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HZljl F  0* .h$*ijWe will first provide an overviewawRegge calculus and its use in numerical class #Dquantum gravity. a(second part[thei|concenta on structing* various c tensors. a si lattice. doiby Ta Tgression from 2D to 3Don 4 DJ[s.4emphasis here J(o introduce0hybrid volume)>red:)"se:0 are a combin, of elements !GsB! dual21circum%qic $5758U defini ri)lco- Einstein-IKeA@|three different ways, notably us!our newz%ADlud)|s!QO(approximate.4omorphism symmeaMrel)tAQeo tracK0Bianchi identAVZvPK ZFTIndex/Slide5/Slide5.iwaV4/*#ccbcd "*(0B$ Transitionnone?)(*2: :(b:c:d: b] da % x*#   >  BZC $NDC%l%@A-?0(8$h*b.=T H _6C 0>D<## @<** (2 :Pr  Rj*2j1<^D@ B"58J!b!n@ 0@ >Dq<*e* M]PrEp!I{ Fj1"e!$@ B"516-B! jjjU-rE- PVsCC h D~D6 At0CZbj  I4*  Q8 qC@8D%% ;27a Aj7"]7A67AZ7U9(N*E=*.QCJ!NO  mv$  =m) -QDdi!A6- D$Zb%: 6AZAp NDC!2 >j".C2V6 %ܕNb>)2 @CXaU!)2)>)@6)V)=)~*qT13D Simplicial Ricci T< of a Dual Edge * u Jy.` , PK ZF|2p p Index/Slide2.iwal !X/|**}||}}|Ȁ||} "*(0B$ Transitionnone?)4?*|2|:4:}::Ȁ||||JK P$}|j*2j >  gD@ B%ˆ|%@A-?0(8"5$8J B,Zjjjg @UXDoPC C05 XHCHCZ| - |)4*|   $o:BDD!3$h*b.=T !^ N!6C 0>D<|&|! ( *{{{}.~Collaborators Paul Alsing, AFRL/RI Rory Conboye and Shannon Ray, FAU Shing-Tung Yau and Jonathan McDonald, Harvard David X. Gu, Stony Brook Support: USAF Grants #FA8750-11-2-0089, #FA5750-15-2-0047 and the Griffiss Institute under VFRP Recent Manuscripts P. Al J. M u & WAM “The Simplicial Ricci Tensor’” Class. Quantum Grav. 28 (2011) 155007 (17pp). arXiv: 1107.2458v1 WAM, J.R, P.MX.D)/ S-T Yau,FFlow,Lomm. Math. Phys. 3294) 579. (1302.0804 Po0, M. Corne, X }= , S. E C. Tison,``B2@:An Example of a Neck Pinch Singularity in 3D,'' Geom., ImagingEZ0Comp. 1 no, 3303-331.)K!E  &,``EquivalencRo$Hamilton's- for 3Dpetries%b.,33-366.*Z aƨ~  $ H v ~   D  24 <    : ,BO  `({  <.  PZ S}r  4en     *       8  } d" http://arxiv.org/abs/u} 2$*Y} ѸX ٸ@YB@j.@2V}}ն<*ť*Q250U PY5=R! l 0ApAv&%5h C4Zljl:[6[" @&2 =j":[@6[V|[<*6JZ 5l - 6GD~ B/β5-zb (uDZb | mK<*| ||$*&> 8ion built arounAA  grantF]|the VFRP program administered by!GrB< d. Dprinciples in this |@are W. A. Miller,FAUD.uk- ?4׸)$2536CF5B-A952-459E-A870-7E7C6B24B6D7 }Af* |2 qC@8De  a828Aj8"}8A  AZ8$||UGL*|m:5yRyJ|ZJ M,  .''r %Laa!L2L =jL}}=J |*{{i8 Acknowledgeme: and Refer s5g{RgB EO {.k h] *npq^7Dc`U5Cj,pbqjpPK ZFC1 Index/Slide1.iwa $^50*$ "*(0B$ Transitionnone?)L*2::: 0::Z[1 %l*2"" >  8FADC 8o[C%%@A-?0(8"5$8J CPZ"b" 4*   $C6DE AA$h*b.=T  _:A 0A >D<   \L*#* /2 :J #Pr" #  d<    <* $* hThe purpose of this talk lo utilize our new definition .Je discrete Ricci tensor for a piecewise simplicial manifold and to derive tFL8flow (DRF) equav Y0d-dimensional.] lattice. ذmathematics used comes from Regge calculus orcivalently DEC. Advantage robust numerical tool2 > 9 ⶖC]93! !b= = gCj B:sB6 BZ a*a`Atheorya polyhedraej4related geometIFmethods are attractive not only in their own right. They pave the way a general Psurfaces. Surely, it[c@always that we maqfe9emiaurvedKia#aboutby passato& limiHowever, bsB@ dusE searchingj ilar>>:.'' 
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These areSRF equats. * A02 :Pr en -JjQe!|$@ B05)anAD(''U,*ICQ: ADT Simplicial Ricci FlowUL for any compact piecewise flat si?manifold, , can be expressed as an6X(each edge, ,UJ BM4  d e6   A  ! "]A B.hD0Yb 5Mb#v*y C#BA2sB8p5*y a y myMV#Z"}f"Am  J(*# .BG FDC9bao}=b "y C.J0##u$*i _Overview+pRegge Calculus Curvature on a.\Geometry Toward Discrete.in D>2*m{$ aw>J.}m:##еA!|С# BqG#DaGdCAE}G^u5mBPaBaBaB #Fa~33H.5 O!$6 '.y%=bA.=b"}o.V''u* RRF D" : W 9 dual-. (SRF)6,ngmX,,af 0circumcentric lattic) , *m( Z-+ *- .(  .(>#2( ) 4 @ A}:r #:rh PL@G ]brr2r=bb ""P 2bVb$##UdS oR< "J6| 2;:; #2;qC@8 D Aa.=f; "9; 2xV;$##5; L*')= 5+NJ '.=6'  @     . #BܡA@ 2\ =j\"=\@.V\#.<*5SJ~^PK ZF ?Index/MasterSlide-3.iwaL#* "*(0B$ Transitionnone?)*2RBlank^D94c*( ! stickyNoteParagraphStyleID % level2HeadlinePar>'d tableCellText-0-bodyK4 $ titlejH(  shape9I  ](Name-0-text 1jZ'E footerRowN ! 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'-0F#  #    boxW   JBd + $kBGObjectPlaceholderJ-%5v JE9W!:c- head204v N!A(% BrBeL 0 )kBGSlideNumberv!3vA-0 |Column ( !&2 6*:0>0Ck05DC06=f0"06V!.T> 6VV  &4C CPK ZFR;;Index/Slide4/Slide3.iwa7P.)* "*(0B$ Transitionnone?)(*2: :::H O  |j*2j >  eD@@ B%%@A-?0(8"5$8J BHZjjj  D*# ## *Here we apply this 2D formula to a subdivided icosahedra triangulation projected out onto a sphere of radius R. The calc:\shows the convergence ofdaverage Gaussian curvature@its correspondingC(tinuum valuG 1/R^2. T@work is an exampl#>}^discussed by Cheeger-M\”uller and Scharader (published c.~70’s). * 2 : !Pr UJqEJMMPA9C هC23CM5MMDC@DZbj (IT0*#  q>8DA A!$h*b.=T a" _:A 0A >D0##u L*'i 5VJ1'Z ' )7@   # J3l-p!MpA&C35- CZljl% J*## VNCkAfGC۔f(## U,v*,2Q?!% Nfق*, !.nCZAAA,  #ZYj]nC6nCN #FC)!"jaZ%NDmA2=j"_."6V~#:$ 6$NwZ$PK ZFr66Index/Slide4/Slide2.iwa2jZGB*6  / ! ?(0Jvv2{Gz]v2 F d"*(0B$ Transitionnone- ?) 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CB(ity appears%NNDC2=jN_=NC6NCZN.N*1N`%IMPORTANT POINTS & PREGN(QUESTIONS* m }1i B $  2( PK ZF"L7  Index/Slide7.iwa B o* "*(0B$ Transitionnone?)(*2: :),Pq*2 >  CB C)D%%@A-?0(8"58JDDDZbj (   4*   qC@8D AA$h*b.=T ( _>A 0A >D< \L*#* /2 :J #Pr" #  d,  :N6N CNDB2N =jN"=NB6NBZN$5NS*)LTIMPORTANT OBSERVATION5^R^ZS:;>;BZC5;C;F; Cf%+:;C2;V.;<*%;5$N$Z$ Mv0* .j M*l8 Just for fun!{VZ{PK ZF0OOIndex/Slide6.iwaKR)$* "*(0B$ Transitionnone?)4*2: :,xl*2 >  CB  IC%%@A-?0(8"58J KC CZj( 4*   9PA B$h*b.=T    _:B 0B >D<   `D*Part 3* 02 :Pr  -  i<*   $* XThe third and final pofs talk is based on new research into the simplicial analog A|e (discrete Ricci flow) DRF equaAs. B *Ptensor was published recently in Classical And Quantum Gravity by Alsing, McDonal Miller@$2011, but . repntsun. . 5uljeljQ2 
(Illustrated for D=3) Examples * i4 4 zNP}X1  8en C M S b y  ):U  :UC%FD=f"=6V .$*)0bOverview of 6 oward D6%in! 
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