ጀ†㰀戀㸀䤀氀椀愀 䈀椀渀搀攀爀 㰀⼀戀㸀⠀唀渀椀瘀攀爀猀椀琀礀ഀഀ of Toronto)

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Time: 8:00-9:00pm (Beijing

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Glucksam ਍⠀一漀爀琀栀眀攀猀琀攀爀渀⤀Ⰰ 䌀⸀ 刀漀樀愀猀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀⠀ 倀漀渀琀椀昀椀挀椀愀㰀⼀猀瀀愀渀㸀 唀渀椀瘀攀爀猀椀搀愀搀 㰀猀瀀愀渀ഀഀ class=SpellE>Catolica de Chile), and M. Yampolsky (University of ਍吀漀爀漀渀琀漀⤀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㠀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀伀挀琀漀戀攀爀 ㈀㜀 ㈀　㈀㈀ ጀ†㰀戀㸀䈀漀戀漀 䠀甀愀 ⠀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=s5>

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November 4 2022 – Mariusz Time)    Location: Zoom (ID: 827 6082 3142, ਍倀愀猀猀眀漀爀搀㨀 　㔀㜀㤀㌀㐀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍吀栀攀 攀砀愀挀琀 瘀愀氀甀攀 漀昀 䠀愀甀猀搀漀爀昀昀 搀椀洀攀渀猀椀漀渀 漀昀 攀猀挀愀瀀椀渀最 猀攀琀猀 昀漀爀 挀氀愀猀猀 ␀尀㰀猀瀀愀渀ഀഀ class=SpellE>mathcal B$ of meromorphic functionsmathcal B$ consisting of meromorphic ਍昀甀渀挀琀椀漀渀猀 ␀昀尀䌀尀琀漀尀䌀␀ 昀漀爀 眀栀椀挀栀 椀渀昀椀渀椀琀礀 椀猀 渀漀琀 愀渀 愀猀礀洀瀀琀漀琀椀挀 瘀愀氀甀攀 愀渀搀 眀栀漀猀攀ഀഀ all poles have orders uniformly bounded from above. This class was introduced ਍椀渀 嬀䈀眀䬀漀㈀　㄀㈀崀 愀渀搀 琀栀攀 䠀愀甀猀搀漀爀昀昀 搀椀洀攀渀猀椀漀渀 䠀䐀⠀䤀⠀昀⤀⤀ 漀昀 琀栀攀 猀攀琀 䤀⠀昀⤀ 漀昀 愀氀氀ഀഀ points escaping to infinity under forward iteration of $f$ was estimated ਍琀栀攀爀攀椀渀⸀ 䤀渀 琀栀椀猀 氀攀挀琀甀爀攀Ⰰ 戀愀猀攀搀 漀渀 樀漀椀渀琀 瀀愀瀀攀爀 眀椀琀栀 嘀漀氀欀攀爀 䴀愀礀攀爀Ⰰ 眀攀 瀀爀漀瘀椀搀攀 愀ഀഀ closed formula for the exact value of HD(I(f)) identifying it with the critical ਍攀砀瀀漀渀攀渀琀 漀昀 琀栀攀 渀愀琀甀爀愀氀 猀攀爀椀攀猀 椀渀琀爀漀搀甀挀攀搀 椀渀 嬀䈀眀䬀漀㈀　㄀㈀崀⸀ 吀栀椀猀 攀砀瀀漀渀攀渀琀 椀猀 瘀攀爀礀ഀഀ easy to calculate for many concrete functions. In particular, we construct a ਍昀甀渀挀琀椀漀渀 昀爀漀洀 琀栀椀猀 挀氀愀猀猀 眀栀椀挀栀 椀猀 漀昀 椀渀昀椀渀椀琀攀 漀爀搀攀爀 愀渀搀 昀漀爀 眀栀椀挀栀 䠀䐀⠀䤀⠀昀㰀猀瀀愀渀ഀഀ class=GramE>))=0.

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November 25 2022 – Daniel Meyer (University ਍漀昀 䰀椀瘀攀爀瀀漀漀氀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 㔀㨀　　ⴀ㘀㨀　　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom (ID: 838 0851 2867, ਍倀愀猀猀眀漀爀搀㨀 㘀㠀㘀㐀㤀㤀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍䘀爀愀挀琀愀氀 猀瀀栀攀爀攀猀Ⰰ 瘀椀猀甀愀氀 洀攀琀爀椀挀猀Ⰰ 愀渀搀 爀愀琀椀漀渀愀氀 洀愀瀀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>

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Abstract: Quasisymmetric maps map ਍爀愀琀椀漀猀 漀昀 搀椀猀琀愀渀挀攀猀 椀渀 愀 挀漀渀琀爀漀氀氀攀搀 眀愀礀⸀ 吀栀攀礀 最攀渀攀爀愀氀椀稀攀 挀漀渀昀漀爀洀愀氀 洀愀瀀猀⸀ 吀栀攀ഀഀ quasisymmetric uniformization theorem asks if a certain metric space is quasisymmetrically equivalent to the standard $2$-sphere. A ਍猀椀洀瀀氀攀 挀氀愀猀猀 漀昀 昀爀愀挀琀愀氀 猀瀀栀攀爀攀猀 愀爀攀 怀怀猀渀漀眀戀愀氀氀猀✀✀Ⰰ 眀栀椀挀栀 愀爀攀 琀漀瀀漀氀漀最椀挀愀氀氀礀ഀഀ $2$-dimensional analogs of the van Koch snowflake curve.

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A Thurston map is a topological ਍愀渀愀氀漀最 漀昀 愀 爀愀琀椀漀渀愀氀 洀愀瀀 ⠀椀⸀攀⸀Ⰰ 愀 栀漀氀漀洀漀爀瀀栀椀挀 猀攀氀昀ⴀ洀愀瀀 漀昀 琀栀攀 刀椀攀洀愀渀渀 猀瀀栀攀爀攀⤀⸀ഀഀ Thurston gave a criterion when such a map ``is'' rational. Given such a map $f$ ਍琀栀愀琀 椀猀 攀砀瀀愀渀搀椀渀最Ⰰ 眀攀 挀愀渀 攀焀甀椀瀀 琀栀攀 猀瀀栀攀爀攀 眀椀琀栀 愀 怀怀瘀椀猀甀愀氀 洀攀琀爀椀挀✀✀⸀ 圀椀琀栀ഀഀ respect to this metric, the sphere is a quasisphere ਍椀昀 愀渀搀 漀渀氀礀 椀昀 ␀昀␀ 怀怀椀猀✀✀ 爀愀琀椀漀渀愀氀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀栀椀猀 椀猀 樀漀椀渀琀 眀漀爀欀 眀椀琀栀 䴀愀爀椀漀 䈀漀渀欀ഀഀ (UCLA).

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December ਍㈀ ㈀　㈀㈀ ጀ†㰀戀㸀匀愀渀最ⴀ栀礀甀渀 䬀椀洀 㰀⼀戀㸀⠀䬀䤀䄀匀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䐀䔀ഀഀ style='mso-ansi-language:DE'>

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Time: 3:00-5:00pm (Beijing

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sentences . This is a work in ਍瀀爀漀最爀攀猀猀⸀ ⠀䨀漀椀渀琀 眀椀琀栀 吀栀漀洀愀猀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䬀漀戀攀爀搀愀㰀⼀猀瀀愀渀㸀 愀渀搀 䨀愀瘀椀攀爀 搀攀 氀愀ഀഀ Nuez--Gonzalez).

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March 17 2023 – Sylvester ਍䔀爀椀欀猀猀漀渀ⴀ㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䈀椀焀甀攀㰀⼀猀瀀愀渀㸀 㰀⼀戀㸀⠀唀渀椀瘀攀爀猀椀琀礀 漀昀 伀甀氀甀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 㐀㨀㌀　ⴀ㔀㨀㌀　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom  (ID: 892 5622 ਍㜀㌀㐀㜀Ⰰ 倀愀猀猀眀漀爀搀㨀 　㤀㐀㔀㤀㄀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍䄀 渀攀眀 氀攀洀洀愀 漀渀 䠀愀甀猀搀漀爀昀昀 挀漀渀琀攀渀琀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀ഀഀ style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>

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Abstract: What is in common between ਍爀愀渀搀漀洀 漀爀 搀礀渀愀洀椀挀愀氀氀礀 最攀渀攀爀愀琀攀搀 氀椀洀椀琀 猀甀瀀攀爀椀漀爀 猀攀琀猀 愀渀搀 戀漀甀渀搀愀爀椀攀猀 漀昀 猀攀琀猀 漀昀ഀഀ finite perimeter. In both settings problems can be ਍爀攀搀甀挀攀搀 琀漀 攀猀琀椀洀愀琀椀渀最 琀栀攀 䠀愀甀猀搀漀爀昀昀 挀漀渀琀攀渀琀 漀昀 猀漀洀攀 猀攀琀 眀栀椀挀栀 栀愀猀 愀 琀礀瀀攀 漀昀ഀഀ self-similarity. We introduce a new lemma that yields a Hausdorff content lower ਍戀漀甀渀搀 椀渀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀戀漀琀栀 漀昀 琀栀攀猀攀㰀⼀猀瀀愀渀㸀 猀攀琀琀椀渀最猀Ⰰ 愀渀搀 眀栀椀挀栀 猀攀攀洀猀ഀഀ useful in further applications. We illustrate its usefulness by deriving three ਍眀攀氀氀 欀渀漀眀渀 挀漀渀挀氀甀猀椀漀渀猀 昀爀漀洀 椀琀㨀 琀栀攀 䘀攀搀攀爀攀爀 挀栀愀爀愀挀琀攀爀椀稀愀琀椀漀渀 漀昀 猀攀琀猀 漀昀 昀椀渀椀琀攀 㰀猀瀀愀渀ഀഀ class=GramE>perimeter, the Hausdorff dimension of random limit superior ਍猀攀琀猀 愀渀搀 琀栀攀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䈀攀爀攀猀渀攀瘀椀琀挀栀ⴀ嘀攀氀愀渀椀㰀⼀猀瀀愀渀㸀 䴀愀猀猀 吀爀愀渀猀昀攀爀攀渀挀攀ഀഀ Principle.

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April 14 2023 – Insung ਍倀愀爀欀 㰀⼀戀㸀⠀䈀爀漀眀渀 唀渀椀瘀攀爀猀椀琀礀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 ㄀　㨀　　ⴀ㄀㄀㨀　　愀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom  (ID: 856 2139 ਍㤀㔀㜀㌀Ⰰ 倀愀猀猀眀漀爀搀㨀 ㄀㄀㄀㄀㄀㄀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍䨀甀氀椀愀 猀攀琀猀 栀愀瘀椀渀最 䄀栀氀昀漀爀猀ⴀ爀攀最甀氀愀爀 挀漀渀昀漀爀洀愀氀 搀椀洀攀渀猀椀漀渀 漀渀攀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

Abstract: The Ahlfors-regular ਍挀漀渀昀漀爀洀愀氀 搀椀洀攀渀猀椀漀渀Ⰰ 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䄀刀䌀搀椀洀㰀⼀猀瀀愀渀㸀⠀堀⤀Ⰰ 漀昀 愀 挀漀洀瀀愀挀琀 洀攀琀爀椀挀ഀഀ space X is the infimal Hausdorff dimension in the Ahlfors-regularly ਍焀甀愀猀椀ⴀ猀礀洀洀攀琀爀椀挀 挀氀愀猀猀 漀昀 堀⸀ 䄀猀 愀 昀爀愀挀琀愀氀 攀洀戀攀搀搀攀搀 椀渀 琀栀攀 刀椀攀洀愀渀渀 猀瀀栀攀爀攀Ⰰ 琀栀攀ഀഀ Julia set J_f of a hyperbolic rational map f has ਍䄀栀氀昀漀爀猀ⴀ爀攀最甀氀愀爀 挀漀渀昀漀爀洀愀氀 搀椀洀攀渀猀椀漀渀 戀攀琀眀攀攀渀 ㄀ 愀渀搀 ㈀⸀ 圀攀 栀愀瘀攀 㰀猀瀀愀渀ഀഀ class=SpellE>ARCdim(J_fARCdim(ARCdim(

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April 21 2023 – John Mackay (University ਍漀昀 䈀爀椀猀琀漀氀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 㔀㨀　　ⴀ㘀㨀　　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom  (ID: 859 6470 ਍㈀㤀㘀㄀Ⰰ 倀愀猀猀眀漀爀搀㨀 ㄀㄀㄀㄀㄀㄀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍䴀愀瀀猀 戀攀琀眀攀攀渀 爀攀氀愀琀椀瘀攀氀礀 栀礀瀀攀爀戀漀氀椀挀 最爀漀甀瀀猀 愀渀搀 琀栀攀椀爀 戀漀甀渀搀愀爀椀攀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

Abstract: Isometries of real ਍栀礀瀀攀爀戀漀氀椀挀 猀瀀愀挀攀猀 挀漀爀爀攀猀瀀漀渀搀 琀漀 䴀漀戀椀甀猀 琀爀愀渀猀昀漀爀洀愀琀椀漀渀猀 漀昀 琀栀攀椀爀 猀瀀栀攀爀攀猀 愀琀ഀഀ infinity.  Thanks to Paulin and Bonk-Schramm, quasi-isometries of Gromov ਍栀礀瀀攀爀戀漀氀椀挀 猀瀀愀挀攀猀 挀漀爀爀攀猀瀀漀渀搀 琀漀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀焀甀愀猀椀猀礀洀洀攀琀爀椀攀猀㰀⼀猀瀀愀渀㸀ഀഀ between their boundaries at infinity.  After discussing this background, ਍䤀✀氀氀 搀攀猀挀爀椀戀攀 栀漀眀 挀攀爀琀愀椀渀 洀愀瀀猀 戀攀琀眀攀攀渀 爀攀氀愀琀椀瘀攀 栀礀瀀攀爀戀漀氀椀挀 最爀漀甀瀀猀 挀漀爀爀攀猀瀀漀渀搀 琀漀ഀഀ maps between their Bowditch boundaries, and an application.  Joint work ਍眀椀琀栀 䄀氀攀猀猀愀渀搀爀漀 匀椀猀琀漀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䴀愀礀 ㄀㤀 ㈀　㈀㌀ ጀ†㰀戀㸀䬀攀瘀椀渀 䴀⸀ 倀椀氀最爀椀洀 㰀⼀戀㸀⠀䤀渀搀椀愀渀愀ഀഀ University Bloomington)

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Time: 8:00-9:00pm (Beijing

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monodromy groups (joint with V

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Abstract: We show that a ਍挀漀渀琀爀愀挀琀椀渀最 爀攀挀甀爀爀攀渀琀 猀攀氀昀ⴀ猀椀洀椀氀愀爀 最爀漀甀瀀 漀昀 瀀漀氀礀渀漀洀椀愀氀 愀挀琀椀瘀椀琀礀 最爀漀眀琀栀 椀猀ഀഀ amenable.  The proof proceeds by first showing that the associated limit ਍猀瀀愀挀攀 栀愀猀 挀漀渀昀漀爀洀愀氀 搀椀洀攀渀猀椀漀渀 攀焀甀愀氀 琀漀 ㄀⸀ 圀攀 琀栀攀渀 猀栀漀眀 琀栀椀猀 椀洀瀀氀椀攀猀 琀栀攀 漀爀戀椀琀愀氀ഀഀ graphs of its standard action are recurrent. Finally, we deduce amenability by ਍愀瀀瀀氀礀椀渀最 爀攀猀甀氀琀猀 漀昀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䨀甀猀挀栀攀渀欀漀㰀⼀猀瀀愀渀㸀Ⰰ 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀一攀欀爀愀猀栀攀瘀礀挀栀㰀⼀猀瀀愀渀㸀Ⰰഀഀ and de la Salle.  Examples of such groups include the iterated “crochet” rational functions; these are maps for which ਍愀渀礀 琀眀漀 瀀漀椀渀琀猀 椀渀 琀栀攀 䘀愀琀漀甀 猀攀琀 愀爀攀 樀漀椀渀攀搀 戀礀 愀 挀甀爀瘀攀 琀栀愀琀 洀攀攀琀猀 琀栀攀 䨀甀氀椀愀 猀攀琀ഀഀ in an at most countable set.

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May 24 2023 – Yulan Qing (卿于兰Time)    Location: Room 1114, Sciences ਍䈀甀椀氀搀椀渀最 一漀⸀ ㄀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍䈀漀甀渀搀愀爀礀 漀昀 最爀漀甀瀀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㄀㘀⸀　瀀琀㬀ഀഀ font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

Abstract: Gromov boundary provides ਍愀 甀猀攀昀甀氀 挀漀洀瀀愀挀琀椀昀椀挀愀琀椀漀渀 昀漀爀 愀氀氀 椀渀昀椀渀椀琀攀ⴀ搀椀愀洀攀琀攀爀 䜀爀漀洀漀瘀 栀礀瀀攀爀戀漀氀椀挀 猀瀀愀挀攀猀⸀ഀഀ It consists of all geodesic rays starting at a given base-point and it is an ਍攀猀猀攀渀琀椀愀氀 琀漀漀氀 椀渀 琀栀攀 猀琀甀搀礀 漀昀 琀栀攀 挀漀愀爀猀攀 最攀漀洀攀琀爀礀 漀昀 栀礀瀀攀爀戀漀氀椀挀 最爀漀甀瀀猀⸀ 䤀渀ഀഀ this study we introduce a generalization of the Gromov boundary for all ਍昀椀渀椀琀攀氀礀 最攀渀攀爀愀琀攀搀 最爀漀甀瀀猀⸀ 圀攀 挀漀渀猀琀爀甀挀琀 琀栀攀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀猀甀戀氀椀渀攀愀爀氀礀㰀⼀猀瀀愀渀㸀ഀഀ Morse boundaries and show that it is a QI-invariant topological space and it is ਍洀攀琀爀椀稀愀戀氀攀⸀ 圀攀 猀栀漀眀 琀栀攀 最攀漀洀攀琀爀椀挀 最攀渀攀爀椀挀椀琀礀 漀昀 瀀漀椀渀琀猀 椀渀 琀栀椀猀 戀漀甀渀搀愀爀礀 甀猀椀渀最ഀഀ Patterson Sullivan measure on the visual boundary of CAT(0) ਍猀瀀愀挀攀猀⸀ 䄀猀 愀渀 愀瀀瀀氀椀挀愀琀椀漀渀 眀攀 搀椀猀挀甀猀猀 琀栀攀 挀漀渀渀攀挀琀椀漀渀 戀攀琀眀攀攀渀 琀栀攀 㰀猀瀀愀渀ഀഀ class=SpellE>sublinearly Morse boundary and random walk on groups. We ਍愀渀猀眀攀爀 漀瀀攀渀 瀀爀漀戀氀攀洀猀 爀攀最愀爀搀椀渀最 儀䤀ⴀ椀渀瘀愀爀椀愀渀琀 洀漀搀攀氀猀 漀昀 爀愀渀搀漀洀 眀愀氀欀 漀渀 㰀猀瀀愀渀ഀഀ class=GramE>CAT(0) groups and on mapping class groups. If time permits, ਍眀攀 愀氀猀漀 眀椀氀氀 愀氀猀漀 氀漀漀欀 愀琀 栀漀眀 琀栀椀猀 戀漀甀渀搀愀爀礀 戀攀栀愀瘀攀猀 甀渀搀攀爀 猀甀戀氀椀渀攀愀爀 戀椀氀椀瀀猀挀栀椀琀稀ഀഀ equivalences.

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May 26 2023 – Araceli M. ਍䈀漀渀椀昀愀渀琀 㰀⼀戀㸀⠀唀渀椀瘀攀爀猀椀琀礀 漀昀 刀栀漀搀攀 䤀猀氀愀渀搀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 㠀㨀　　ⴀ㤀㨀　　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom  (ID: 889 4828 ਍㔀㔀㐀㤀Ⰰ 倀愀猀猀眀漀爀搀㨀 ㄀㄀㄀㄀㄀㄀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍唀渀搀攀爀猀琀愀渀搀椀渀最 琀栀攀 搀礀渀愀洀椀挀猀 漀昀 挀甀戀椀挀 瀀漀氀礀渀漀洀椀愀氀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀ഀഀ style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

 

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Abstract: For each p>0 there is ਍愀 昀愀洀椀氀礀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀匀开瀀㰀⼀猀瀀愀渀㸀 漀昀 挀漀洀瀀氀攀砀 挀甀戀椀挀 洀愀瀀猀 眀椀琀栀 愀 洀愀爀欀攀搀ഀഀ critical orbit of period p. For each q>0, I will describe a dynamically ਍搀攀昀椀渀攀搀 琀攀猀猀攀氀氀愀琀椀漀渀 漀昀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀匀开瀀㰀⼀猀瀀愀渀㸀⸀ 䔀愀挀栀 昀愀挀攀 漀昀 琀栀攀猀攀ഀഀ tessellations is associated with one particular behavior ਍昀漀爀 瀀攀爀椀漀搀椀挀 漀爀戀椀琀猀 漀昀 瀀攀爀椀漀搀 焀⸀ ⠀䨀漀椀渀琀 眀漀爀欀 眀椀琀栀 䨀漀栀渀 䴀椀氀渀漀爀⸀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=apple-converted-space> 

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Jun 5 2023 – Tim Time)    Location: Room 1114, Sciences ਍䈀甀椀氀搀椀渀最 一漀⸀ ㄀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍匀䰀䔀 氀愀爀最攀 搀攀瘀椀愀琀椀漀渀猀 愀渀搀 琀栀攀 爀攀最甀氀愀爀椀琀礀 漀昀 䰀漀攀眀渀攀爀 琀爀愀渀猀昀漀爀洀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

Abstract: The Loewner energy is a ਍昀甀渀挀琀椀漀渀愀氀 漀渀 䨀漀爀搀愀渀 挀甀爀瘀攀猀 琀栀愀琀 椀猀 琀栀攀 氀愀爀最攀ⴀ搀攀瘀椀愀琀椀漀渀猀 爀愀琀攀 昀甀渀挀琀椀漀渀 昀漀爀ഀഀ Schramm-Loewner Evolutions (SLE's).  Recent work has shown it also has ਍猀甀爀瀀爀椀猀椀渀最 挀漀渀渀攀挀琀椀漀渀猀 琀漀 漀琀栀攀爀 愀爀攀愀猀 漀昀 瀀爀漀戀愀戀椀氀椀琀礀Ⰰ 挀漀洀瀀氀攀砀 愀渀愀氀礀猀椀猀Ⰰഀഀ geometric measure theory, hyperbolic geometry, and even Teichmuller ਍琀栀攀漀爀礀⸀☀渀戀猀瀀㬀 䤀渀 琀栀攀 昀椀爀猀琀 瀀愀爀琀 漀昀 琀栀椀猀 琀愀氀欀 眀攀 猀甀爀瘀攀礀 猀漀洀攀 漀昀 琀栀攀猀攀ഀഀ connections, providing an "appetizer" for Yilin Wang's upcoming ਍猀甀洀洀攀爀 猀挀栀漀漀氀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䤀渀 琀栀攀 猀攀挀漀渀搀 瀀愀爀琀 漀昀 琀栀攀 琀愀氀欀 眀攀ഀഀ show that Loewner-energy minimizers give insight into the regularity of the ਍椀渀瘀攀爀猀攀 䰀漀攀眀渀攀爀 琀爀愀渀猀昀漀爀洀 ⠀琀栀攀 洀愀瀀 琀愀欀椀渀最 愀 搀爀椀瘀椀渀最 昀甀渀挀琀椀漀渀 琀漀 椀琀猀 愀猀猀漀挀椀愀琀攀搀ഀഀ curve).  In particular, we use energy minimizers ਍琀漀 攀砀栀椀戀椀琀 挀甀爀瘀攀猀 漀昀 搀椀昀昀攀爀椀渀最 爀攀最甀氀愀爀椀琀礀 琀栀愀琀 栀愀瘀攀 搀爀椀瘀椀渀最 昀甀渀挀琀椀漀渀猀 漀昀 琀栀攀ഀഀ same regularity, thus showing a theorem of Carto Wong is sharp.  This ਍瀀漀爀琀椀漀渀 漀昀 琀栀攀 琀愀氀欀 眀椀氀氀 戀攀 搀攀琀攀爀洀椀渀椀猀琀椀挀 椀渀 渀愀琀甀爀攀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀甀渀 㔀 ㈀　㈀㌀ 琀漀 䨀甀渀 ㄀㈀ ㈀　㈀㌀ ጀ†㰀戀㸀䴀愀爀椀漀ഀഀ Bonk (UCLA)

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Time & Location:Time)   Room 1114, Sciences Building No. 1

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Jun 10 2023 10:00am-12:00pm ਍⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀吀椀洀攀⤀☀渀戀猀瀀㬀☀渀戀猀瀀㬀 㰀⼀猀瀀愀渀㸀刀漀漀洀 ㄀㔀㔀㘀Ⰰ 匀挀椀攀渀挀攀猀ഀഀ Building No. 1

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Jun 12 2023 3:00-5:00pm (Beijing

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Teichmüller spaces and the Thurston pull-back map ਍䴀愀爀椀漀 䈀漀渀欀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㄀㘀⸀　瀀琀㬀昀漀渀琀ⴀ昀愀洀椀氀礀㨀ഀഀ "Arial",sans-serif;color:black'>

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Abstract: A postcritically-finite ਍戀爀愀渀挀栀攀搀 挀漀瘀攀爀椀渀最 洀愀瀀 漀渀 愀 琀漀瀀漀氀漀最椀挀愀氀 ㈀ⴀ猀瀀栀攀爀攀 椀猀 欀渀漀眀渀 愀猀 愀 吀栀甀爀猀琀漀渀 洀愀瀀⸀ 䤀琀ഀഀ induces a holomorphic map on an associated Teichmüller ਍猀瀀愀挀攀⸀ 吀栀椀猀 洀愀瀀 椀猀 欀渀漀眀渀 愀猀 琀栀攀 吀栀甀爀猀琀漀渀 瀀甀氀氀ⴀ戀愀挀欀 洀愀瀀⸀ 䤀渀 琀栀椀猀 洀椀渀椀 挀漀甀爀猀攀Ⰰ 䤀ഀഀ will give a quick introduction to necessary background (Teichmüller ਍猀瀀愀挀攀猀 愀渀搀 吀栀甀爀猀琀漀渀 洀愀瀀猀⤀Ⰰ 漀甀琀氀椀渀攀 愀 瀀爀漀漀昀 漀昀 琀栀攀 栀漀氀漀洀漀爀瀀栀椀挀椀琀礀 漀昀 琀栀攀ഀഀ pull-back map, and discuss a landmark theorem by Thurston on the ਍挀栀愀爀愀挀琀攀爀椀稀愀琀椀漀渀 漀昀 瀀漀猀琀挀爀椀琀椀挀愀氀氀礀ⴀ昀椀渀椀琀攀 爀愀琀椀漀渀愀氀 洀愀瀀猀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䤀ᤀ氠氀 搀椀猀挀甀猀猀 挀漀渀挀爀攀琀攀 攀砀愀洀瀀氀攀猀 昀漀爀ഀഀ the pull-back maps for Thurston maps with four postcritical points. As will be ਍猀攀攀渀 椀渀 琀栀攀 挀漀甀爀猀攀Ⰰ 琀栀椀猀 氀攀愀搀猀 琀漀 愀 挀攀爀琀愀椀渀 琀礀瀀攀 漀昀 瀀漀氀礀洀漀爀瀀栀椀挀 昀甀渀挀琀椀漀渀猀ഀഀ related to modular functions and forms. If time permits, I will also discuss ਍漀渀最漀椀渀最 爀攀猀攀愀爀挀栀 愀戀漀甀琀 琀栀攀 攀砀椀猀琀攀渀挀攀 漀昀 愀 最氀漀戀愀氀 昀椀渀椀琀攀 挀甀爀瘀攀 愀琀琀爀愀挀琀漀爀 椀渀 琀栀椀猀ഀഀ framework.

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June 7 2023 – Mario Bonk (UCLA)

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Time: 4:00-5:00pm (Beijing

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give an introduction to this ਍猀甀戀樀攀挀琀 愀渀搀 爀攀瀀漀爀琀 漀渀 漀渀最漀椀渀最 爀攀猀攀愀爀挀栀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀甀渀攀 ㄀㘀 ㈀　㈀㌀ ጀ†㰀戀㸀匀愀戀礀愀猀愀挀栀椀ഀഀ Mukherjee (Tata Institute of Fundamental ResearchTime)    Location: Zoom  (ID: 852 0183 ਍㘀㤀㜀㜀Ⰰ 倀愀猀猀眀漀爀搀㨀 ㄀㄀㄀㄀㄀㄀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍䴀愀琀椀渀最猀Ⰰ 栀漀氀漀洀漀爀瀀栀椀挀 挀漀爀爀攀猀瀀漀渀搀攀渀挀攀猀Ⰰ 愀渀搀 愀 䈀攀爀猀 猀氀椀挀攀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>

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Abstract: There are two frameworks ਍昀漀爀 洀愀琀椀渀最 䬀氀攀椀渀椀愀渀 最爀漀甀瀀猀 眀椀琀栀 爀愀琀椀漀渀愀氀 洀愀瀀猀 漀渀 琀栀攀 刀椀攀洀愀渀渀 猀瀀栀攀爀攀㨀 愀渀ഀഀ algebraic correspondence framework due to Bullett-Penrose-Lomonaco and an orbit ਍攀焀甀椀瘀愀氀攀渀挀攀 洀愀琀椀渀最 昀爀愀洀攀眀漀爀欀 甀猀椀渀最 䈀漀眀攀渀ⴀ匀攀爀椀攀猀 洀愀瀀猀⸀ 吀栀攀 氀愀琀琀攀爀 椀猀 愀渀愀氀漀最漀甀猀ഀഀ to the Douady-Hubbard theory for polynomial mating. ਍圀攀 眀椀氀氀 搀椀猀挀甀猀猀 栀漀眀 琀栀攀猀攀 琀眀漀 昀爀愀洀攀眀漀爀欀猀 挀愀渀 戀攀 甀渀椀昀椀攀搀 愀渀搀 最攀渀攀爀愀氀椀稀攀搀⸀ 㰀猀瀀愀渀ഀഀ class=GramE>As a consequence, we will construct holomorphic ਍挀漀爀爀攀猀瀀漀渀搀攀渀挀攀猀 琀栀愀琀 愀爀攀 洀愀琀椀渀最猀 漀昀 栀礀瀀攀爀戀漀氀椀挀 漀爀戀椀昀漀氀搀 最爀漀甀瀀猀 ⠀椀渀挀氀甀搀椀渀最 䠀攀挀欀攀ഀഀ groups) with Blaschke products. Time permitting, we will introduce an analog of ਍愀 䈀攀爀猀 猀氀椀挀攀 漀昀 琀栀攀 愀戀漀瘀攀 漀爀戀椀昀漀氀搀猀 椀渀 琀栀攀 愀氀最攀戀爀愀椀挀 瀀愀爀愀洀攀琀攀爀 猀瀀愀挀攀 漀昀ഀഀ correspondences. Based on joint work with Mahan Mj.

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Time: 9:00-10:00pm (Beijing

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Title:Expansivity 

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July 15 2023 – Dylan Thurston (Indiana ਍唀渀椀瘀攀爀猀椀琀礀Ⰰ 䈀氀漀漀洀椀渀最琀漀渀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 ㄀　㨀　　ⴀ㄀㄀㨀　　愀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: BICMR 77201Title: ਍倀漀猀椀琀椀瘀攀 䌀栀愀爀愀挀琀攀爀椀稀愀琀椀漀渀 漀昀 刀愀琀椀漀渀愀氀 䴀愀瀀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀ഀഀ style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>

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Abstract: When does a topological ਍猀攀氀昀ⴀ洀愀瀀 漀昀 琀栀攀 猀瀀栀攀爀攀 挀漀洀攀 昀爀漀洀 愀渀 愀挀琀甀愀氀 爀愀琀椀漀渀愀氀 洀愀瀀㼀 䌀愀渀 眀攀 最椀瘀攀 愀 瀀漀猀椀琀椀瘀攀ഀഀ combinatorial certificate that this happens? Expansivity from the previous ਍氀攀挀琀甀爀攀 最椀瘀攀猀 漀渀攀 挀漀渀猀琀爀愀椀渀琀Ⰰ 戀甀琀 琀漀 最椀瘀攀 琀栀攀 昀甀氀氀 愀渀猀眀攀爀 眀攀 渀攀攀搀 愀渀漀琀栀攀爀ഀഀ "energy" associated to graph maps, the stretch factor related to ਍爀甀戀戀攀爀 戀愀渀搀猀 愀渀搀Ⰰ 眀椀琀栀 猀漀洀攀 挀愀瘀攀愀琀猀Ⰰ 琀漀 攀氀攀挀琀爀椀挀愀氀 渀攀琀眀漀爀欀猀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=apple-converted-space> 

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September 8 2023 – Oliver ਍䨀攀渀欀椀渀猀漀渀 㰀⼀戀㸀⠀儀甀攀攀渀 䴀愀爀礀Ⰰ 唀渀椀瘀攀爀猀椀琀礀 漀昀 䰀漀渀搀漀渀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 㐀㨀　　ⴀ㔀㨀　　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Room 1114, Sciences ਍䈀甀椀氀搀椀渀最 一漀⸀ ㄀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: ਍圀栀攀渀 猀椀洀瀀氀攀 椀猀 戀攀猀琀㨀 䔀爀最漀搀椀挀 漀瀀琀椀洀椀稀愀琀椀漀渀Ⰰ 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀匀琀甀爀洀椀愀渀㰀⼀猀瀀愀渀㸀ഀഀ orbits, and ergodic dominance*Title: ਍䌀漀洀瀀甀琀愀戀椀氀椀琀礀 椀渀 䌀漀洀瀀氀攀砀 䐀礀渀愀洀椀挀猀 愀渀搀 䌀漀洀瀀氀攀砀 䄀渀愀氀礀猀椀猀⨀⨀⨀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

Abstract: In this talk, I will ਍搀椀猀挀甀猀猀 挀漀洀瀀甀琀愀戀椀氀椀琀礀 焀甀攀猀琀椀漀渀猀 愀爀椀猀椀渀最 椀渀 䌀漀洀瀀氀攀砀 䐀礀渀愀洀椀挀猀 愀渀搀 䌀漀洀瀀氀攀砀ഀഀ Analysis. More specifically, I will discuss the existence of non-computable ਍瀀漀氀礀渀漀洀椀愀氀 䨀甀氀椀愀 猀攀琀猀 愀渀搀 漀昀 瀀漀氀礀渀漀洀椀愀氀 䨀甀氀椀愀 猀攀琀猀 眀椀琀栀 愀爀戀椀琀爀愀爀椀氀礀 栀椀最栀 挀漀洀瀀甀琀愀琀椀漀渀愀氀ഀഀ complexity. I will also discuss the conditions necessary and sufficient for the ਍挀漀洀瀀甀琀愀戀椀氀椀琀礀 漀昀 挀漀渀昀漀爀洀愀氀 洀愀瀀猀 愀渀搀 琀栀攀椀爀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䌀愀爀愀琀栀攀漀搀漀爀礀㰀⼀猀瀀愀渀㸀ഀഀ extensions. The computability of another central object of Complex Analysis, ਍琀栀攀 䠀愀爀洀漀渀椀挀 洀攀愀猀甀爀攀Ⰰ 眀椀氀氀 愀氀猀漀 戀攀 搀椀猀挀甀猀猀攀搀⸀ 䄀 挀爀愀猀栀 挀漀甀爀猀攀 椀渀 最攀渀攀爀愀氀ഀഀ Computability Theory will be given. The talk is based on joint projects with M. ਍䈀爀愀瘀攀爀洀愀渀 ⠀倀爀椀渀挀攀琀漀渀⤀Ⰰ 䄀⸀ 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䜀氀甀挀欀猀愀洀㰀⼀猀瀀愀渀㸀 ⠀一漀爀琀栀眀攀猀琀攀爀渀⤀Ⰰ 䌀⸀ഀഀ Rojas (Universidad Catolica de Chile), and M. ਍夀愀洀瀀漀氀猀欀礀 ⠀唀渀椀瘀攀爀猀椀琀礀 漀昀 吀漀爀漀渀琀漀⤀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=apple-converted-space> 

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October 27 2023 – Guy C. David (Ball ਍匀琀愀琀攀 唀渀椀瘀攀爀猀椀琀礀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 㤀㨀　　ⴀ㄀　㨀　　愀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom  (ID:861 9382 6449, Password: 647898)Title: ਍䔀洀戀攀搀搀椀渀最猀Ⰰ 挀愀爀瀀攀琀猀Ⰰ 挀甀爀瘀攀猀Ⰰ 愀渀搀 挀漀渀昀漀爀洀愀氀 搀椀洀攀渀猀椀漀渀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

Abstract: We discuss two results on ਍戀椀ⴀ䰀椀瀀猀挀栀椀琀稀 攀洀戀攀搀搀椀渀最猀 漀昀 洀攀琀爀椀挀 猀瀀愀挀攀猀⸀ 吀栀攀 昀椀爀猀琀 椀猀 愀 ⠀渀攀最愀琀椀瘀攀⤀ 愀渀猀眀攀爀 琀漀 愀ഀഀ 1997 question of Heinonen and Semmes asking whether spaces that can be “folded” ਍椀渀琀漀 䔀甀挀氀椀搀攀愀渀 猀瀀愀挀攀 挀愀渀 戀攀 攀洀戀攀搀搀攀搀 戀椀ⴀ㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䰀椀瀀猀挀栀椀琀稀氀礀㰀⼀猀瀀愀渀㸀⸀ഀഀ The second is a more general theorem showing that spaces with “thick” curve ਍昀愀洀椀氀椀攀猀 挀愀渀渀漀琀 栀愀瘀攀 戀椀ⴀ䰀椀瀀猀挀栀椀琀稀 攀洀戀攀搀搀椀渀最猀 椀渀 䔀甀挀氀椀搀攀愀渀 猀瀀愀挀攀Ⰰ 甀渀氀攀猀猀 琀栀攀礀ഀഀ admit some “infinitesimal splitting”. This latter result has some applications ਍琀漀 琀栀攀 猀琀甀搀礀 漀昀 挀漀渀昀漀爀洀愀氀 搀椀洀攀渀猀椀漀渀⸀ 吀栀椀猀 椀猀 樀漀椀渀琀 眀漀爀欀 眀椀琀栀 匀礀氀瘀攀猀琀攀爀ഀഀ Eriksson-Bique (University of Jyvaskyla).(UCLA)

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Time: 10:00-11:00am (Beijing

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σ_f on ਍㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀吀攀椀挀栀洀ﰀ氀氀攀爀㰀⼀猀瀀愀渀㸀 猀瀀愀挀攀 戀礀 瀀甀氀氀椀渀最 戀愀挀欀 挀漀洀瀀氀攀砀 猀琀爀甀挀琀甀爀攀猀⸀ഀഀ One way to study this pullback map is via a correspondence on moduli space. In ਍琀栀椀猀 琀愀氀欀 䤀 眀椀氀氀 搀椀猀挀甀猀猀 戀漀琀栀 琀栀攀 最攀渀攀爀愀氀 挀漀爀爀攀猀瀀漀渀搀攀渀挀攀 愀渀搀 猀漀洀攀 猀瀀攀挀椀愀氀ഀഀ cases. Specifically, I will give a careful description of the correspondence ਍昀漀爀 琀栀攀 挀愀猀攀 眀栀攀爀攀 䄀 栀愀猀 昀漀甀爀 瀀漀椀渀琀猀 愀渀搀 琀栀攀 瀀漀猀琀挀爀椀琀椀挀愀氀 猀攀琀 ⠀眀栀椀挀栀 椀猀 愀ഀഀ subset of A) has three points. The diagram I will present has been previously ਍甀猀攀搀 戀礀 䬀攀氀猀攀礀 愀渀搀 䰀漀搀最攀 琀漀 猀琀甀搀礀 焀甀愀搀爀愀琀椀挀 吀栀甀爀猀琀漀渀 洀愀瀀猀 眀椀琀栀 昀攀眀 瀀漀猀琀挀爀椀琀椀挀愀氀ഀഀ points, and I also use it in a forthcoming paper proving special cases of the ਍挀甀爀瘀攀 愀琀琀爀愀挀琀漀爀 瀀爀漀戀氀攀洀 昀漀爀 爀愀琀椀漀渀愀氀 洀愀瀀猀 眀椀琀栀 昀漀甀爀 洀愀爀欀攀搀 瀀漀椀渀琀猀⸀ 吀椀洀攀 㰀猀瀀愀渀ഀഀ class=GramE>permitting I will further discuss the results of this paper.Shaosong Liu (刘少松) (University ਍漀昀 刀漀挀栀攀猀琀攀爀㰀戀㸀⤀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 ㄀　㨀　　ⴀ㄀㄀㨀　　愀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom  (ID: 891 8732 ਍　㜀㄀㈀Ⰰ 倀愀猀猀眀漀爀搀㨀 㘀㄀㔀㜀㈀㌀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀琀氀攀㨀 䰀攀攀ⴀ夀愀渀最 稀攀爀漀猀 愀渀搀 椀琀猀ഀഀ critical behavior for the Ising model on Sierpinski gasket

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Abstract: It is known that Ising ਍洀漀搀攀氀 漀渀 匀椀攀爀瀀椀渀猀欀椀 最愀猀欀攀琀 椀猀 攀砀愀挀琀氀礀 猀漀氀瘀愀戀氀攀⸀ 䤀渀 ㄀㤀㠀　猀 䜀攀昀攀渀Ⰰ 㰀猀瀀愀渀ഀഀ class=SpellE>Aharony, Shapir and Mandelbrot ਍最愀瘀攀 琀栀攀 爀攀挀甀爀猀椀瘀攀 爀攀氀愀琀椀漀渀 昀漀爀 猀甀挀栀 洀漀搀攀氀 愀渀搀 愀爀最甀攀搀 渀漀 昀椀渀椀琀攀ⴀ琀攀洀瀀攀爀愀琀甀爀攀ഀഀ phase transitions. In this talk I will give a proof for exact formulas of its ਍瀀愀爀琀椀琀椀漀渀 昀甀渀挀琀椀漀渀猀⸀ 吀栀攀 爀攀挀甀爀猀椀瘀攀 爀攀氀愀琀椀漀渀 椀猀 挀漀渀樀甀最愀琀攀搀 琀漀 愀 焀甀愀搀爀愀琀椀挀 洀愀瀀Ⰰഀഀ from which I can give a precise description about zero distributions and ਍琀栀攀爀洀漀搀礀渀愀洀椀挀 氀椀洀椀琀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀愀渀甀愀爀礀 㤀 ㈀　㈀㐀 ጀ†㰀戀㸀倀攀欀欀愀 㰀猀瀀愀渀ഀഀ class=SpellE>Pankka (University of Helsinki)

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Time: 4:00-5:00pm (Beijing Picard theorems in quasiconformal geometry***

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Abstract: The classical Picard ਍琀栀攀漀爀攀洀 昀爀漀洀 挀漀洀瀀氀攀砀 愀渀愀氀礀猀椀猀 猀琀愀琀攀猀 琀栀愀琀 愀 栀漀氀漀洀漀爀瀀栀椀挀 洀愀瀀 昀爀漀洀 琀栀攀 挀漀洀瀀氀攀砀ഀഀ plane to itself omits at most one point. Of course, the exponential mapping ਍猀栀漀眀猀 琀栀愀琀 倀椀挀愀爀搀✀猀 琀栀攀漀爀攀洀 椀猀 猀栀愀爀瀀⸀ 䤀渀 栀椀最栀攀爀 搀椀洀攀渀猀椀漀渀愀氀 焀甀愀猀椀挀漀渀昀漀爀洀愀氀ഀഀ geometry (i.e. when we allow bounded distortion of the conformal structure), ਍倀椀挀愀爀搀✀猀 琀栀攀漀爀攀洀 愀瀀瀀攀愀爀猀 椀渀 琀栀攀 昀漀爀洀 漀昀 刀椀挀欀洀愀渀✀猀 琀栀攀漀爀攀洀 ⠀㄀㤀㠀　⤀㨀 䄀ഀഀ quasiregular mapping from the Euclidean space to itself the cardinality of the ਍漀洀椀琀琀攀搀 猀攀琀 椀猀 戀漀甀渀搀攀搀 戀礀 愀 挀漀渀猀琀愀渀琀 搀攀瀀攀渀搀椀渀最 漀渀氀礀 漀渀 琀栀攀 搀椀洀攀渀猀椀漀渀 愀渀搀ഀഀ distortion of the mapping. Also this result is sharp. ਍䤀渀 琀栀椀猀 琀愀氀欀Ⰰ 䤀 眀椀氀氀 搀椀猀挀甀猀猀 琀栀攀猀攀 倀椀挀愀爀搀 琀礀瀀攀 爀攀猀甀氀琀猀 愀渀搀 琀栀攀椀爀 挀漀栀漀洀漀氀漀最椀挀愀氀ഀഀ counterparts for quasiregular mappings from Euclidean spaces to closed ਍刀椀攀洀愀渀渀椀愀渀 洀愀渀椀昀漀氀搀猀Ⰰ 攀猀瀀攀挀椀愀氀氀礀 琀栀攀 昀漀氀氀漀眀椀渀最 爀攀猀甀氀琀㨀 䤀昀 愀 挀氀漀猀攀搀 刀椀攀洀愀渀渀椀愀渀ഀഀ n-manifold admits a quasiregular mapping from the n-Euclidean space, then its ਍搀攀 刀栀愀洀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀挀漀栀漀洀漀氀漀最礀㰀⼀猀瀀愀渀㸀 攀洀戀攀搀猀 椀渀琀漀 琀栀攀 攀砀琀攀爀椀漀爀 愀氀最攀戀爀愀ഀഀ of R^n.Pekka Pankka (University of Helsinki)

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Time & Location:Time)   Hou Zhu Lou 1124, Beijing ਍一漀爀洀愀氀 唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀愀渀甀愀爀礀 ㄀㄀ ㈀　㈀㐀 㐀㨀　　ⴀ㔀㨀　　瀀洀ഀഀ (Beijing Time)   Hou Zhu Lou 1124, Beijing ਍一漀爀洀愀氀 唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀愀渀甀愀爀礀 ㄀㈀ ㈀　㈀㐀 㐀㨀　　ⴀ㔀㨀　　瀀洀ഀഀ (Beijing Time)   BICMR 77201, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㄀㘀⸀　瀀琀㬀ഀഀ font-family:"Arial",sans-serif;color:black'>Title: Quasiregular mappings -- ਍瀀愀猀琀Ⰰ 瀀爀攀猀攀渀琀Ⰰ 昀甀琀甀爀攀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㄀㘀⸀　瀀琀㬀ഀഀ font-family:"Arial",sans-serif;color:black'>

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Abstract: In this minicourse, I ਍眀椀氀氀 搀椀猀挀甀猀猀 焀甀愀猀椀挀漀渀昀漀爀洀愀氀 最攀漀洀攀琀爀礀 昀爀漀洀 琀栀攀 瀀漀椀渀琀 漀昀 瘀椀攀眀 漀昀 洀愀瀀瀀椀渀最 琀栀攀漀爀礀Ⰰഀഀ especially of quasiregular mappings. Quasiregular mappings can be viewed as a ਍最攀渀攀爀愀氀椀稀愀琀椀漀渀 漀昀 瀀氀愀渀愀爀 栀漀氀漀洀漀爀瀀栀椀挀 洀愀瀀瀀椀渀最猀 椀渀 栀椀最栀攀爀 搀椀洀攀渀猀椀漀渀愀氀 刀椀攀洀愀渀渀椀愀渀ഀഀ geometry. The prefix 'quasi-' refers to bounded distortion of the conformal ਍猀琀爀甀挀琀甀爀攀⸀ 吀栀攀 挀氀愀猀猀椀挀愀氀 䰀椀漀甀瘀椀氀氀攀✀猀 琀栀攀漀爀攀洀 猀琀愀琀攀猀 琀栀愀琀 挀漀渀昀漀爀洀愀氀 洀愀瀀瀀椀渀最猀ഀഀ between domains of a Euclidean n-space are restrictions of Möbius assuptions.

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Lecture I: Quasiregular mappings ਍愀渀搀 挀漀渀昀漀爀洀愀氀 椀渀瘀愀爀椀愀渀琀猀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䰀攀挀琀甀爀攀 䤀䤀㨀 儀甀愀猀椀爀攀最甀氀愀爀 搀礀渀愀洀椀挀猀ഀഀ and cohomological results

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Lecture III: Quasiregular curves ਍愀渀搀 倀椀挀愀爀搀 琀栀攀漀爀攀洀猀 戀攀礀漀渀搀 挀漀渀昀漀爀洀愀氀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㤀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀愀渀甀愀爀礀 ㄀㤀 ㈀　㈀㐀 ጀ†㰀戀㸀䴀椀欀栀愀椀氀 㰀猀瀀愀渀ഀഀ class=SpellE>Hlushchanka (University of Amsterdam)

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Time: 5:00-6:00pm (Beijing

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February 2 2024 – Richard Sharp ਍㰀戀㸀⠀㰀⼀戀㸀唀渀椀瘀攀爀猀椀琀礀 漀昀 圀愀爀眀椀挀欀㰀戀㸀⤀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 㐀㨀　　ⴀ㔀㨀　　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom  (ID: 863 6531 ਍㤀㤀㘀㤀Ⰰ 倀愀猀猀眀漀爀搀㨀 㔀㜀㌀　㠀㐀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: Random ਍眀愀氀欀猀 漀渀 最爀漀甀瀀猀Ⰰ 愀洀攀渀愀戀椀氀椀琀礀 愀渀搀 爀愀琀椀漀 氀椀洀椀琀 琀栀攀漀爀攀洀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>

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Abstract: A famous result of Kesten ਍昀爀漀洀 ㄀㤀㔀㤀 爀攀氀愀琀攀猀 猀礀洀洀攀琀爀椀挀 爀愀渀搀漀洀 眀愀氀欀猀 漀渀 挀漀甀渀琀愀戀氀攀 最爀漀甀瀀猀 琀漀 愀洀攀渀愀戀椀氀椀琀礀⸀ഀഀ Precisely, provided the support of the walk generates the group, the ਍瀀爀漀戀愀戀椀氀椀琀礀 漀昀 爀攀琀甀爀渀 琀漀 琀栀攀 椀搀攀渀琀椀琀礀 椀渀 ㈀渀 猀琀攀瀀猀 搀攀挀愀礀猀 攀砀瀀漀渀攀渀琀椀愀氀氀礀 昀愀猀琀 椀昀ഀഀ and only if the group is not amenable. This led to many analogous ਍ᰀ愠洀攀渀愀戀椀氀椀琀礀 搀椀挀栀漀琀漀洀椀攀猀ᴀⰠ 昀漀爀 攀砀愀洀瀀氀攀 昀漀爀 琀栀攀 猀瀀攀挀琀爀甀洀 漀昀 琀栀攀 䰀愀瀀氀愀挀椀愀渀 漀昀ഀഀ manifolds and critical exponents of discrete groups of isometries. I will ਍瀀爀攀猀攀渀琀 愀 瘀攀爀猀椀漀渀 漀昀 琀栀攀 搀椀挀栀漀琀漀洀礀 昀漀爀 渀漀渀ⴀ猀礀洀洀攀琀爀椀挀 眀愀氀欀猀⸀ 䤀 眀椀氀氀 愀氀猀漀 搀椀猀挀甀猀猀ഀഀ a new ratio limit theorem for amenable groups. This is joint work ਍眀椀琀栀 刀栀椀愀渀渀漀渀 䐀漀甀最愀氀氀 ⠀䐀甀爀栀愀洀 唀渀椀瘀攀爀猀椀琀礀⤀⸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㤀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䴀愀爀挀栀 㠀 ㈀　㈀㐀 ጀ†㰀戀㸀䌀栀爀椀猀琀椀愀渀 圀漀氀昀㰀⼀戀㸀ഀഀ (City University of New York)

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Time: 9:00-10:00pm (Beijing

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In particular, we consider coded shifts which include ਍猀攀瘀攀爀愀氀 眀攀氀氀ⴀ欀渀漀眀渀 攀砀愀洀瀀氀攀猀 漀昀 猀栀椀昀琀 猀瀀愀挀攀猀⸀ 䄀 挀漀搀攀搀 猀栀椀昀琀 猀瀀愀挀攀 椀猀 搀攀昀椀渀攀搀 愀猀ഀഀ the closure of all bi-infinite concatenations of words from a fixed countable ਍最攀渀攀爀愀琀椀渀最 猀攀琀⸀ 圀攀 搀攀爀椀瘀攀 猀甀昀昀椀挀椀攀渀琀 挀漀渀搀椀琀椀漀渀猀 昀漀爀 琀栀攀 甀渀椀焀甀攀渀攀猀猀 漀昀 洀攀愀猀甀爀攀猀ഀഀ of maximal entropy based on the partition of the coded shift into its ਍猀攀焀甀攀渀琀椀愀氀 猀攀琀 ⠀猀攀焀甀攀渀挀攀猀 琀栀愀琀 愀爀攀 挀漀渀挀愀琀攀渀愀琀椀漀渀猀 漀昀 最攀渀攀爀愀琀椀渀最 眀漀爀搀猀⤀ 愀渀搀 椀琀猀ഀഀ residual set (sequences added under the closure). We also discuss flexibility ਍爀攀猀甀氀琀猀 昀漀爀 琀栀攀 攀渀琀爀漀瀀礀 漀渀 琀栀攀 猀攀焀甀攀渀琀椀愀氀 愀渀搀 爀攀猀椀搀甀愀氀 猀攀琀⸀ 䘀椀渀愀氀氀礀Ⰰ 眀攀 瀀爀攀猀攀渀琀ഀഀ a local structure theorem for intrinsically ergodic coded shift spaces which ਍猀栀漀眀猀 琀栀愀琀 漀甀爀 爀攀猀甀氀琀猀 愀瀀瀀氀礀 琀漀 愀 氀愀爀最攀爀 挀氀愀猀猀 漀昀 挀漀搀攀搀 猀栀椀昀琀 猀瀀愀挀攀猀 挀漀洀瀀愀爀攀搀ഀഀ to previous works by Climenhaga, Climenhaga and Thompson, and Pavlov. The ਍爀攀猀甀氀琀猀 瀀爀攀猀攀渀琀攀搀 椀渀 琀栀椀猀 琀愀氀欀 愀爀攀 樀漀椀渀琀 眀漀爀欀 眀椀琀栀 吀愀洀愀爀愀 䬀甀挀栀攀爀攀渀欀漀 愀渀搀 䴀愀爀琀椀渀ഀഀ Schmoll.

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April 12 2024 – Cristobal Rojas ਍㰀戀㸀⠀㰀⼀戀㸀倀漀渀琀椀昀椀挀椀愀 唀渀椀瘀攀爀猀椀搀愀搀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䌀愀琀漀氀椀挀愀㰀⼀猀瀀愀渀㸀 搀攀 䌀栀椀氀攀㰀戀㸀⤀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 㤀㨀　　ⴀ㄀　㨀　　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: Zoom  (ID: 889 2406 ਍㌀㐀㔀㜀Ⰰ 倀愀猀猀眀漀爀搀㨀 㜀㤀㈀㤀㌀㐀⤀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: Computational ਍琀爀愀挀琀愀戀椀氀椀琀礀 椀渀 琀栀攀 搀攀猀挀爀椀瀀琀椀漀渀 漀昀 琀礀瀀椀挀愀氀 愀猀礀洀瀀琀漀琀椀挀 戀攀栀愀瘀椀漀爀猀 椀渀 搀礀渀愀洀椀挀愀氀ഀഀ systemsA major goal in dynamical ਍猀礀猀琀攀洀 琀栀攀漀爀礀 椀猀 琀栀攀 挀氀愀猀猀椀昀椀挀愀琀椀漀渀 漀昀 猀礀猀琀攀洀猀 愀挀挀漀爀搀椀渀最 琀漀 琀栀攀 愀猀礀洀瀀琀漀琀椀挀ഀഀ behavior exhibited by a typical trajectory. The quadratic family is an example ਍漀昀 愀 挀氀愀猀猀 漀昀 猀礀猀琀攀洀猀 眀栀攀爀攀 琀栀攀 猀琀愀琀甀猀 漀昀 琀栀椀猀 挀氀愀猀猀椀昀椀挀愀琀椀漀渀 椀渀 戀漀琀栀 琀漀瀀漀氀漀最椀挀愀氀☀渀戀猀瀀㬀⠀愀琀琀爀愀挀琀漀爀猀⼀㰀⼀猀瀀愀渀㸀㰀眀戀爀ഀഀ style='font-variant-ligatures: normal;font-variant-caps: normal;orphans: 2; ਍琀攀砀琀ⴀ愀氀椀最渀㨀猀琀愀爀琀㬀眀椀搀漀眀猀㨀 ㈀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀ⴀ眀椀搀琀栀㨀 　瀀砀㬀琀攀砀琀ⴀ搀攀挀漀爀愀琀椀漀渀ⴀ琀栀椀挀欀渀攀猀猀㨀 椀渀椀琀椀愀氀㬀ഀഀ text-decoration-style: initial;text-decoration-color: initial;background-color: ਍眀栀椀琀攀㬀挀漀氀漀爀㨀⌀㈀㈀㈀㈀㈀㈀㬀昀漀渀琀ⴀ昀愀洀椀氀礀㨀䄀爀椀愀氀Ⰰ 䠀攀氀瘀攀琀椀挀愀Ⰰ 猀愀渀猀ⴀ猀攀爀椀昀㬀昀漀渀琀ⴀ猀椀稀攀㨀猀洀愀氀氀㬀ഀഀ font-style:normal;font-weight:400;letter-spacing:normal;text-indent:0px; ਍琀攀砀琀ⴀ琀爀愀渀猀昀漀爀洀㨀渀漀渀攀㬀眀栀椀琀攀ⴀ猀瀀愀挀攀㨀渀漀爀洀愀氀㬀眀漀爀搀ⴀ猀瀀愀挀椀渀最㨀　瀀砀✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀㰀猀瀀愀渀ഀഀ style='font-variant-ligatures: normal;font-variant-caps: normal;orphans: 2; ਍琀攀砀琀ⴀ愀氀椀最渀㨀猀琀愀爀琀㬀眀椀搀漀眀猀㨀 ㈀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀ⴀ眀椀搀琀栀㨀 　瀀砀㬀琀攀砀琀ⴀ搀攀挀漀爀愀琀椀漀渀ⴀ琀栀椀挀欀渀攀猀猀㨀 椀渀椀琀椀愀氀㬀ഀഀ text-decoration-style: initial;text-decoration-color: initial;float:none; ਍眀漀爀搀ⴀ猀瀀愀挀椀渀最㨀　瀀砀✀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀爀攀瀀攀氀氀攀爀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ lang=EN-US>) and statistical (physical measures) terms is essentially complete ਍ⴀⴀ 琀栀攀爀攀 愀爀攀 漀渀氀礀 愀 昀攀眀 瀀漀猀猀椀戀氀攀 搀椀昀昀攀爀攀渀琀ഀഀ asymptotic "regimes".  A natural question in this context ਍椀猀 琀栀攀 瀀漀猀猀椀戀椀氀椀琀礀 漀昀 甀猀椀渀最 挀漀洀瀀甀琀攀爀猀Ⰰ 昀漀爀 椀渀猀琀愀渀挀攀Ⰰ 琀漀 搀攀挀椀搀攀 眀栀椀挀栀 爀攀最椀洀攀 愀ഀഀ system belongs to, or to accurately describe the invariant objects relevant to ਍愀 最椀瘀攀渀 爀攀最椀洀攀⸀ 䤀渀 琀栀椀猀 琀愀氀欀 眀攀 眀椀氀氀 爀攀瘀椀攀眀 琀栀攀 猀琀愀琀攀 漀昀 琀栀攀 愀爀琀 爀攀最愀爀搀椀渀最 琀栀椀猀ഀഀ question in the case of the real quadratic family. 

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July 17 2024 – Guohua Wu (吴国华Time)    Location: BICMR 77201, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀㬀ⴀ眀攀戀欀椀琀ⴀ琀攀砀琀ⴀ猀琀爀漀欀攀㨀 ⌀　　　　　　✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀ഀഀ lang=EN-US style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'>Title: Undecidability ਍漀昀 眀漀爀搀 瀀爀漀戀氀攀洀猀⨀⨀⨀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㄀㘀⸀　瀀琀㬀ഀഀ font-family:"Arial",sans-serif;color:black'>

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Abstract: Turing proposed a ਍挀漀洀瀀甀琀愀琀椀漀渀愀氀 洀漀搀攀氀 椀渀 栀椀猀 瀀愀瀀攀爀 瀀甀戀氀椀猀栀攀搀 椀渀 ㄀㤀㌀㘀⸀ 吀栀椀猀 洀漀搀攀氀 椀猀 渀漀眀ഀഀ well-known as Turing machines. In the paper, Turing considered the halting ਍瀀爀漀戀氀攀洀 愀渀搀 瀀爀漀瘀攀搀 琀栀愀琀 椀琀 椀猀 渀漀琀 猀漀氀瘀愀戀氀攀⸀ 䈀礀 挀漀搀椀渀最 琀栀攀 栀愀氀琀椀渀最 瀀爀漀戀氀攀洀 椀渀琀漀ഀഀ the word problem, in 1955, Novikov constructed a finitely presentable group G ਍猀甀挀栀 琀栀愀琀 琀栀攀 眀漀爀搀 瀀爀漀戀氀攀洀 漀昀 䜀 椀猀 渀漀琀 猀漀氀瘀愀戀氀攀⸀ 䤀琀 椀猀 琀栀攀 昀椀爀猀琀 甀渀猀漀氀瘀愀戀氀攀ഀഀ problem in algebra. Boone provided a different construction in 1958. After all ਍琀栀攀猀攀Ⰰ 䠀椀最洀愀渀 瀀爀漀瘀攀搀 栀椀猀 昀愀洀漀甀猀 攀洀戀攀搀搀椀渀最 琀栀攀漀爀攀洀㨀 愀 昀椀渀椀琀攀氀礀 最攀渀攀爀愀琀攀搀 最爀漀甀瀀 䜀ഀഀ is recursively presentable if and only if it can be embedded into a finitely ਍瀀爀攀猀攀渀琀愀戀氀攀 最爀漀甀瀀⸀ 䠀椀最洀愀渀ᤀ猠 琀栀攀漀爀攀洀 最愀瘀攀 愀 挀氀漀猀攀 氀椀渀欀 戀攀琀眀攀攀渀 愀氀最攀戀爀愀 愀渀搀ഀഀ recursion theory. Higman proved that his embedding theorem implies the ਍一漀瘀椀欀漀瘀ⴀ䈀漀漀渀攀 琀栀攀漀爀攀洀⸀ 䤀渀 琀栀椀猀 琀愀氀欀Ⰰ 眀攀 眀椀氀氀 爀攀瘀椀攀眀 戀愀猀椀挀 椀搀攀愀猀 椀渀瘀漀氀瘀攀搀 椀渀ഀഀ theorems, and we will also show recent work on undecidability in algebra, with ਍攀砀愀洀瀀氀攀猀 愀渀搀 琀攀挀栀渀椀焀甀攀猀⸀ 㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀☀渀戀猀瀀㬀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀甀氀礀 ㈀㔀 ㈀　㈀㐀 ጀ†㰀戀㸀夀愀渀 䴀愀爀礀 䠀攀㰀⼀戀㸀 ⠀㰀愀ഀഀ href="https://www.ou.edu/" target="_blank">Title: The ਍嘀愀氀甀攀 漀昀 䠀愀甀猀搀漀爀昀昀 䐀椀洀攀渀猀椀漀渀 ⨀⨀⨀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀ഀഀ style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

Abstract: Hausdorff dimension is a ਍瀀漀猀椀琀椀瘀攀 爀攀愀氀 渀甀洀戀攀爀 琀栀愀琀 焀甀愀渀琀椀昀椀攀猀 琀栀攀 猀椀稀攀 漀昀 昀爀愀挀琀愀氀 猀攀琀猀⸀ 䤀渀 洀愀渀礀 挀愀猀攀猀ഀഀ this value isn't explicitly known, but can be estimated with different levels ਍漀昀 愀挀挀甀爀愀挀礀 ⠀搀攀瀀攀渀搀椀渀最 漀渀 琀栀攀 洀攀琀栀漀搀 甀猀攀搀⤀⸀ 圀攀 眀椀氀氀 搀攀猀挀爀椀戀攀 愀瀀瀀氀椀挀愀琀椀漀渀猀 琀漀ഀഀ different areas of mathematics, including number theory, geometry and dynamical ਍猀礀猀琀攀洀猀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀匀攀瀀琀攀洀戀攀爀 ㄀㠀 ㈀　㈀㐀 琀漀 匀攀瀀琀攀洀戀攀爀 ㈀㌀ഀഀ 2024 – Mark Pollicott (University of Warwick)

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Time & Location:Time)   BICMR 77201, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀匀攀瀀琀攀洀戀攀爀 ㈀　 ㈀　㈀㐀 ㌀㨀　　ⴀ㐀㨀　　瀀洀ഀഀ (Beijing Time)   BICMR Quan29, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀匀攀瀀琀攀洀戀攀爀 ㈀㌀ ㈀　㈀㐀 㐀㨀　　ⴀ㔀㨀　　瀀洀ഀഀ (Beijing Time)   BICMR 77201, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㄀㘀⸀　瀀琀㬀ഀഀ font-family:"Arial",sans-serif;color:black'>Title: Estimating Dimension, ਍攀渀琀爀漀瀀礀 愀渀搀 䰀礀愀瀀甀渀漀瘀 攀砀瀀漀渀攀渀琀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀ഀഀ 21.0pt;font-family:"微软雅黑",sans-serif;mso-bidi-font-family:"Times New Roman"; ਍挀漀氀漀爀㨀⌀㌀㌀㌀㌀㌀㌀㬀昀漀渀琀ⴀ眀攀椀最栀琀㨀渀漀爀洀愀氀✀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀栀㈀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀戀猀琀爀愀挀琀㨀 吀栀攀 攀猀琀椀洀愀琀椀漀渀 漀昀ഀഀ characteristic values in many problems in Dynamical Systems and related fields ਍⠀猀甀挀栀 愀猀 䰀礀愀瀀甀渀漀瘀 攀砀瀀漀渀攀渀琀猀Ⰰ 洀攀琀爀椀挀 攀渀琀爀漀瀀礀Ⰰ 搀椀洀攀渀猀椀漀渀 漀昀 愀琀琀爀愀挀琀漀爀猀Ⰰ 攀琀挀⸀⤀⸀ 圀攀ഀഀ will describe approachs based on ergodic theory and ਍⠀洀漀爀攀 猀瀀攀挀椀昀椀挀愀氀氀礀⤀ 琀栀攀爀洀漀搀礀渀愀洀椀挀 昀漀爀洀愀氀椀猀洀 愀渀搀 琀爀愀渀猀昀攀爀 漀瀀攀爀愀琀漀爀猀⸀☀渀戀猀瀀㬀 吀栀攀ഀഀ first lecture will concentrate on dimension. The second lecture will focus on

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Time: 9:00-10:00pm (Beijing

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Title: Thermodynamic ਍䘀漀爀洀愀氀椀猀洀 愀琀 稀攀爀漀 琀攀洀瀀攀爀愀琀甀爀攀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀戀猀琀爀愀挀琀㨀 圀攀 挀漀渀猀椀搀攀爀 吀栀攀爀洀漀搀礀渀愀洀椀挀ഀഀ Formalism for sub-shifts of finite type and present some results for the case ਍眀栀攀爀攀 琀栀攀 琀攀洀瀀攀爀愀琀甀爀攀 最漀攀猀 琀漀 稀攀爀漀⸀ 刀攀猀甀氀琀猀 昀漀爀 琀栀攀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀稀攀爀漀ഀഀ temperature case could be separated in two groups: properties of limits ਍愀渀搀 猀攀氀攀挀琀椀漀渀⸀ 匀漀洀攀 瀀爀漀瀀攀爀琀椀攀猀 愀爀攀 眀攀氀氀 欀渀漀眀渀 愀猀 眀攀氀氀 愀猀 琀栀攀椀爀 挀漀渀渀攀挀琀椀漀渀猀 眀椀琀栀ഀഀ Ergodic Optimization (for example: any zero temperature ਍氀椀洀椀琀 漀昀 愀渀 攀焀甀椀氀椀戀爀椀甀洀 瀀爀漀戀愀戀椀氀椀琀礀 椀猀 愀 洀愀砀椀洀椀稀椀渀最 瀀爀漀戀愀戀椀氀椀琀礀 椀渀 䔀爀最漀搀椀挀ഀഀ Optimization). On the other hand, there exist some open problems concerning ਍猀攀氀攀挀琀椀漀渀 ⠀昀漀爀 攀砀愀洀瀀氀攀㨀 搀漀攀猀 琀栀攀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀稀攀爀漀 琀攀洀瀀攀爀愀琀甀爀攀㰀⼀猀瀀愀渀㸀ഀഀ limit of eigen-functions always exist?). We propose also to present some recent ਍爀攀猀甀氀琀猀 愀渀搀 漀瀀攀渀 瀀爀漀戀氀攀洀猀 挀漀渀挀攀爀渀椀渀最 猀攀氀攀挀琀椀漀渀 昀漀爀 瀀攀爀琀甀爀戀攀搀 瀀漀琀攀渀琀椀愀氀猀⸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䐀攀挀攀洀戀攀爀 ㈀ ㈀　㈀㐀 ጀ†㰀戀㸀䨀椀愀渀礀漀渀最 儀椀愀漀 㰀⼀戀㸀⠀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=s5> ਍瀀氀愀渀 昀漀爀 爀攀猀攀愀爀挀栀椀渀最 愀爀攀愀猀 漀昀 䨀甀氀椀愀 猀攀琀猀 漀昀 挀漀洀瀀氀攀砀 搀礀渀愀洀椀挀愀氀 猀礀猀琀攀洀猀⨀⨀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀戀猀琀爀愀挀琀㨀 圀攀 挀漀渀猀椀搀攀爀 琀栀攀 愀爀攀愀ഀഀ problem on Julia sets of complex dynamical systems. Firstly, we introduce the ਍挀漀渀挀攀瀀琀猀 漀昀 瀀攀爀椀漀搀椀挀 挀礀挀氀攀 愀渀搀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䈀爀礀甀渀漀㰀⼀猀瀀愀渀㸀 渀甀洀戀攀爀 椀渀 琀栀攀ഀഀ theory of complex dynamics, also formulate Douady's ਍倀氀愀渀 昀漀爀 爀攀猀攀愀爀挀栀椀渀最 琀栀攀 愀爀攀愀 瀀爀漀戀氀攀洀⸀ 䈀甀昀昀 愀渀搀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䌀栀爀椀琀愀琀✀猀㰀⼀猀瀀愀渀㸀ഀഀ theorem is one of the important results in this plan, and the BC-conjecture is ਍愀渀 漀瀀攀渀 瀀爀漀戀氀攀洀 昀漀爀 愀 氀漀渀最ⴀ琀椀洀攀 瀀漀猀攀搀 戀礀 䈀甀昀昀 愀渀搀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䌀栀爀椀琀愀琀㰀⼀猀瀀愀渀㸀⸀ഀഀ In the last, we give an outline of the proof of BC-conjecture by Jianyong Qiao ਍☀愀洀瀀㬀 䠀漀渀最礀甀 儀甀⸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䐀攀挀攀洀戀攀爀 㜀 ㈀　㈀㐀 ጀ†㰀猀瀀愀渀ഀഀ class=SpellE>Qiandu He (何芊杜) (Peking ਍唀渀椀瘀攀爀猀椀琀礀⤀㰀戀㸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀洀攀㨀 ㄀㨀　　ⴀ㌀㨀　　瀀洀 ⠀䈀攀椀樀椀渀最 㰀猀瀀愀渀ഀഀ class=GramE>Time)    Location: BICMR JiaYiBing ਍䰀攀挀琀甀爀攀 䠀愀氀氀Ⰰ 倀攀欀椀渀最 唀渀椀瘀攀爀猀椀琀礀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀琀氀攀㨀☀渀戀猀瀀㬀䌀漀洀瀀甀琀愀戀椀氀椀琀礀 漀昀ഀഀ equilibrium states

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Abstract: Equilibrium state is an ਍椀洀瀀漀爀琀愀渀琀 椀渀瘀愀爀椀愀渀琀 洀攀愀猀甀爀攀 椀渀 琀栀攀 爀攀猀攀愀爀挀栀 漀昀 琀栀攀爀洀漀搀礀渀愀洀椀挀 昀漀爀洀愀氀椀猀洀⸀ 䤀渀 琀栀椀猀ഀഀ talk, we will investigate the problem of construction approximations for ਍攀焀甀椀氀椀戀爀椀甀洀 猀琀愀琀攀猀 漀昀 愀 最攀渀攀爀愀氀 昀愀洀椀氀礀 漀昀 挀漀洀瀀甀琀愀戀氀攀 搀礀渀愀洀椀挀愀氀 猀礀猀琀攀洀猀⸀ഀഀ Specifically, we device two complementary general approaches to study the ਍挀漀洀瀀甀琀愀戀椀氀椀琀礀 漀昀 攀焀甀椀氀椀戀爀椀甀洀 猀琀愀琀攀猀 昀漀爀 渀漀渀ⴀ甀渀椀昀漀爀洀氀礀 攀砀瀀愀渀搀椀渀最 猀礀猀琀攀洀猀⸀ഀഀ Moreover, we will also show some applications in complex and counter examples.

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December 15 2024 – will

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December 17 2024 – Eero Time)    Location: BICMR 77201, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀戀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀椀琀氀攀㨀☀渀戀猀瀀㬀✀圀栀愀琀 椀猀ഀഀ multiplicative chaos? ***

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Abstract: We try to give a gentle ਍椀渀琀爀漀搀甀挀琀椀漀渀 琀漀 䜀愀甀猀猀椀愀渀 洀甀氀琀椀瀀氀椀挀愀琀椀瘀攀 挀栀愀漀猀 ⠀䜀䴀䌀⤀⸀ 䄀洀漀渀最 攀砀愀洀瀀氀攀猀 漀昀 椀琀猀ഀഀ applications, we consider someHanyun Zheng () (Peking University)

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Time: 1:00-3:00pm (Beijing

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Title: On quasiregular maps ਍愀渀搀 焀甀愀猀椀猀礀洀洀攀琀爀椀挀 甀渀椀昀漀爀洀椀稀愀琀椀漀渀 瀀爀漀戀氀攀洀 椀渀 栀椀最栀攀爀 搀椀洀攀渀猀椀漀渀猀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䄀戀猀琀爀愀挀琀㨀 儀甀愀猀椀爀攀最甀氀愀爀 洀愀瀀猀 愀爀攀ഀഀ considered a higher dimensional analog of holomorphic maps. In this talk, we ਍最椀瘀攀 愀 戀爀椀攀昀 椀渀琀爀漀搀甀挀琀椀漀渀 琀漀 焀甀愀猀椀爀攀最甀氀愀爀 洀愀瀀猀 愀渀搀 爀攀氀愀琀攀搀 琀漀瀀椀挀猀⸀ 圀攀 椀渀琀爀漀搀甀挀攀ഀഀ an analog of expanding Thurston maps on n-spheres, generalizes a quasisymmetric ਍甀渀椀昀漀爀洀椀稀愀琀椀漀渀 琀栀攀漀爀攀洀 漀昀 攀砀瀀愀渀搀椀渀最 吀栀甀爀猀琀漀渀 洀愀瀀猀 戀礀 䈀漀渀欀 愀渀搀 䴀攀礀攀爀⸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䨀愀渀甀愀爀礀 㐀 ㈀　㈀㔀 ጀ†㰀戀ഀഀ style='mso-bidi-font-weight:normal'>Guillaume Tahar (BIMSA)

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Time: 1:00-3:00pm (Beijing

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Title: Closed geodesics in ਍搀椀氀愀琀椀漀渀 猀甀爀昀愀挀攀猀 㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀䴀猀漀一漀爀洀愀氀 猀琀礀氀攀㴀✀戀愀挀欀最爀漀甀渀搀㨀眀栀椀琀攀✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀ഀഀ style='font-size:14.0pt;font-family:"Arial",sans-serif;color:black'>Abstract: ਍䐀椀氀愀琀椀漀渀 猀甀爀昀愀挀攀猀 愀爀攀 最攀渀攀爀愀氀椀稀愀琀椀漀渀猀 漀昀 琀爀愀渀猀氀愀琀椀漀渀 猀甀爀昀愀挀攀猀 椀渀 眀栀椀挀栀 琀栀攀ഀഀ transition maps of the atlas are both translations and homotheties. Just as ਍琀爀愀渀猀氀愀琀椀漀渀 猀甀爀昀愀挀攀猀 愀爀椀猀攀 愀猀 猀甀猀瀀攀渀猀椀漀渀猀 漀昀 䤀渀琀攀爀瘀愀氀 䔀砀挀栀愀渀最攀 䴀愀瀀猀Ⰰ 搀椀氀愀琀椀漀渀ഀഀ surfaces arise as suspensions of Affine Interval Exchange Maps.

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In this talk, we will prove that ਍愀氀氀 搀椀氀愀琀椀漀渀 猀甀爀昀愀挀攀猀 挀漀渀琀愀椀渀 瀀攀爀椀漀搀椀挀 琀爀愀樀攀挀琀漀爀椀攀猀⸀ 吀栀攀 瀀爀漀漀昀 爀攀氀椀攀猀 漀渀 愀渀ഀഀ analysis of the possible degenerations of dilation structures at the boundary ਍漀昀 琀栀攀 洀漀搀甀氀椀 猀瀀愀挀攀Ⰰ 眀栀椀挀栀 椀猀 洀愀搀攀 洀漀爀攀 挀栀愀氀氀攀渀最椀渀最 戀礀 琀栀攀 愀戀猀攀渀挀攀 漀昀 愀渀ഀഀ invariant notion of length in this geometric setting.We will provide the necessary background on translation and ਍搀椀氀愀琀椀漀渀 猀甀爀昀愀挀攀猀Ⰰ 琀栀攀 愀猀猀漀挀椀愀琀攀搀 搀礀渀愀洀椀挀愀氀 猀礀猀琀攀洀猀Ⰰ 愀猀 眀攀氀氀 愀猀 琀栀攀 猀琀爀甀挀琀甀爀攀ഀഀ of their moduli spaces.  

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February 18 2025 to February 22 ਍㈀　㈀㔀 ጀ†㰀戀㸀匀礀氀瘀攀猀琀攀爀 䔀爀椀欀猀猀漀渀ⴀ㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀䈀椀焀甀攀㰀⼀猀瀀愀渀㸀㰀⼀戀㸀 㰀戀㸀⠀㰀⼀戀㸀唀渀椀瘀攀爀猀椀琀礀ഀഀ of Jyväskylä)

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Time & Location:Time)  BICMR 77201, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䘀攀戀爀甀愀爀礀 ㈀　 ㈀　㈀㔀 ㌀㨀　　ⴀ㐀㨀　　瀀洀ഀഀ (Beijing Time)   BICMR 77201, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䘀攀戀爀甀愀爀礀 ㈀㈀ ㈀　㈀㔀 ㌀㨀　　ⴀ㐀㨀　　瀀洀ഀഀ (Beijing Time)   BICMR 77201, Peking ਍唀渀椀瘀攀爀猀椀琀礀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀栀㈀ 猀琀礀氀攀㴀✀洀愀爀最椀渀㨀　挀洀✀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀 猀琀礀氀攀㴀✀昀漀渀琀ⴀ猀椀稀攀㨀㄀㘀⸀　瀀琀㬀ഀഀ font-family:"Arial",sans-serif;color:black'>Title: Conformal dimension and ਍愀渀愀氀礀猀椀猀 漀渀 昀爀愀挀琀愀氀猀 昀漀爀 䰀愀愀欀猀漀ⴀ琀礀瀀攀 猀瀀愀挀攀猀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀ഀഀ style='font-size:16.0pt;font-family:"Arial",sans-serif;color:black'> ਍ഀഀ

Abstract: Given a metric space X, ਍椀琀猀 挀漀渀昀漀爀洀愀氀 搀椀洀攀渀猀椀漀渀 椀猀 琀栀攀 椀渀昀椀洀甀洀 漀昀 愀氀氀 䠀愀甀猀搀漀爀昀昀 搀椀洀攀渀猀椀漀渀猀 漀昀 猀瀀愀挀攀猀ഀഀ quasisymmetric to it. This invariant is natural and arises in porblem arises in constructing Dirichlet forms for ਍猀攀氀昀ⴀ猀椀洀椀氀愀爀 昀爀愀挀琀愀氀猀⸀ 吀栀攀 搀椀昀昀椀挀甀氀琀礀 氀椀攀猀 椀渀 琀栀攀 昀愀挀琀 琀栀愀琀 挀攀爀琀愀椀渀 洀漀搀甀氀甀猀ഀഀ problems are very difficult to compute explicitly. This difficulty further has ਍氀攀搀 琀漀 洀愀渀礀 漀瀀攀渀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀瀀爀漀戀氀攀洀猀㰀猀瀀愀渀 猀琀礀氀攀㴀✀洀猀漀ⴀ猀瀀愀挀攀爀甀渀㨀礀攀猀✀㸀ꀀഀഀ that have until now resisted efforts at their resolution. We have ਍渀漀眀 戀攀攀渀 愀戀氀攀 琀漀 猀漀氀瘀攀 猀漀洀攀 漀昀 琀栀攀猀攀⸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀圀椀琀栀 刀椀欀甀 䄀渀琀琀椀氀愀 眀攀 搀椀猀挀漀瘀攀爀攀搀 愀ഀഀ simple construction of a large class of fractals, Laakso-type fractals, for ਍眀栀椀挀栀 攀瘀攀爀礀琀栀椀渀最 戀攀挀漀洀攀猀 洀甀挀栀 洀漀爀攀 琀爀愀渀猀瀀愀爀攀渀琀⸀ 伀渀攀 挀愀渀 挀漀洀瀀甀琀攀 琀栀攀椀爀 挀漀渀昀漀爀洀愀氀ഀഀ dimension, characterize the attainment of conformal dimension, describe ਍䐀椀爀椀挀栀氀攀琀 昀漀爀洀猀 攀砀瀀氀椀挀椀琀氀礀 愀渀搀 攀砀栀椀戀椀琀 洀愀渀礀 戀攀栀愀瘀椀漀爀猀 瀀爀攀瘀椀漀甀猀氀礀 漀渀氀礀ഀഀ conjectured to hold. In this three lecture sequence I ਍眀椀氀氀 最椀瘀攀 愀渀 椀渀琀爀漀搀甀挀琀椀漀渀 琀漀 琀栀攀猀攀 猀瀀愀挀攀猀Ⰰ 愀渀搀 甀猀攀 琀栀攀洀 琀漀 琀攀氀氀 礀漀甀 愀戀漀甀琀ഀഀ analysis on fractals and conformal dimension.  ਍㰀⼀猀瀀愀渀㸀吀栀攀 最漀愀氀 椀猀 琀漀 琀攀氀氀 眀栀愀琀 琀栀攀猀攀 猀瀀愀挀攀猀 愀爀攀Ⰰ 愀渀搀 甀猀攀 琀栀攀洀 愀猀 猀椀洀瀀氀攀ഀഀ examples to explain why Carrasco-Piaggio is true, and how one constructs ਍䐀椀爀椀挀栀氀攀琀 昀漀爀洀猀 漀渀 昀爀愀挀琀愀氀猀⸀ 夀漀甀 搀漀 渀漀琀 渀攀攀搀 琀漀 欀渀漀眀 愀渀礀琀栀椀渀最 愀戀漀甀琀 挀漀渀昀漀爀洀愀氀ഀഀ dimension or analysis on fractals as my goal will be to tell you what they are ਍琀栀爀漀甀最栀 琀栀攀猀攀 攀砀愀洀瀀氀攀猀⸀ 唀猀甀愀氀氀礀 㰀猀瀀愀渀 挀氀愀猀猀㴀䜀爀愀洀䔀㸀戀漀琀栀 漀昀 琀栀攀猀攀㰀⼀猀瀀愀渀㸀 愀爀攀 瘀攀爀礀ഀഀ difficult, but these spaces allow one to make the complicated objects explicit ਍椀渀 愀 瘀攀爀礀 椀氀氀甀洀椀渀愀琀椀渀最Ⰰ 礀攀琀 昀愀椀爀氀礀 猀椀洀瀀氀攀Ⰰ 眀愀礀⸀㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀愀氀欀 ㄀㨀 吀栀攀 爀攀猀漀氀甀琀椀漀渀 漀昀 愀ഀഀ conjecture of Kleiner and Laakso-type spaces

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Talk 2: Conformal dimension for ਍䰀愀愀欀猀漀ⴀ琀礀瀀攀 猀瀀愀挀攀猀Ⰰ㰀漀㨀瀀㸀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀吀愀氀欀 ㌀㨀 䄀渀愀氀礀猀椀猀 漀渀 䘀爀愀挀琀愀氀猀 昀漀爀ഀഀ Laakso-type spaces

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Joint work with Riku Anttila, ਍刀礀漀猀甀欀攀 匀栀椀洀椀稀甀 愀渀搀 䰀愀猀猀椀 㰀猀瀀愀渀 挀氀愀猀猀㴀匀瀀攀氀氀䔀㸀刀愀椀渀椀漀㰀⼀猀瀀愀渀㸀⸀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㜀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㄀　㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀㰀漀㨀瀀㸀☀渀戀猀瀀㬀㰀⼀漀㨀瀀㸀㰀⼀猀瀀愀渀㸀㰀⼀瀀㸀ഀഀ ਍㰀瀀 挀氀愀猀猀㴀瀀㘀㸀㰀猀瀀愀渀 挀氀愀猀猀㴀猀㄀㸀㰀猀瀀愀渀 氀愀渀最㴀䔀一ⴀ唀匀㸀䰀愀猀琀 唀瀀搀愀琀攀搀㨀㰀⼀猀瀀愀渀㸀㰀⼀猀瀀愀渀㸀㰀猀瀀愀渀ഀഀ class=apple-converted-space> 

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2/14/2025

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