Elliptic Functions
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13 Elliptic Function
13 Elliptic Function Recall that a function g : C → Cˆ is an elliptic function if it is meromorphic and there exists a lattice L = {mω1 + nω2 | m, n ∈ Z} such that g(z + ω)= g(z) for all z ∈ C and all ω ∈ L where ω1,ω2 are complex numbers that are R-linearly independent. We have shown that an elliptic function cannot be holomorphic, the number of its poles are finite and the sum of their residues is zero. Lemma 13.1 A non-constant elliptic function f always has the same number of zeros mod- ulo its associated lattice L as it does poles, counting multiplicites of zeros and orders of poles. Proof: Consider the function f ′/f, which is also an elliptic function with associated lattice L. We will evaluate the sum of the residues of this function in two different ways as above. Then this sum is zero, and by the argument principle from complex analysis 31, it is precisely the number of zeros of f counting multiplicities minus the number of poles of f counting orders. Corollary 13.2 A non-constant elliptic function f always takes on every value in Cˆ the same number of times modulo L, counting multiplicities. Proof: Given a complex value b, consider the function f −b. This function is also an elliptic function, and one with the same poles as f. By Theorem above, it therefore has the same number of zeros as f. Thus, f must take on the value b as many times as it does 0. -
Halloweierstrass
Happy Hallo WEIERSTRASS Did you know that Weierestrass was born on Halloween? Neither did we… Dmitriy Bilyk will be speaking on Lacunary Fourier series: from Weierstrass to our days Monday, Oct 31 at 12:15pm in Vin 313 followed by Mesa Pizza in the first floor lounge Brought to you by the UMN AMS Student Chapter and born in Ostenfelde, Westphalia, Prussia. sent to University of Bonn to prepare for a government position { dropped out. studied mathematics at the M¨unsterAcademy. University of K¨onigsberg gave him an honorary doctor's degree March 31, 1854. 1856 a chair at Gewerbeinstitut (now TU Berlin) professor at Friedrich-Wilhelms-Universit¨atBerlin (now Humboldt Universit¨at) died in Berlin of pneumonia often cited as the father of modern analysis Karl Theodor Wilhelm Weierstrass 31 October 1815 { 19 February 1897 born in Ostenfelde, Westphalia, Prussia. sent to University of Bonn to prepare for a government position { dropped out. studied mathematics at the M¨unsterAcademy. University of K¨onigsberg gave him an honorary doctor's degree March 31, 1854. 1856 a chair at Gewerbeinstitut (now TU Berlin) professor at Friedrich-Wilhelms-Universit¨atBerlin (now Humboldt Universit¨at) died in Berlin of pneumonia often cited as the father of modern analysis Karl Theodor Wilhelm Weierstraß 31 October 1815 { 19 February 1897 born in Ostenfelde, Westphalia, Prussia. sent to University of Bonn to prepare for a government position { dropped out. studied mathematics at the M¨unsterAcademy. University of K¨onigsberg gave him an honorary doctor's degree March 31, 1854. 1856 a chair at Gewerbeinstitut (now TU Berlin) professor at Friedrich-Wilhelms-Universit¨atBerlin (now Humboldt Universit¨at) died in Berlin of pneumonia often cited as the father of modern analysis Karl Theodor Wilhelm Weierstraß 31 October 1815 { 19 February 1897 sent to University of Bonn to prepare for a government position { dropped out. -
On Fractal Properties of Weierstrass-Type Functions
Proceedings of the International Geometry Center Vol. 12, no. 2 (2019) pp. 43–61 On fractal properties of Weierstrass-type functions Claire David Abstract. In the sequel, starting from the classical Weierstrass function + 8 n n defined, for any real number x, by (x) = λ cos (2 π Nb x), where λ W n=0 ÿ and Nb are two real numbers such that 0 λ 1, Nb N and λ Nb 1, we highlight intrinsic properties of curious mapsă ă which happenP to constituteą a new class of iterated function system. Those properties are all the more interesting, in so far as they can be directly linked to the computation of the box dimension of the curve, and to the proof of the non-differentiabilty of Weierstrass type functions. Анотація. Метою даної роботи є узагальнення попередніх результатів автора про класичну функцію Вейерштрасса та її графік. Його можна отримати як границю послідовності префракталів, тобто графів, отри- маних за допомогою ітераційної системи функцій, які, як правило, не є стискаючими відображеннями. Натомість вони мають в деякому сенсі еквівалентну властивість до стискаючих відображень, оскільки на ко- жному етапі ітераційного процесу, який дає змогу отримати префра- ктали, вони зменшують двовимірні міри Лебега заданої послідовності прямокутників, що покривають криву. Такі системи функцій відіграють певну роль на першому кроці процесу побудови підкови Смейла. Вони можуть бути використані для доведення недиференційованості функції Вейєрштрасса та обчислення box-розмірності її графіка, а також для по- будови більш широких класів неперервних, але ніде не диференційовних функцій. Останнє питання ми вивчатимемо в подальших роботах. 2010 Mathematics Subject Classification: 37F20, 28A80, 05C63 Keywords: Weierstrass function; non-differentiability; iterative function systems DOI: http://dx.doi.org/10.15673/tmgc.v12i2.1485 43 44 Cl. -
The Chiral De Rham Complex of Tori and Orbifolds
The Chiral de Rham Complex of Tori and Orbifolds Dissertation zur Erlangung des Doktorgrades der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg im Breisgau vorgelegt von Felix Fritz Grimm Juni 2016 Betreuerin: Prof. Dr. Katrin Wendland ii Dekan: Prof. Dr. Gregor Herten Erstgutachterin: Prof. Dr. Katrin Wendland Zweitgutachter: Prof. Dr. Werner Nahm Datum der mundlichen¨ Prufung¨ : 19. Oktober 2016 Contents Introduction 1 1 Conformal Field Theory 4 1.1 Definition . .4 1.2 Toroidal CFT . .8 1.2.1 The free boson compatified on the circle . .8 1.2.2 Toroidal CFT in arbitrary dimension . 12 1.3 Vertex operator algebra . 13 1.3.1 Complex multiplication . 15 2 Superconformal field theory 17 2.1 Definition . 17 2.2 Ising model . 21 2.3 Dirac fermion and bosonization . 23 2.4 Toroidal SCFT . 25 2.5 Elliptic genus . 26 3 Orbifold construction 29 3.1 CFT orbifold construction . 29 3.1.1 Z2-orbifold of toroidal CFT . 32 3.2 SCFT orbifold . 34 3.2.1 Z2-orbifold of toroidal SCFT . 36 3.3 Intersection point of Z2-orbifold and torus models . 38 3.3.1 c = 1...................................... 38 3.3.2 c = 3...................................... 40 4 Chiral de Rham complex 41 4.1 Local chiral de Rham complex on CD ...................... 41 4.2 Chiral de Rham complex sheaf . 44 4.3 Cechˇ cohomology vertex algebra . 49 4.4 Identification with SCFT . 49 4.5 Toric geometry . 50 5 Chiral de Rham complex of tori and orbifold 53 5.1 Dolbeault type resolution . 53 5.2 Torus . -
Continuous Nowhere Differentiable Functions
2003:320 CIV MASTER’S THESIS Continuous Nowhere Differentiable Functions JOHAN THIM MASTER OF SCIENCE PROGRAMME Department of Mathematics 2003:320 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 03/320 - - SE Continuous Nowhere Differentiable Functions Johan Thim December 2003 Master Thesis Supervisor: Lech Maligranda Department of Mathematics Abstract In the early nineteenth century, most mathematicians believed that a contin- uous function has derivative at a significant set of points. A. M. Amp`ereeven tried to give a theoretical justification for this (within the limitations of the definitions of his time) in his paper from 1806. In a presentation before the Berlin Academy on July 18, 1872 Karl Weierstrass shocked the mathematical community by proving this conjecture to be false. He presented a function which was continuous everywhere but differentiable nowhere. The function in question was defined by ∞ X W (x) = ak cos(bkπx), k=0 where a is a real number with 0 < a < 1, b is an odd integer and ab > 1+3π/2. This example was first published by du Bois-Reymond in 1875. Weierstrass also mentioned Riemann, who apparently had used a similar construction (which was unpublished) in his own lectures as early as 1861. However, neither Weierstrass’ nor Riemann’s function was the first such construction. The earliest known example is due to Czech mathematician Bernard Bolzano, who in the years around 1830 (published in 1922 after being discovered a few years earlier) exhibited a continuous function which was nowhere differen- tiable. Around 1860, the Swiss mathematician Charles Cell´erieralso discov- ered (independently) an example which unfortunately wasn’t published until 1890 (posthumously). -
On Hodge Theory and Derham Cohomology of Variétiés
On Hodge Theory and DeRham Cohomology of Vari´eti´es Pete L. Clark October 21, 2003 Chapter 1 Some geometry of sheaves 1.1 The exponential sequence on a C-manifold Let X be a complex manifold. An amazing amount of geometry of X is encoded in the long exact cohomology sequence of the exponential sequence of sheaves on X: exp £ 0 ! Z !OX !OX ! 0; where exp takes a holomorphic function f on an open subset U to the invertible holomorphic function exp(f) := e(2¼i)f on U; notice that the kernel is the constant sheaf on Z, and that the exponential map is surjective as a morphism of sheaves because every holomorphic function on a polydisk has a logarithm. Taking sheaf cohomology we get exp £ 1 1 1 £ 2 0 ! Z ! H(X) ! H(X) ! H (X; Z) ! H (X; OX ) ! H (X; OX ) ! H (X; Z); where we have written H(X) for the ring of global holomorphic functions on X. Now let us reap the benefits: I. Because of the exactness at H(X)£, we see that any nowhere vanishing holo- morphic function on any simply connected C-manifold has a logarithm – even in the complex plane, this is a nontrivial result. From now on, assume that X is compact – in particular it homeomorphic to a i finite CW complex, so its Betti numbers bi(X) = dimQ H (X; Q) are finite. This i i also implies [Cartan-Serre] that h (X; F ) = dimC H (X; F ) is finite for all co- herent analytic sheaves on X, i.e. -
AN EXPLORATION of COMPLEX JACOBIAN VARIETIES Contents 1
AN EXPLORATION OF COMPLEX JACOBIAN VARIETIES MATTHEW WOOLF Abstract. In this paper, I will describe my thought process as I read in [1] about Abelian varieties in general, and the Jacobian variety associated to any compact Riemann surface in particular. I will also describe the way I currently think about the material, and any additional questions I have. I will not include material I personally knew before the beginning of the summer, which included the basics of algebraic and differential topology, real analysis, one complex variable, and some elementary material about complex algebraic curves. Contents 1. Abelian Sums (I) 1 2. Complex Manifolds 2 3. Hodge Theory and the Hodge Decomposition 3 4. Abelian Sums (II) 6 5. Jacobian Varieties 6 6. Line Bundles 8 7. Abelian Varieties 11 8. Intermediate Jacobians 15 9. Curves and their Jacobians 16 References 17 1. Abelian Sums (I) The first thing I read about were Abelian sums, which are sums of the form 3 X Z pi (1.1) (L) = !; i=1 p0 where ! is the meromorphic one-form dx=y, L is a line in P2 (the complex projective 2 3 2 plane), p0 is a fixed point of the cubic curve C = (y = x + ax + bx + c), and p1, p2, and p3 are the three intersections of the line L with C. The fact that C and L intersect in three points counting multiplicity is just Bezout's theorem. Since C is not simply connected, the integrals in the Abelian sum depend on the path, but there's no natural choice of path from p0 to pi, so this function depends on an arbitrary choice of path, and will not necessarily be holomorphic, or even Date: August 22, 2008. -
ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi)
Math 213a (Fall 2021) Yum-Tong Siu 1 ELLIPTIC FUNCTIONS (Approach of Abel and Jacobi) Significance of Elliptic Functions. Elliptic functions and their associated theta functions are a new class of special functions which play an impor- tant role in explicit solutions of real world problems. Elliptic functions as meromorphic functions on compact Riemann surfaces of genus 1 and their associated theta functions as holomorphic sections of holomorphic line bun- dles on compact Riemann surfaces pave the way for the development of the theory of Riemann surfaces and higher-dimensional abelian varieties. Two Approaches to Elliptic Function Theory. One approach (which we call the approach of Abel and Jacobi) follows the historic development with motivation from real-world problems and techniques developed for solving the difficulties encountered. One starts with the inverse of an elliptic func- tion defined by an indefinite integral whose integrand is the reciprocal of the square root of a quartic polynomial. An obstacle is to show that the inverse function of the indefinite integral is a global meromorphic function on C with two R-linearly independent primitive periods. The resulting dou- bly periodic meromorphic functions are known as Jacobian elliptic functions, though Abel was actually the first mathematician who succeeded in inverting such an indefinite integral. Nowadays, with the use of the notion of a Rie- mann surface, the inversion can be handled by using the fundamental group of the Riemann surface constructed to make the square root of the quartic polynomial single-valued. The great advantage of this approach is that there is vast literature for the properties of the Jacobain elliptic functions and of their associated Jacobian theta functions. -
Abelian Varieties
Abelian Varieties J.S. Milne Version 2.0 March 16, 2008 These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version of the notes was distributed during the teaching of an advanced graduate course. Alas, the notes are still in very rough form. BibTeX information @misc{milneAV, author={Milne, James S.}, title={Abelian Varieties (v2.00)}, year={2008}, note={Available at www.jmilne.org/math/}, pages={166+vi} } v1.10 (July 27, 1998). First version on the web, 110 pages. v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph shows the Tasman Glacier, New Zealand. Copyright c 1998, 2008 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Contents Introduction 1 I Abelian Varieties: Geometry 7 1 Definitions; Basic Properties. 7 2 Abelian Varieties over the Complex Numbers. 10 3 Rational Maps Into Abelian Varieties . 15 4 Review of cohomology . 20 5 The Theorem of the Cube. 21 6 Abelian Varieties are Projective . 27 7 Isogenies . 32 8 The Dual Abelian Variety. 34 9 The Dual Exact Sequence. 41 10 Endomorphisms . 42 11 Polarizations and Invertible Sheaves . 53 12 The Etale Cohomology of an Abelian Variety . 54 13 Weil Pairings . 57 14 The Rosati Involution . 61 15 Geometric Finiteness Theorems . 63 16 Families of Abelian Varieties . -
Lecture 9 Riemann Surfaces, Elliptic Functions Laplace Equation
Lecture 9 Riemann surfaces, elliptic functions Laplace equation Consider a Riemannian metric gµν in two dimensions. In two dimensions it is always possible to choose coordinates (x,y) to diagonalize it as ds2 = Ω(x,y)(dx2 + dy2). We can then combine them into a complex combination z = x+iy to write this as ds2 =Ωdzdz¯. It is actually a K¨ahler metric since the condition ∂[i,gj]k¯ = 0 is trivial if i, j = 1. Thus, an orientable Riemannian manifold in two dimensions is always K¨ahler. In the diagonalized form of the metric, the Laplace operator is of the form, −1 ∆=4Ω ∂z∂¯z¯. Thus, any solution to the Laplace equation ∆φ = 0 can be expressed as a sum of a holomorphic and an anti-holomorphid function. ∆φ = 0 φ = f(z)+ f¯(¯z). → In the following, we assume Ω = 1 so that the metric is ds2 = dzdz¯. It is not difficult to generalize our results for non-constant Ω. Now, we would like to prove the following formula, 1 ∂¯ = πδ(z), z − where δ(z) = δ(x)δ(y). Since 1/z is holomorphic except at z = 0, the left-hand side should vanish except at z = 0. On the other hand, by the Stokes theorem, the integral of the left-hand side on a disk of radius r gives, 1 i dz dxdy ∂¯ = = π. Zx2+y2≤r2 z 2 I|z|=r z − This proves the formula. Thus, the Green function G(z, w) obeying ∆zG(z, w) = 4πδ(z w), − should behave as G(z, w)= log z w 2 = log(z w) log(¯z w¯), − | − | − − − − near z = w. -
Pathological” Functions and Their Classes
Properties of “Pathological” Functions and their Classes Syed Sameed Ahmed (179044322) University of Leicester 27th April 2020 Abstract This paper is meant to serve as an exposition on the theorem proved by Richard Aron, V.I. Gurariy and J.B. Seoane in their 2004 paper [2], which states that ‘The Set of Dif- ferentiable but Nowhere Monotone Functions (DNM) is Lineable.’ I begin by showing how certain properties that our intuition generally associates with continuous or differen- tiable functions can be wrong. This is followed by examples of functions with surprisingly unintuitive properties that have appeared in mathematical literature in relatively recent times. After a brief discussion of some preliminary concepts and tools needed for the rest of the paper, a construction of a ‘Continuous but Nowhere Monotone (CNM)’ function is provided which serves as a first concrete introduction to the so called “pathological functions” in this paper. The existence of such functions leaves one wondering about the ‘Differentiable but Nowhere Monotone (DNM)’ functions. Therefore, the next section of- fers a description of how the 2004 paper by the three authors shows that such functions are very abundant and not mere fancy counter-examples that can be called “pathological.” 1Introduction A wide variety of assumptions about the nature of mathematical objects inform our intuition when we perform calculations. These assumptions have their utility because they help us perform calculations swiftly without being over skeptical or neurotic. The problem, however, is that it is not entirely obvious as to which assumptions are legitimate and can be rigorously justified and which ones are not. -
1 Complex Theory of Abelian Varieties
1 Complex Theory of Abelian Varieties Definition 1.1. Let k be a field. A k-variety is a geometrically integral k-scheme of finite type. An abelian variety X is a proper k-variety endowed with a structure of a k-group scheme. For any point x X(k) we can defined a translation map Tx : X X by 2 ! Tx(y) = x + y. Likewise, if K=k is an extension of fields and x X(K), then 2 we have a translation map Tx : XK XK defined over K. ! Fact 1.2. Any k-group variety G is smooth. Proof. We may assume that k = k. There must be a smooth point g0 G(k), and so there must be a smooth non-empty open subset U G. Since2 G is covered by the G(k)-translations of U, G is smooth. ⊆ We now turn our attention to the analytic theory of abelian varieties by taking k = C. We can view X(C) as a compact, connected Lie group. Therefore, we start off with some basic properties of complex Lie groups. Let X be a Lie group over C, i.e., a complex manifold with a group structure whose multiplication and inversion maps are holomorphic. Let T0(X) denote the tangent space of X at the identity. By adapting the arguments from the C1-case, or combining the C1-case with the Cauchy-Riemann equations we get: Fact 1.3. For each v T0(X), there exists a unique holomorphic map of Lie 2 δ groups φv : C X such that (dφv)0 : C T0(X) satisfies (dφv)0( 0) = v.