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3 Mar 2021: Geometry IB – 2020/21 – Sheet 4: Hyperbolic surfaces and Gauss-Bonnet [Circa Lectures 19–24].

26 Mar 2021: ρ(4, 4) =δ. 24(p12 p11 4p10 3p8 4p7 p6 4p5 3p4 4p2 p 1),. ... Thus. α(n,d) = pnσS(n). Nσ α(n,d | σ), (24). 13. andα(n,d | σ) = N1σ.

18 Jan 2021: 24. E-homology. Combining the vanishing line for E-homology with calculations ofSuslin for GL2(A), we obtain the following chart for E-homology:.

12 Jan 2021: INVERSE PROBLEMS FOR CONNECTIONS. GABRIEL P. PATERNAIN. Abstract. We discuss various recent results related to the inverse problem ofdetermining a unitary connection from its parallel transport along geodesics. 1. Introduction. Let (M,g) be a

15 Oct 2021: g|t=0 = dt2 h (3.24)tg|t=0 = 2k (3.25). provided > 0 is sufficiently small. ... 5. To establish local uniqueness we show that given any development of (Σ,h,k) itis possible to construct wave coordinates such that (3.24), (3.25) hold.

15 Oct 2021: 24 Chapter 2 Distributions. 2.2 Derivatives of distributions. Things are looking good for Property ii) because the dual space to a vector space isnaturally a vector space.

21 Sep 2021: 24) 2{i,j,b1,. ,bs, i} 4{i, i,b1,. ,bs,j} 2{j,i,b1,. ,bs, i} = 4A(i,j).If r,s > 1 then we instead obtain. ... bs by a2,. ,ar,a1in (24). This gives the factor (1)r. We deduce the result for [[ ]] from that for [ ] as before.

15 Oct 2021: Definition 24. Suppose M is at least k 1 regular. i) A Ckvector field is a Ckmap X : M TM such that at every point p M, wehave X(p)

15 Oct 2021: A spacetime is a four dimensional Lorentzian manifold. 24. 2.1 The metric and causal geometry 25. ... 2.24). Taking (2.22)(2.23)(2.24) and noting a cancellation between terms with onederivative falling on Z and one on W , we arrive at the result.

18 Jun 2021: Theorem 2.6.7. (Seifert) 2g(K) > deg K(t). 24 Knots, Polynomials, and Categorification.