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Google's PageRank vfexvzTOea^Rg; 4. v^\t~g^(uN[Ed"}-NS0 7s_:_ 74TIl_ 7Mf[`NHyers Ulam3z['`vi_ v^N[Q{|wQSOe zvQ3z['`0 Te N[EQS Hyers Ulam3z['`v;NT`` gNHNPtTSU\vzz0 74T f 7M\f[`NOP_Re z(Wuirf[0irtf[0Pgeyf[I{f[y-Nv^(u f[`NOP_Re zv8^(uRgb]OYϑel0SlelI{ege zv['`NS'Yenя'``0  7D[R`_e zv{l0 S<EfNR.Le Veque, "Numerical Methods for conservation Laws" 7"~^ 7Gauss-Bonnet and Poincare-Hopf Revisited This is an undergraduate reading course concluding with a senior thesis. Except in very rare cases, a senior thesis in mathematics is not expected to be an original contribution to mathematical research. However, originality of presentation is expected. One should study several sources of the subject until he/she thoroughly writes his/her own thesis. We are planning to revisit two famous results in both geometry and topology. These are Gauss-Bonnet and Poincare-Hopf theorems. Prerequisite are basic knowledge of differential geometry and algebraic topology. Topics are also able to be 'mixed & matched' or adapted to the level of the audience's knowledge and interest. References: [1] K. Burns and M. Gidea, Differential geometry and topology with a view to dynamical systems. Chapman and Hall/CRC, Boca Raton, 2005. [2] M.P. do Carmo, Differential forms and applications, Springer-Verlag, New York, 1994 [3]J.W. Milnor, Topology from the differentiable viewpoint. Princeton Univ. Press, Princeton, 1997 [4]W. Zhang, Lectures on Chern-Weil Theory and Witten deformations, World Scientific, Singapore, 2001 7 _k 7xGaloist(WQUOT{/gNpe Nv^(uR:NQ*NN 9hnctQTR b 1. NpetepeW@x 2. p-adicpeWW@xTGalois h:yRek 3. p-adic _Re zW@x 4. -iWf~eQ 5. QUOGaloistTW,g 7H2206H4206 N 6-8HGX301N 2-4H2106BHGX501N 6-8HGX106H2207N 6-8H2104AV 6-8H2210H4208H4408H2205H2208H2217H2215H2111H2209H2113AN 11-13d^(uyf[-NvS ,gs;Nr^wQ g[Èofvpef[irtS0 ^gf[uYt@bf[vpef[t/fegnN[E gTvxvzbg^SN(ueg㉳Q0@w͑W{Q f[u`T㉳QR0 7OP_Re zSvQc6Rt YegnNirt0Rf[0uir0Sf[0Pge0~Nm0{tI{f[yvsaSN(uOP_Re z\O:Npef[!jWegc0OP_Re z-Nvc6R/fcǏS_v c6RQpe O_OP_Re zvngNNHQ~[vBl08^vc6RSbc'`0‰'`0G['`NSgOc6RI{0 ,gs;NxNN~xQOP_Re zv['`tNSc6Rt wQSOSbyr_ t Cauchy-Kowalevskaya[t Holmgren[t c'`S‰'` ^\/Oyr/UN'`el [vPelI{0 zW@xpete z ;NSe.s [1] Coron J.-M., Control and Nonlinearity, 2007. [2] Evans, Lawrence C., Partial differential equations, 1998. [3] Li Tatsien, Controllability and Observability for Quasilinear Hyperbolic Systems, 2010. [4] Lions J.-L., Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev., 30(1988), 1-68. [5] Russell D. 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