for Lie olds group : the to going basics Friday Fish - PowerPoint PPT Presentation

classes Characteristic for Lie olds group : the to going basics Friday Fish 2410712020 ,

① - fixed M Motivagcio - s Mppalbdle classes :P ,VB→c4BE HMM ) Characteristic Chern ESM classes Cpk v. B → :

② - fixed M Motivagcio EFM Cite Chern CKCEIEHTM ) Cpk classes Bms v. → : - Well Chern theory Classifying space B - wjyxjei Rei Choose 7 : ' Qu Quinn EE f :NTM→EndfE ) I curvature → I I A- dwttfw ,w ] t B M n > - - detziztrttf-EG.CH ' i CEHFB ) i i ) f Crye )EH" ( M ) ,

③ Motivation : ( Marius Rui ) Characteristic CA ) : classes EE Rep for - YA ) : : Hay (g) cry E) EH " Image HIA ) → VE →

④ Motivation : ( Marius Rui ) Characteristic CA ) : classes EE Rep for Construction :P TIAKTCEI TIE ) → : → : The Local Etu TIA ) frame e agitated of the . := I wite C' IAI ④ gln . . ( Ew ) :=trlw÷weIEH" " ' (A) we Car → . LIRE Es travel tray ) # de X ud X trlwf )= des V trave ) - - weft I ? ? ? ) Replace by we

⑤ Definition : § .EE/2epal5l:49:ExTEy ) ⇐ G÷µ§¥E rt ÷ , . ' HIGH HIGLIED HCG ) Defa :

⑥ co chains Definition at GIGUE ) : Hakan ' HCGLIED Defa HCG ) E=MxEn→GLlEt56ln=Gln( G) roof - id Locally : - , :E×→Eyh E As It tt M → * tallest C' C Gen ) Ccg )

⑦ G- Lie BG Cohomology and of → group : R Cl B. G ) Wto k ) - s Gh : , um ) RGhN C- ( Gln ) - Computation C of Coger , → matrices ) ( Polar decomposition of ch formulas classes local for ur . .

⑧ BG Cohomology of A G ! Ch P µ , ¥ IF . classes " Gppaecrbdle : f*HtBG ) - Shulman Model for HTBG ) Bott complex E. G : : simplicial - Eat riieoai IIE " 't ' .ci n' Ep*G=G as Epa → → - - T Td T Edyta ) -4 r 't EOG ) IYER ) → - Td Td Td .GS#CEnG1Er4EaG ) ICE t -

⑨ BG Cohomology of - Shulman - simplicial : Model for HTBG ) Bolt complex E. G : ryiaa tIEoGhry " EpG=G' AG as , . , t → . = GP T BPG T Td .GS#CEnG1Ed4EaG)--iix:r9-tEpGHtt-tEpG)Lx:tttEpf)-ir9-fEpG Edyta ) -8 I 't EOG ) I' l Eeg ) → ix. - ↳ → xeg : Td Td Td ICE ) - algebra DIE . G ) - dog of → - . G) basic )= HINE HIBG ) U ICE A . G) Keri Kerk → - . basic

⑧ - Well Chern theory algebra algebra Lie 9- Well us : 'f=SPg*④NP¥y* WP piezo - algebra - deg on Soy Ix trivial contraction onto ; ' g → : - L×=adI Xeg , g*¥r' =D 'lEoG ) Lela ) (G) F- connection - left . MC inv -

① - Well Chern theory WP '9-=SPg*④N¥* algebra Well : - algebra :W÷W 9- dg , Lx ix. W ' ' " Xtg W : → -0,1 Lieu ) =D 'lEoG ) (G) of connection F- - Wog -4 site .G ) . spaces Thmttexeev - Meinrenken ) g- dg .nu 'S SHBG ) - algebras 9- da $99 cohom In .

④ - Well Chern theory tell Wog -9 - sp 9- dg DENKI F- Lie LG ) : sites ) . - Compact G Connected Thmttexeev - Mein renken ) , - algebras 9- da h fogginess Is ' I BG ) a equiv . . Cohoon In . - connected Thru I S G KCG . ) . compact subgroup Max , . Then heck ) k with ( Wg ) r . equiv ICB SHE . G) r ah .GL Is - basic . bas .

⑤ - Well Chern theory 9*-71161-7 Leela ) ICE Wg g- : . Thru ( S G Gmax . ) .BG/lsah.egniv KC , R=Lie( k ) . compact connected - , ( Walk . basic 'll Altar - basic

④ - Well Chern theory - heck ) Thru ( S G Gmax . ) KC . compact R connected - , , ( Walk ftp.GIlsah.egniv ICE . G) r.ba 1 . basic . . . Ifan :%4kbasrIlEoGlr ftp.GKCCG ) 2 basic . - S Van Est Integration I Melnrenken . )

④ Description of Had Glen cochalns at he UhI=un=fAegHA*= - At Uh )cGLn . compact Max , lighten "" . basic I ) Clan , Polar decomposition p=fAEgLn/A= # I - ALI t ALI 9 Ln A want = top : -

④ Description of Had Glen cochalns at Polar decamp - At P=fAE9lnlA= # I Uni -_fAeglnIA*= . : A --Az#tAtzA 9 Ln want = top : b. WIN [ Ulm , PKD , UIHDCUIN C. [ D. DKUIN a. , , PTR # IN Lemma If b. g=R④P ⇒ : = inv te basic - . HIND 'T r Itb Hlg & c ⇒ inv . .

④ Description of Had Glen cochalns at It lgln.VN/=MP*)wini-inv g Ln want = top : - t detttI-AI-a.IG.tn Characteristic coeffcge-15-glil.nu : Cat AHHHH 159 milk . .cn ] . . → Carlan :C sight r map : .net/Tt-blnlmvTKgt--cleOIzq.i ' Arkan ) Eeg .it/tii..sAzg.i)--EEttrlAhn' ' , TESLA - I

⑧ Description of Had Glen cochalns at : High .VN/=fNDYuim.FnulN9ln)ulht- gln win = top basic T a AE - A A*=A → Carlan :( 551ft r map : .net/Tt-'9lnlmvTKqI--cleOIzq.i .hn/Hl9ln.UlnD=AlUi.Us....hzn- Eeg .it/tii..sAzg.i)--EE7rlArni ' Ahearn ) ' , 8ES2q - I - ' IQs " login = 1155¥ )umm=H' . , Ip ) Luzg it * it

④ Description of Had Glen cochalns at I . bas .fm/UlIClG1nliHlGlnl--Nv..Vs....lkn..lh2q lightning ⇒ Vzq - I - I e :P Polar PUM ; P decomposition Glen - - A =e×U NEVIN XEP : , E GLYNN =p p ex extent ⇐ x a . EXUM :=et×UM to IR t → ,

⑧ Description of Had Glen cochalns at I . bas , " I → ' g In =p D) IN win ) 29--1 → @ → Gln PUM -_ lightskin = Rcolnlucmltrlss ) E C' ( Gln ) : - ICP ) ? " Vi atop vdexulfiraltrpll um , .mu tE¥' .FM/UIClGlnliHlGlnt--Nv..Vs....lkn..l/Nj#)umHnrU29-- A=e×v : etxep 's .HN#xlFfE.tH=trlxl ✓ L - I ' / ex extent ⇐ × a . EXUM :=et×UM te IR t → ,

④ e C' (5) vs . Class 1st IE ) The ch ' ¥ GIGUE ) ) ) Hudak HIGLIED Defa HCG ) : - Gink ) GLIEII " Gen E=MxE Locally : - - , → Eyes : Ex E As R -4 taken Mcginty ; C' C Gent Ccg ) YE , U¥EIkI=trlwkLt5 vitEHgf-trfx.gl I C' (A) Ag=e×9UjEagTEtg , ,xgtD wy-wzatftfwz.at ) in P

④ Questions : Ch . classes homotopy for rep to → Up . I Adjoint ) group olds classes Intrinsic for → Applications →