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Jack Thorne
https://www.dpmms.cam.ac.uk/~jat58/17 May 2024: On the average number of 2-Selmer elements of elliptic curves over F. ... Elliptic curves over Q. are modular. J. Eur. Math. Soc. 21 (2019), no. -
Is Cambridge biased?
https://www.dpmms.cam.ac.uk/~wtg10/biased.html5 May 2004: Finally, suppose that the average of f is significantly higher over A than it is over B. ... This probability increases with t, and the increase is very considerable when t is a couple of standard deviations higher than the average of f over B. -
MATHEMATICAL TRIPOS: PART IA Lent 2024 PROBABILITY JRNExample Sheet…
https://www.dpmms.cam.ac.uk/study/IA/Probability/2023-2024/pex3.pdf11 Jan 2024: 3. Let X be a Poisson random variable of parameter λ (0, ). (a) By optimizing the estimate of Question 2(b) over β, show that, for all x λ,. ... yet appeared? (b) What is the probability that the entire culture dies out eventually? -
The beginnings of Galois theory
https://www.dpmms.cam.ac.uk/~wtg10/galois.html29 Jan 2002: It says that the dimension of Q(b) over Q equals the dimension of Q(a) over Q times the dimension of Q(b) over Q(a). ... But what is the dimension of Q(b) over Q(a)? It means the dimension of the vector space of all linear combinations of powers of b, -
Michaelmas Term 2023-24 Numbers and Sets: Examples Sheet 1 ...
https://www.dpmms.cam.ac.uk/study/IA/Numbers%2BSets/2023-2024/numset1_2023.pdf12 Oct 2023: Write down its negation insymbolic form. (Here m, n, a, b should be understood as ranging over all natural numbers.). ... m n a b (n m) [(a = 1) (b = 1) (ab 6= n)]. -
Michaelmas Term 2023 Julian Sahasrabudhe Linear Algebra: Example…
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2023-2024/lin-alg-ex4-2023.pdf24 Nov 2023: 1. The square matrices A and B over the field F are congruent if B = PT AP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(. -
Solving cubic equations
https://www.dpmms.cam.ac.uk/~wtg10/cubic.html25 Apr 2002: 2. =b? To answer that, we need to put it back in the original form, by multiplying out the bracket and taking b over to the left hand side. ... If we wish to carry this approach over to the cubic, then we should have clear answers to the following two -
Michaelmas Term 2023 Julian Sahasrabudhe Linear Algebra: Example…
https://www.dpmms.cam.ac.uk/study/IB/LinearAlgebra/2023-2024/lin-alg-ex2-2023.pdf30 Oct 2023: 5. Let A and B be n n matrices over a field F. ... Which over R? 8. Find the eigenvalues and give bases for the eigenspaces of the following complex matrices: 1 1 00 3 20 1 0. -
Michaelmas Term 2023 T.A. Fisher Galois Theory: Example Sheet ...
https://www.dpmms.cam.ac.uk/study/II/Galois/2023-2024/galois_theory-23-1.pdf10 Oct 2023: Suppose that α has minimal polynomial f over Q, and let β = α4.Find a, b, c Q such that β = a bα cα2. ... The elements α andβ are then said to be algebraically dependent over K.]. -
Publications | Department of Pure Mathematics and Mathematical…
https://www.dpmms.cam.ac.uk/publications?page=3317 Jul 2024: 2019). 148,. 103. (doi: 10.1090/proc/14749). Cut-off phenomenon for the axb Markov chain over a finite field. ... M Gross, B Siebert. (2019). …. 34. …. Department of Pure Mathematics and Mathematical Statistics. -
PART II REPRESENTATION THEORYSHEET 4 Unless otherwise stated, all ...
https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2016-2017/repex4.pdf10 Jan 2017: PART II REPRESENTATION THEORYSHEET 4. Unless otherwise stated, all vector spaces are finite-dimensional over C. ... : a,b,x Fp. of matrices over the finite field Fp (p prime). -
DIFFERENTIAL GEOMETRY (PART III) MICHAELMAS 2021 EXAMPLE SHEET 2 ...
https://www.dpmms.cam.ac.uk/files/study/III/DifferentialGeometry/2021-2022/DGSheet2.pdf1 Nov 2021: DIFFERENTIAL GEOMETRY (PART III) MICHAELMAS 2021. EXAMPLE SHEET 2. 1. Let T , U, and V be finite-dimensional vector spaces over a field K. ... Show that M M is trivial.(b) Show that for all n we have TSn R = Rn1 over Sn. -
ALGEBRAIC TOPOLOGY (PART III) MICHAELMAS 2019 EXAMPLE SHEET 4 ...
https://www.dpmms.cam.ac.uk/study/III/Algebraic%20Topology/2019-2020/2019Ex4.pdf4 Dec 2019: vector bundles over B, and that Ui is a. ... 9. Construct a 3-dimensional real vector bundle over S4 which has no nonvanishing section. -
Automata & Formal LanguagesMichaelmas Term 2023 Part II of ...
https://www.dpmms.cam.ac.uk/study/II/AutomataAndFormalLanguages/2023-2024/M23_AFL_ES2.corrected.pdf23 Oct 2023: Prof. Dr. B. Löwe. Example Sheet #2. (14) Work over the alphabet Σ = {0, 1} and for n 1 consider. ... 16) Let L and M be languages over an alphabet Σ and consider the set equation X = LX M. -
1 Algebraic Topology Part III, 2015-16: Sheet 4 1. ...
https://www.dpmms.cam.ac.uk/study/III/Algebraic%20Topology/2018-2019/AT(III)2018-sheet4.pdf21 Nov 2018: form over R is equivalent to a direct sum of copies of the form(. ... b) Prove that if E and F are oriented vector bundles over a space X, their Euler classessatisfy eEF = eE eF. -
1 Algebraic Topology Part III, 2018-19: Sheet 3 1. ...
https://www.dpmms.cam.ac.uk/study/III/Algebraic%20Topology/2018-2019/AT(III)2018-sheet3.pdf11 Nov 2018: bundle of realrank one) over the circle, and prove that bundle is non-trivial. ... 7. Let p : E B be a fibre bundle over a path-connected space B with fibre F = p1(b).Suppose that H(F) is finitely generated and free, and that -
ON MODULAR UNITS A. J. Scholl Introduction. In [6], ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/u.pdf29 Jan 2010: More generally, for any compact open subgroup K of Gf , there is a modular curve MK defined over Qwith. ... Thus over Z[1/q], uq is nonvanishing, and so in particular uq O(Y0(q)Q) O. -
ANALYSIS II—EXAMPLES 4 Mich. 2014 The questions marked with ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2014-2015/14sheet4revised.pdf27 Feb 2015: Nj=1 ‖γ(tj) γ(tj1)‖ where the sup is taken over all. finite partitions 0 = t0 < t1 <. < tN = 1. (i) Give an example for which (γ) =. If γ is continuously ... b) = inf (γ), wherethe inf is taken over all continuous γ: [0, 1] E with γ(0) = a, -
Relaxing stratification Thomas ForsterDPMMS, CMS Wilberforce…
https://www.dpmms.cam.ac.uk/~tef10/relaxing.pdf29 Mar 2011: affordsus no variables (as it might be ‘F’ and ‘G’) to range over the second orderobjects captured by the branching quantifiers which would enable us to say‘F = G’. ... 8). Let us instantiate ‘u’ for the lower-case variables in (8) -
A note on trilinear forms for reducible representations and ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/trilinear.pdf29 Jan 2010: The vectors {ug | g G} span Sp over k, since φ is a generator forIndGB δ. ... We work in the category MQ Q of Chow motives over Qwith coefficients in Q. -
TOPICS IN SET THEORY: Example Sheet 3 1 Department ...
https://www.dpmms.cam.ac.uk/study/III/2014-15/TopicsinSetTheory/TST%20Example%20Sheet%203%202014.pdf19 Dec 2014: A filter F over κ is normal if for every {Xα F : α < κ}, the diagonal. ... ii) Prove Ulam’s theorem: there is no κ+ -saturated κ+ -complete ideal over κ+. -
Independence for Partition Regular Equations
https://www.dpmms.cam.ac.uk/~par31/preprints/indeppr-abs.html9 Dec 2005: A matrix A is said to be partition regular (PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all ... Those matrices which are partition regular over the positive integers were completely -
Remarks on special values of L-functions Anthony J. Scholl* ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/remarks.pdf29 Jan 2010: groups:. Conjecture B. Let E be a motive over Z. Then. ... d′i = d. ′. By the above,. each B′i is a motive over Z, so is a submotive of the universal extension B†i. -
The Thought-experiment
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/1-2.html13 Aug 2009: Let us try some examples. Let L be the set of strings over the alphabet {a, b} which have the same number of as as bs. ... Let L be the set of strings over the alphabet {a, b} which have an even number of as and an even number of bs. -
Grammars
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/3-0.html13 Aug 2009: S aSa. S bSb. Sε. generates the language of palindromes over the alphabet Σ = {a, b}. ... Examples here are 'swim', 'walk', 'cat' and 'dog'. Notice that there is a grammar that generates the language of palindromes over Σ = {a, b} even though this -
More about bombs
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/2-1-1.html13 Aug 2009: The thought-experiment swiftly persuades us that the set of palindromes over an alphabet Σ is not regular (unless Σ contains only one character of course!). ... Let L be the language of palindromes over {a, b}. It isn't regular, but there is no obvious -
Exercises
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/3-0-1.html13 Aug 2009: 3.0.1 Exercises. Provide a context-free grammar for regular expressions over the alphabet {a, b}. -
Languages Recognised by Machines
https://www.dpmms.cam.ac.uk/~tef10/cam_only/langs-and-automata/1-1.html13 Aug 2009: The language {aa, ab, ba, bb} is the language of two-letter words over Σ. ... for characters in alphabets;. We tend to use lower-case letters from near the end of the alphabet - like 'u', 'v' and 'w'- for variables to range over strings;. -
Mirror symmetry and the SYZ conjecture
https://www.dpmms.cam.ac.uk/~mg475/current.pdf16 Jul 2014: Mark Gross Mirror symmetry and the SYZ conjecture. In this case, the SYZ fibration over B has a singular fibre over thesingular point of B, a pinched torus, and the total ... space of thefibration over B has two holomorphic disks, one as depicted:. -
REMARKS ON GLUING PUNCTURED LOGARITHMIC MAPS MARK GROSS Abstract. ...
https://www.dpmms.cam.ac.uk/~mg475/gluing.pdf6 Mar 2023: of punctured curves and pre-stable or stable punctured log maps f : C/W X orf : C/W X defined over B. ... stacks. We remind the reader that allproducts in this square are taken over B. -
INTRODUCTION TO PLECTIC COHOMOLOGY J. NEKOVÁŘ AND A. J. ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/introplec.pdf18 Feb 2015: H = B is the multiplicative group of a quaternion division algebra B over F (sothat Y is compact). ... over R or C).• (Log-)syntomic cohomology Hisyn(,sQp(j)) (for varieties over a p-adic. -
Vanishing cycles and non-classical parabolic cohomologyA. J.…
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/van.pdf29 Jan 2010: E! X, and D is the divisor over which fails to be smooth. ... X is dened over K. This determines direct and inverse image maps: Wk̀! -
Michaelmas Term 2014 SJW Linear Algebra: Example Sheet 2 ...
https://www.dpmms.cam.ac.uk/~sjw47/LinAlg14-2.pdf27 Oct 2014: 0 3 1. . 3. Let A and B be nn matrices over a field F. ... Show that A can bediagonalised. 11. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. -
Geometry-of-numbers over number fields and the density of ADEfamilies …
https://www.dpmms.cam.ac.uk/~mo512/nf.pdf23 May 2024: Gm(F ) over B(F ), so that it suffices to count elements with invariants in Σ to prove Theorem1.2. ... Then, the set Λ (B(1)) is a fundamental domain for the action onO over B(F), which we take as our Σi,. -
TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 00, Number 0,…
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/mfdrc.pdf21 Apr 2013: A) g : X X′ is étale over Y ′(B) Γ(X,OX) = K is a field. ... ar,. where sum ranges over complete set of representatives of congruence classes modulop in Q(ζm). -
Part III: Differential geometry (Michaelmas 2004) Alexei Kovalev…
https://www.dpmms.cam.ac.uk/~agk22/vb.pdf11 Nov 2019: The ψβα’s can be taken from some vector bundle E over B, then P will be‘constructed from E’. ... The construction can be reversed, so as to start from a principalG-bundle P over a base manifold B and obtain the vector bundle E over B. -
ON THE SMOOTH LOCUS OF AFFINE SCHUBERT VARIETIES GEORGIOS ...
https://www.dpmms.cam.ac.uk/~rz240/Min_deg.pdf28 Jan 2024: We fix T B Ga maximal torus and Borel pair in G defined over F which are in good positionrelative to s, and let I = Gal(F/F). ... We let B Tbe the Borel subgroup defined over F determined by BH and let X(T). -
Michaelmas Term 2015 SJW Linear Algebra: Example Sheet 4 ...
https://www.dpmms.cam.ac.uk/~sjw47/LinAlg15-4.pdf20 Nov 2015: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(. -
THE IMPORTANCE OF MATHEMATICS W. T. Gowers It is ...
https://www.dpmms.cam.ac.uk/~wtg10/importance.pdf12 Sep 2002: The size of A A is 20, which is over. three times larger than the size of A. ... B B = {2, 3, 4, 5, 6, 7, 8, 10, 12} ,. which has size 9 and is therefore only just over twice the size of B. -
Mich. 2016 NUMBERS AND SETS – EXAMPLES 1 PAR ...
https://www.dpmms.cam.ac.uk/~par31/ns/1.pdf14 Oct 2016: If not, then get on with the questions below and don’t turn over. ... 0a) Write down a definition of multiplication for natural numbers. Prove that multipli-cation is distributive over addition: that is, for all natural numbers a, b and c we havea(b -
Michaelmas Term 2014 SJW Linear Algebra: Example Sheet 4 ...
https://www.dpmms.cam.ac.uk/~sjw47/LinAlg14-4.pdf4 Dec 2014: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(. -
TOPICS IN SET THEORY: Example Sheet 4 1 Department ...
https://www.dpmms.cam.ac.uk/study/III/2014-15/TopicsinSetTheory/TST%20Example%20Sheet%204%202014.pdf19 Dec 2014: ZFC. (a) G is generic in P over M ;(b) GD 6= for every dense open set D P in M ;(c) GC 6= for every dense subset C P in ... The intended interpretationof the second-order variables ranges over arbitrary subsets of the domain. -
Elliptic Curves and Modularity Jack A. Thorne Abstract We ...
https://www.dpmms.cam.ac.uk/~jat58/EMS.pdf29 Sep 2023: elliptic curves over Q in 2001 by Breuil, Conrad,Diamond, and Taylor [8]. ... 2] P. B. Allen, C. Khare, J. A. Thorne, Modularity of GL2(Fp)-representations over CMfields, arXiv:1910.12986. -
A NOTE ON THE FAMILY SIGNATURE THEOREM OSCAR RANDAL-WILLIAMS ...
https://www.dpmms.cam.ac.uk/~or257/FamilySignature.pdf23 Jan 2019: F ; Z)/tors over B.Then. ... ξπ := (Hd/2(F; R) Hd/2(F; R)) C K0(B).In both cases these virtual vector bundles are defined over BO(H,λ), giving a. -
M 2003 NUMBERS AND SETS – EXAMPLES SHEET 2 ...
https://www.dpmms.cam.ac.uk/~wtg10/nasex2.pdf20 Oct 2004: in symbolic form. (For this question the letters m, n, a, b should be understood as ranging. ... over all positive integers - for instance, m really means m N.). -
Michaelmas Term 2016 SJW Linear Algebra: Example Sheet 4 ...
https://www.dpmms.cam.ac.uk/~sjw47/lin_alg-16-4.pdf21 Nov 2016: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(. -
The boundary of the Eisenstein symbol by Norbert Schappacher1 ...
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/bdy_of_eisenstein.pdf29 Jan 2010: defined over F. Then there are commutative diagrams :. (1.2.1). Q[Q]0n1 Θ. ... Gm Z/NZ µµµN Z/NZ (Z/NZ)2. (ζaN,b) 7 (a,b). defined over Q(ζN ). 18. 7.3 The main theorem enables us to calculate the effect of -
Modular curves and Ne19 eron models of generalized Jacobians
https://www.dpmms.cam.ac.uk/~ajs1005/preprints/neronfinal.pdf10 Feb 2023: and Φ(J) = ker(b)/ im(a).If X is semistable (meaning that Xs is smooth over k apart from double points with. ... v) (v′) Z[C]0,. the sum taken over all vertices v′ adjacent to v.In general, we have an isomorphism of Gal(k̄/k)-modules Φ(J) = ker(b)/ -
Michaelmas Term 2023 T.A. Fisher Galois Theory: Example Sheet ...
https://www.dpmms.cam.ac.uk/~taf1000/galois_theory-23-1.pdf10 Oct 2023: Suppose that α has minimal polynomial f over Q, and let β = α4.Find a, b, c Q such that β = a bα cα2. ... The elements α andβ are then said to be algebraically dependent over K.]. -
prelim13.dvi
https://www.dpmms.cam.ac.uk/~jmeh1/Teaching/prelim13.pdf10 Oct 2013: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 2. Determine which of the following sets of sequences of real numbers (xn) form vector spaces over R.(a) xn is bounded.(b) xn
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