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  2. Rigidity Theorems for Hyperbolic Groups Alexis Marchand Abstract The…

    https://www.dpmms.cam.ac.uk/~aptm3/docs/maths/2020-RigidityTheoremsHyperbolicGroups.pdf
    6 May 2020: d. (p, Im c|[b′,b]. ) D0. Therefore there exist s [a,a′] and t [b′,b] such that d (p,c(s)) ,d (p,c(t)) D0. ... By definition, form,n 1, there exists bm,n B such that amRnbm,n.
  3. COMPLEX ANALYSIS EXAMPLES 1, LENT 2020 Neshan Wickramasekera. Please…

    https://www.dpmms.cam.ac.uk/study/IB/ComplexAnalysis/2019-2020/2ca_IB_ex1_2020.pdf
    28 Jan 2020: Regarding T as a map from C into C by identifyingR2 with C in the usual way, show that there exist unique complex numbers A, B such thatfor every z C, ... Can such a map everbe a Möbius transformation? 9. Let U C be open and let f = uiv : U C.
  4. GroupsOscar Randal-Williams…

    https://www.dpmms.cam.ac.uk/~or257/teaching/IAGroups/2019/groups.pdf
    2 Dec 2020: identity). (G3) For each a G there exists a b G such that a b = e. ... ii) If a G then e a = a. (iii) If a,b,b′ G are such that a b = e and a b′ = e, then b = b′.
  5. Mapping Class GroupsLectures by Henry WiltonNotes by Alexis Marchand…

    https://www.dpmms.cam.ac.uk/~aptm3/docs/lecture-notes/PartIII-MappingClassGroups.pdf
    30 May 2020: Abigon for α,β is an embedded (closed) disc D S such that D(αβ) = D = ab where a αand b β are arcs. ... Therefore, after iterating, we eventually findψ Zd 6 Mod(S) such that φαi ψ αi.
  6. Rigidity Theorems for Hyperbolic Groups Alexis Marchand Abstract The…

    https://www.dpmms.cam.ac.uk/~aptm3/docs/maths/2020-RigidityTheoremsHyperbolicGroups-Abridged.pdf
    6 May 2020: By definition, form,n 1, there exists bm,n B such that amRnbm,n. ... R such that dX (ai,b) =dA (ai,αi) ε2 dB (βi,b).
  7. Category TheoryLectures by Peter JohnstoneNotes by Alexis Marchand…

    https://www.dpmms.cam.ac.uk/~aptm3/docs/lecture-notes/PartIII-CategoryTheory.pdf
    8 Jun 2020: A representation ofthis functor, if it exists, is called a product of A and B: it consists of an object AB together withmorphisms AB π1 A and AB π2 B such ... iI of objects of (A G) indexed by a set I, such that every B ob (A G) has amorphism Bi B in
  8. Michaelmas Term 2020 Linear Algebra: Example Sheet 2 of ...

    https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2020-2021/example-sheet-2.pdf
    19 Oct 2020: 5. (a) Suppose that f Matn,n(F) is such that f(AB) = f(BA) for all A,B Matn,n(F) and f(I) = n.Show that f is the ... fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj.
  9. Algebraic TopologyLectures by Jacob RasmussenNotes by Alexis Marchand …

    https://www.dpmms.cam.ac.uk/~aptm3/docs/lecture-notes/PartIII-AlgebraicTopology.pdf
    28 May 2020: F :X I Y (where I = [0, 1]) such that F(, 0) = f0 and F(, 1) = f1. ... Let f : Sn Sn and q Sn be such that f1 (f(q)) = {q1,.
  10. Riemann Surfaces Henry Wilton∗ Michaelmas 2020 Contents 1 Analytic ...

    https://www.dpmms.cam.ac.uk/~hjrw2/RS%20lectures.pdf
    30 Nov 2020: f. There is a chart (φ,U) about p withφ(p) = 0 such that. ... of isolated zeroes, there is an analytic functiong with g(0) 0 such that.
  11. ms.dvi

    https://www.dpmms.cam.ac.uk/~ik355/PAPERS/ms.pdf
    5 Jun 2020: Q(B(Xn1 ;D)) that such a match indeed occurs. In the case when no distortion is allowed, Q(B(Xn1 ;D)). ... the supremum is taken over all integers r 1 and all pairs of events A and B such that B 2 (Y 1rk),.
  12. Metric EmbeddingsLectures by András ZsákNotes by Alexis Marchand…

    https://www.dpmms.cam.ac.uk/~aptm3/docs/lecture-notes/PartIII-MetricEmbeddings.pdf
    3 Jun 2020: iii) f is a bilipschitz embedding if there exist a,b > 0 such that. ... A {1,. ,n} with |A| = t such that A(s) = {B A, |B| = s}is monochromatic.Definition 2.17 (Trees).
  13. TOPICS IN ANALYSIS (Lent 2020): Example Sheet 3 Comments, ...

    https://www.dpmms.cam.ac.uk/study/II/TopicsinAnalysis/2019-2020/topics-sheet3.pdf
    26 Feb 2020: n}, let the non-negative numbers A(n)k andthe ‘nodes’ x. (n)k [a,b] be given such that for each polynomial P , the error. ... such that for every f C([1, 1]), there exists a unique series.
  14. paper.dvi

    https://www.dpmms.cam.ac.uk/~ik355/PAPERS/jtp.pdf
    5 Jun 2020: Theorem Almost sure invariance principle Let X be a nitevalued stationary process such. ... fB t t g such that for any. R t B t O t as.
  15. Michaelmas Term 2020 Linear Algebra: Example Sheet 4 of ...

    https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2020-2021/example-sheet-4.pdf
    9 Nov 2020: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 13. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.
  16. Profinite GroupsLectures by Gareth WilkesNotes by Alexis Marchand…

    https://www.dpmms.cam.ac.uk/~aptm3/docs/lecture-notes/PartIII-ProfiniteGroups.pdf
    10 Jun 2020: jJ Gj such that pj f = qj, and we have f(Z) L. ... Let (Gj)jJ be a surjective inverse system such that G = limjJ Gj.
  17. Adaptive and non-linear MCMC algorithms

    https://www.dpmms.cam.ac.uk/~ik355/AWMCMC/AWMCMC_talks/EM.pdf
    6 Jun 2020: I A family of transition kernels {Pθ,θ Θ} such that, for all θ Θ,the target distribution π? ... Θ and a set A such that P(A) = 1 and.
  18. Part III Computability and Logic: 24 Lectures in 2020/21 ...

    https://www.dpmms.cam.ac.uk/~tef10/cam_only/partiiicomputability2020.pdf
    10 Oct 2020: b) If A is finite show that we can find such a B with B A. ... a) Can you define a ordered pair such that ρ(〈x,y〉) = max(ρ(x),ρ(y)) 1?(b) Can you define a ordered pair such that ρ(〈x,y〉) = max(
  19. ALGEBRAIC GEOMETRY, SHEET III: LENT 2020 Throughout this sheet, ...

    https://www.dpmms.cam.ac.uk/study/II/AlgebraicGeometry/2019-2020/AGExample3.pdf
    3 Mar 2020: Con-struct a morphism of smooth curves X P1 such that the preimage of consists ofexactly n points, with ramification indices given by a1,. ... Conclude thatf,g,f g,f λg must be perfect squares. (b) Prove the following: If f,g are polynomials in k[t]
  20. finalTR.dvi

    https://www.dpmms.cam.ac.uk/~ik355/PAPERS/finalTR.pdf
    5 Jun 2020: b) There exist (distinct) points fx0;x1; : : : ;xkg in I such that, for every permutation ofthe indices f0;1; : : : ;kg with not equal to the identity, we have. ... Then by Lemma 1 there is a (unique) < 0 such that (a) holds, and letting W be dened as in
  21. Lent Term 2020 T.A. Fisher Groups Rings and Modules: ...

    https://www.dpmms.cam.ac.uk/study/IB/GroupsRings%2BModules/2019-2020/grm-20-3.pdf
    24 Feb 2020: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elementsa and b in R is an element d in R such that d divides both ... a and b, and if c dividesboth a and b then c divides d.

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