Course presentation - Name, syllabus, today’s plan; - Plan:

1) Present resources; objectives (direct/indirect benefits) - Textbook, class discussions, web board … - Additional readings: Remarkable Mathematicians, The Mathematics Century 2) How the course is organized: - Syllabus (see bellow!) - Chapter 19! 1950-7/12/2005 (Cat. th., pert. th., DE & graph series, F. diagrams, Q. computing etc.) - Emphasis on 19th-20th century (due to exp. growth of knowledge) 3) Questionnaire / background test;

4) For next time: read Ch.11 & wr. outline; bookmark favorites (W4) 5) Math complements – free discussion ? - More on Recitation-Lecture Format (2 parts): 1st hour: discuss the textbook (transparency & outlines ready!) - The math, the event: his/her story, type of math (art/science) Who (did), what, when & why? - Relevance to other areas of math or to other sciences - Compare with today’s level of sophistication - A 10 min. break - 2nd hour: math complements & meta mathematics - Completions: - More numbers: 0, -x, a/b, square roots, roots (lim & complex numbers) - More functions: distributions - More limits: Hilbert spaces, complete S-T - More groups: quantum groups - Math design (top-down versus bottom up; analysis versus algebra as methods!) - Geometry implies/requires applications! (Modeling w. DE & Lie groups, mechanics etc.) - Interface & implementation (e.g. TV user / producer; Th. Hyp., concl. & proof) - Research and publication process - Projects: 1: Math.: Art or Science? Mathematics out of curiosity or necessity? Examples: Fermat’s Last Theorem (Pythagora -> Fermat or -> Minkovsky/Einstein) [Extra-credit: 2. Completions – extending a math tool to overcome obstructions in solving a problem. Example: number systems N-W-Z-Q-R-C-H-O; distributions (Dirac’s delta …)] - What to prepare before class: - Reading notes (or outline) for the assigned section (In advance!) - A few favorite points to talk/mention in class (participation; 2-3 min.)

Other Ideas & Topics for Discussion - To better understand the use of history in teaching and understanding mathematics read Annex 2 from textbook: 1) Anecdotes, 2) Content & focusing on ideas, 3) Development of mathematics ideas. - Math: Queen (art) or Slave (put to work; science) ? - Art: e.g. 2^(2^n+1) - Curiosity fueled the development of mathematics (although killed the cat). - Science: designing new (mathematical) models; - Sometimes the needed math was there (Calabi-Yau manifolds & string theory) - Nice math problems (e.g. mouse and cat) I could always solve, and needed problems I could never solve (e.g. some elliptic integrals from physics etc.) - Some vital problems with solutions (completions): - “No can do!”; “we’ll make it work” & completions - Example: distributions (δ-functions, convolution unit, derivatives of piece-wise diff. fn.) - Can we differentiate cont. functions? (Distribution sense) - Details versus ideas: - Learn the ideas, never mind the details - Programs: interface & implementation - Details are usually author dependent! - Example: from Newton F=ma to Einstein T=kG - Low level/high level languages (set theory & category theory) - Getting busy: “What if …” (e.g. change 2 into 3, then n in Pythagorean numbers eq.) - Mathematical “dimensions”: space-time-depth

Mat321: “History of Mathematics: 1600-present” Resources and Organisation

•••• Instructor : Lucian Ionescu

- Office hours: MTWR 9:30-10:25 a.m., STV 303G, 8-7167 - Appointments: in person or by e-mail, LMIones@ilstu.edu - Web site: \www.ilstu.edu\~lmiones\mat321

•••• Text: A history of mathematics: an introduction, V. Katz, 2nd

ed.

• Objectives: - To understand how mathematics evolved and to learn about the

forefront figures responsible for this. - To inform you about various areas of mathematics and the

relationship with other sciences.

•••• Prerequisites: Mat 147 •••• Syllabus

1) Course format - Read sections before class; - 1st part: discussion of main figures & contributions - 2nd part: complements & meta-mathematic ☺☺☺☺ 2) Homework & Projects 3) Evaluation

• What is “Mathematics”? Discuss Project 1 - Mathematics: Art or Science? Examples - Fermat last theorem - Four color problem (& computer proofs) - Differential geometry and general relativity - Riemann surfaces & string theory Discuss Project 2 - Mathematics R&D: Extensions and Completions.

Examples: - Adding an identity, - “Can we subtract 3 from 2?” Providing inverses/opposites - “Can we divide 2 into 3 equal parts?” - Square roots & field extensions etc. - Is the step function differentiable? - Is Dirac’s delta function a function? - Unique factorization: add ideals … - Completing a metric space (e.g. Q -> R) - etc.

More topics to be discussed eventually (when borred):

- Learning mathematics: “big picture” and “last tin y detail”

- Mathematicians: user, looser or producer? ☺☺☺☺ … if u can

- What’s your favorite math?

- How do you know it’s “geometry, algebra or analysis” (use your taste, of course! ☺☺☺☺)

Math 321 – Feedback Name: 1. What is your favorite mathematics area? (e.g. Calculus, Algebra, History of math, etc.) 2. What science, if any, interests you most? (Physics, Computer science, Biology, etc.) 3. Please list relevant courses taken (e.g. Abstract algebra, Calculus, etc.) 4. Are you more interested in math or math history? 5. If interested in history of math, what aspects are you interested in? (e.g. Biography of mathematicians, events and “stories”, famous problems, new mathematical areas/trends etc.) - Comments and suggestions:

(From A History of Mathematics, V. J. Katz, 3rd edition)

Ch. 14 Algebra, Geometry and Probability in 17th century

14.1 Theory of equations - Notations: aaaaa (!?) - Albert Giraud: FTA & “Viete’s relations” (symm. poly.) - Decartes & equation solving (reducing the degree) 14.2 Analytic geometry – what is it? Curves / loci <-> equations - Pierre de Fermat (1601-1665) (unpublished) - Rene Descartes (1596-1650) (published) - Algebraic geometry (why waiting?): equations <-> ideals and algebras example: i=[x] in R[x]/(x2+1) - Jan de Witt (extending Fermat’s unpublished work)

14.3 Elementary probability - Early start by Cardano & Tartaglia: forgotten - ~1660: “Sponsors”: de Mere; “Consultant”: Blaise Pascal - European thought: - understand stable frequencies - Determine reasonable degree of belief - Modern probabilities (probability space): Events & probability measure - Pascal’s Triangle – combinatorial coefficients - Pascal: uses math induction, solves Mere’s problem - Christian Huygens & expectation values - Average amount one would win (outcome) - Other “observables”: random variables

14.4 Number theory Pierre de Fermat (1601-1665) - French lawyer (Hobby: math) - Number theory contributions recognized next century - Marsenne primes: 2p-1 - Fermat’s Little Th. (modern: Zp is a field; |U(p)|=p-1) - Fermat (false) conj. 2^(2^n)+1 are primes. - Leonard Euler (math/phys etc.) - Fermat’s Last Theorem (famous) - Generalizing Pythagorean numbers - Art/science? (Gauss’s opinion) - What method was systematically used? - Were European math. interested in Fermat’s work? (# th.) - Math as a game (What if …) - Einstein point of view about games (QM & God)

14.5 Projective geometry - Roots in art (proj. of obj. on a plane) - Girard Desarques (1591-1661) - “Def.” PG studies properties of conics invariant under proj. - What is a projection? (linear, affine) - Qualitative study (synthetic geometry) rather than quantitative (analytic geometry) - Klein’s program (study obj./properties invariant to a group of transformations; i.e. symmetry!) - Points at infinity – completion; - 1-point compactification (R->S1); “compact spaces” are good! - Pascal’s Th. (hexagon in a conic)

Ch. 15 The Beginning of Calculus - Newton & Leibniz - created calculus:

differentiation & integration - Leibnitz rule = product rule for algebras - Newton-Leibniz correspondence

- Secrecy versus sharing; - Dissemination of results; open source programing

- Geometric aspects of curves: tangent, normal, subtangent, subnormal, abscissa, ordinate (Exercise in analytic geometry) 15.1 Tangents and extrema - Kepler 1615 (min/max problem: parallelepiped in sphere) - Fermat’s method of adequality:

x1,x2 replaced by x, x+e (Taylor expansion) - Descartes & method of normals - Idea: tangency point <-> double root - Critical of Fermat - Algorithms of Hudde & Sluse - Simplifying Descartes & Fermat - Sluse’s method: implicit differentiation (recall)

15.2 Areas and volumes - Main idea: break up regions in smaller pieces of known measure - Alternative: “Divide and conquer” (Greeks, Romans etc. before Descartes’s method of solving a problem) - Infinitesimals and indivisibles - Galileo: indivisibles: n-dim fig. Made of (n-1)-dim. - Kepler: infinitesimals: very thin/small

method of counting area of the circle - Cavalieri - Complete theory of indivisibles - Slicing principle (method of cross-sections) & comparing with a known volume - Cavalieri’s principle: L1(h)=k L2(h) => A1=k A2 - Results: ∫

bkdxx

0

(also by F, P, Roberval, Torricelli)

- Evangelista Torricelli (1608-1647) - Vol. Of an unbounded solid - Used proofs by contradiction - Discovered the principle of barometer - Fermat and area under higher parabolas y=axk - Precursor of Riemann sums - Difficulties to generalize to rational exponents. - John Wallis – derived similar integrasion formulas - Roberval – area under cycloid - Gregory of St. Vincent – area under xy=1

15.3 Rectification of curves (tangents) & FTC - Van Heuraet – finding length <-> area under a curve (∫) - wealthy person; travel & study (wouldn’t you?) - Isaac Barrow & James Gregory – organized material about tangents and areas - J. Gregory’s FTC: related tangent (d) and areas (∫) - Main improvement: conceived area as a function of a variable - I. Barrow’s FTC had (the usual) 2 parts - They were not the inventors of calculus: presentation was in geometric form.

Ch. 16 Newton and Leibniz 16.1 Isaac Newton - Organized results on Calculus in a few years => 1000 pages book of today’s Calculus (good?/bad? Kac’s Q.C. – no lim.) - Theory & results: P.S., binomial theorem - Calculus of fluxions (v=s’; s dot; i.e. math with physics intuition behind, like geometry), - Applications: osculating circle, procedures for finding areas - Mathematics of the heavens: - Principia (in geometric language, not analysis) - Concept of limit made precise - Calculus was developed well before the physics - De motu corporum (calculus models in mechanics) - Newton’s laws:

Fa=-Fr (allows to divide systems), F=ma (phys. & math. Sides)

- Separates experimental results from math, e.g. Hook’s law, gravity etc. (defines and interface between math & phys, application & implementation etc.) - Compare with Einstein’s general relativity: T=kG

16. 2 G. W. Leibniz - Studied relation between sums and differences (telescoping sums as total change = sum of small changes) - Introduced d & ∫; ds2=dx2+dy2, - Equation of the tangent line (local coord.) dy=f’(x)dx - The calculus of differentials: dc=0, d(v+y)=dv+dy, product rule, power rule - Leibniz rule: differential algebras - Determining min/max: dy=0, horizontal tangent - Concavity depends on 2nd differences; inflection points - FTC & DE: dz/dx=y (law of tangency) � z=∫ydx (quadrature) - Interested in DE; e.g. DE for sine: x’’=-x - Infinitesimals: convenient formalism; existence? Leibniz: “irrelevant” - Newton-Leibniz priority controversy - N’s approach: v & s (mechanic interpretation) - Leibniz: - & + (“pure” math approach) - Accusations of plagiarism from N & Bernoulli brothers=> - English & continental mathematicians ceased the

exchange of ideas; English didn’t benefit from Cont. R&D

16.3 First Calculus texts - Leiniz approach led to analysis: - L”hospital’s Analysis of Small Infinitesimals - Bernoulli was working for L’Hospital - English side: Dittion & Hayes: developed N’s theory of fluxions (appeal to mechanical intuition) Conclusions (1600-1800) Math emerged sporadically, in several places, at the hobby/trade level; secrecy & envy were more common then collaborations; dissemination of results was the exception (manuscripts rather then publications); non-specialist learns mostly Euclid-Newton stuff!

Part IV: Modern Mathematics Ch. 17 Analysis in the 18th Century - Leonard Euler – most prolific mathematician in history! - Mathematician were interested in solving DE (modeling mechanical systems); new functions discovered (sol. of DE) - Bernoulli set problems for Europe’s mathematicians - T. Simpson & C. Maclaurin in England - Maria G. Agnessi in Italy - J-L Lagrange tried to eliminate infinitesimals & limits, basing Calculus on power series - Modern interpretation of infinitesimals: - Geometric: differential forms / exterior algebra - Algebraic: quotient algebras k[x,dx]/(dx2) 17.1 Differential equations - J. & J. Bernoulli brothers understood Leibniz’s methods - Catenary eq. dy/dx=ds/a; solved by J. Bernoulli - Brahistochrone: curve of quickest descent (vertical plane; minimize time, not distance) - Translating Newton’s Synthetic Method of Fluxions into the Method of Differentials (Modern choice: differentials) - DE & trig. Fn .: sin(θ) viewed as a segment rather than PS. - Leonard Euler: - Classifies fn.: algebraic & transcendental - Introduced integrating factors to solve DE - Used sin & cos for higher order DE

- Solved constant coef. Linear DE: reduced to alg. eq. (characteristic equation)

- Modern point of view: consider the commutative alg. of diff. Op. acting on functions. - Logarithms of Negative and Complex Numbers - Bernoulli confused about logarithms of negatives - Euler explained: eix=cos(x)+i sin(x) - log has infinity many branches (exp:C->C* has a kernel) - The Calculus of Variations - Started from problems like min/max of integrals

- Minimizing: I(y)= ∫b

a

dxyyxF )',,( (fixed end point variation)

- Example 1: line minimizes distance F=(1+y’2)1/2 17.2 The Calculus of Several Variables - Differential calculus of 2 variables: not f(x,y), but rather families of curves {Cα(x)} (3D vertical sections) - Leibnitz discovered interchangeability theorem: dα∫fdx=∫ dαfdx - Total differentials - Nicolaus Bernoulli introduced the total differential: dy=pdx+qdα Double integrals and Multiple integration - Developed by Euler – 1769; defined double integral by generalizing antiderivatives to two variables - Change of variables in double integrals by Euler: dA=Jdxdy although J is the Jacobia; proof: formal argument

- Lack of axiomatic foundation (a la Euclid’s geometry) bothered contemporary mathematicians. - Designing mathematical theory versus implementation. - Equality of mixed 2nd partials is proved by Euler: ∂xy=∂yx (known to Bernoulli) - Alexis-Claude Clairaut (1713-1765): found integrability condition for DE pdx+qdy=0 (“Exactness condition”):py=qx, and discovered the method of solving them. - Change of variables in multiple integration PDE: the wave equation - ½ 18th century – Euler, d’Alambert, D. Bernoulli - Jean Le Rond d’Alambert (1717-1783) – derived the wave eq. ytt=kyxx & solution y=F(t+x)+G(t-x); - Boundary conditions y(0)=y(L)=0 => G=-F - Euler derived the same result 2 years later - Bernoulli wrote the solution in terms of sin & cos (separation of variables; later: Fourier series for eigenvalue problems) 17.3 Calculus texts - (England) Thomas Simpson’s Treatise of fluxions – Newton’s approach, i.e. using PS to solve DE. - (Scotland) Colin Maclaurin’s Treatise of fluxions (!)

(Recall: Maclaurin series ) - (Europe – Italy) Maria Agnessi’s Institutione analitiche – Leibniz influence; child of a wealthy merchant ; Euler’s introductio: Analysis, Calculus (Diff. & Int.)

17.4 Foundations of Calculus - Euclid’s Elements – “model of how mathematics should be done”; there was no logical basis for central part of calculus - Top mathematicians: Newton, Leibniz, Euler had a strong intuitive feel for the subject; new that what they were doing is correct - “Translation”: designing the blue prints of the theory, knowing that it can be rigorously implemented - George Berkeley’s Th Analyst – criticism of arguments - Maclaurin defended calculus: Treatise of flusions; still missed the concept of limit. - Euler and limits: “0:0=2:1” (“a la” L’hospital) - d’Alambert gave a definition of limit - Lagrange used power series extensively: defined f’(x) by looking at the PS: f(x+I)=f(x)+I p(x)+…=> f’(x)=p(x) - Vague definition of functions - Any function had a PS representation (!?) - Joseph-Louis Lagrange (1736-1813) gave a better proof (praised by Euler) - Classical dynamics: Lagrangean, action, principle of minimal action, equations of dynamics: Euler-Lagrange equations. - 1932 Einstein’s idea: implement gravity as a change of the metric; geodesics=motion under gravity - ~ 1940 Kaluza-Klein theory: trying to unify electromagnetism & gravity by adding extra-dimensions to space-time.

Ch. 18 Probability and Statistics in the 18th century - Probability: Law of Large Numbers (frequency -> expectation), normal distributions etc. 18.1 Probability - Game theory and aleatory contracts; maritime insurance contracts - J. Bernoulli and “Ars Conjectandi” - Law of Large Numbers (frequency -> expectation val.) - Sum of integral powers; involves B-numbers - De Moivre and the doctrine of chances 1718 (more precise results than Bernoulli, but still no statistical inference) - Bayes and statistical inference (deviation, distribution etc.) - Axiomatic definition of probability - Laplace made further progress, computed some Gaussian integrals. 18.2 Statistical Inference 18.3 Applications of Probability

Ch. 19 Algebra and Number Theory in the Eighteenth Century - Algebra: one of main goals – solving polynomial eq. - Geometry: - Prove the parallel postulate

(=> non-euclidean geometries & differential geometry) - Applications of analysis in geometry

(analytic geometry evolved into differential geometry) 19.1 Algebra texts - Few developments in algebra; rather systematization of earlier materials - Major algebra texts: Newton, Maclaurin, Euler Newton’s Arithmetica Universalis: - Lectured on algebra at Cambridge for 10 year; 1683 decided to write them down. Starts with + and x. Maclaurin’s Treatise of Algebra - Algebra= generalized arithmetic; ex. +a-a=0 etc. - Solves systems by elimination (Cramer’s rules)

(determinants appeared ~ 1700) - Solved polynomial equations (Cardano’s formula,

Descartes rule of signs, Newton’s method of approximating solutions)

Euler’s Introduction to Algebra -Better then N’s & Maclaurin’s - Def. algebra: science which teaches you how to determine unknown quantities from known quantities. (Digression: Algebra versus Analysis; what about Geometry?)

- Starts w. rules of signs etc.; introd. C numbers;fails to note possible difficulties (√-1 √-1=-1 & √-1 √-1=√(-1)(-1)=√1=1 !?)

- Gives general solutions to arithmetic problems rather then single solutions.

- Explains the method of elimination – systems of eq. (noticed systems may be: det., undet., incompatible)

- Solved number theoretic problems: #s mod n; defined class ops.; basic ideas of ring theory are present. (Recall: groups, rings and fields) - Defined quadratic residue; p≠0 prime s.t. p is a square modulo q. 19.2 Advances in the Theory of Equations - Lagrange and the solution of polynomial equations - Reviewed methods for n≤4 - Noticed the role of roots of unity in solving the cubic eq. and the role of permuting the roots (=> Galois theory later) - Ferrari’s method to solve 4th degree eq. - Generalized to n≥5 some results: roots of 1 & permut. 19.3 Number Theory … 19.4 Mathematics in the Americas - Scientific societies:

- American Phylosophical Soc., - American Academy of Arts & Sci.

- John Winthorpe at Hrvard – taught math. - David Rittenhouse - astronomer - Benjamin Franklin – influence in founding U. of Pensylvania - Thomas Jefferson has studied math. - In general: only a practical concern for math.

Ch. 20 Geometry in the Eighteenth Century - New directions: analytic geometry using calculus tools - Continuation of old trend: axiomatic geometry, trying to prove the parallel postulate 20.1 Clairaut’s “Elements of Geometry” 1741 - Advocating a natural approach: presented geometry as a science of measurement, not axiomatic as in Euclid’s approach. - Emphasized: curiosity enabled people to solve new problems. 20.2 The parallel postulate: - Saccheri: thought he proved it by eliminating other possibilities: sum of angles in a quadrilateral <, =, > 360 (depends on the space curvature!) - Lambert & parallel postulate – Theorie der Parallelinien 1766

- By contradiction: as a consequence, the difference 360-sum of angles depends on area

- Considered geometry on the sphere (+ curvature) & started to consider geometry on sphere of imaginary radius (- curvature).

20.3 Analytic and Differential Geometry Clairaut and space curves - 18th c. geometry benefited from analysis - Curves as intersection of surfaces; - Defined tangent and normal to surfaces Euler and space curves and surfaces – 1775 - Defined direction cosines - Arc length, curvature and radius of curvature (beginning of differential geometry) Gaspard Monge (1746-1818) - Systematized results of analytis and dif. geom.. - 1st time point-slope eq. - Geometrie descriptive: no alg., pure geometry - Application de l’analyse to geometrie 1807 (How to apply analysis to geometry) - Used extensively projections: on coordinate planes and axes - Eq. of tangent plane to a surface:

dz=Zxdx+… d-form, z-z0=Zx(x-x0)+Zy(y-y0) - Eq. of the normal 20.4 Euler and the Beginning of Topology - A “little problem”: is there a stroll passing over each bridge once? (picture) - Euler’s methodology: generalize first & then solve! (also Grothendieck; irrelevant details make it harder etc.) - “Space” contractible to a graph; Hamiltonian path - Topology and Topological Spaces (Brief)

20.5 French revolution and mathematics education - Math.: in academies (founded by monarchs), not in universities (didn’t provide an advanced math education) - Universities were dominated by philosophers. - Military schools provided math & science education - After revolution: Ecole Polytechnique - Descriptive geometry by Monge - Standards developed - Model for college of engineering in Europe & US

21. Algebra and Number Theory in the 19th century - 1800’s algebra: solving equations - 1900’s algebra: study (& design) new mathematical structures Carl Gauss’s Disquisitiones Arithmetica (number theory)

- Generalizing arithmetica (# th.) to other “# systems” - Gauss integers Z[i] and other “integers” without the unique factorization property (UF) - Gauss studied cyclotomic fields & integers Z[ω], ωn=1

- Kummer introduced “ideal” numbers to “rescue” UF at the level of ideals (Ex. of extension/completion) - Abel proved eq. n≥5 can’t be solved by √ - Evariste Galois sketched the relation between solvability and permutation of roots: the study of subgroups of the group of the equation. - 1854 Arthur Cayley defined abstract groups; notion later developed by Walter van Dyck and Heinrich Weber - Leopold Kronecker & Richard Dedekind def. field numbers - H. Webed defined abstract field - 1800~1833 George Peacock & A. de Morgan generalized properties of integers and axiomatized the basic ideas of algebra

- 1843 W. R. Hamilton discovered quaternions (NC field=div. Alg): gen. By i,j,k w. rel. i2=j2=k2=-1 - Quaternions 4-dim; physicists used 3D: algebra of vectors developed by O. Heaviside & J. W. Gibbs: i,j,k, dot & cross products. - More general binary operations used by Hamilton in physics and G. Boole to the study of logic (Boolean algebra) - Theory of matrices (term introduced by Sylvester 1850) - Determinant have been used earlier - Cayley developed the algebra of matrices (“implementation”) - Cauchy studied eigenvalues (“design”); Camille Jordan & G. Frobenius developed the idea (R&D: diagionalization and canonical Jordan form) - Frobenius organized the theory of matrices in the form it has today (>100 old)

21.1 Number Theory - Legendre published work on number theory 1798 - Carl F. Gauss: investigations in arithmetic Gauss and Congruences - Za, Euler function, units; p prime Lagrange Th. - Wilson Th. (p-1)!=-1 mod p - Gauss integers, norm & factorization;

- Ex. 4n+1 not prime in Z[I] - Analog of F. Th. Of Ariyhmetic Fermat Last Theorem and Unique Factorization - Fermat Last Th. (art or science?); the challenge: if UFD => proof OK. - Attempts to prove: - Euler “Algebra” 1770 for n=3 - Sophie Germain – special cases - Legendre n=5, Dirichlet n=14, Gabriel Lambe n=7 - Lame’s proof relied on UF in Z[α],αn=1 - Liouville’s objection - Kummer’s article 1844: UF fails in some Z[α]. Kummer and ideal numbers - Cyclotomic integers, conjugates f(αi), norm of f(α) (multiplic.) - Prime are irreducible elements - Introducing “ideal complex numbers” which factor in “ideal prime numbers” (trying to extend/complete numbers to get UF)

Dedekind (1831-1916) and ideals - 1871 D. found the correct def. for an integer in a domain of # - Kummer defined only divisibility by “ideal numbers” - Dedekind defined:

- Algebraic number f(θ)=0 in Q[x], algebraic integer - Algebraic number fields Γθ, divisibility, prime,

irreducible, Euclidean domain (domain w. norm s.t. Remainder Th. holds) => Euclid’s alg. => UF

- Ideals (divisibility w. ideal numbers => “axiomatic properties”: closed under + & x Γθ)

- Principal ideals, divisibility J|I � I⊂J - Prime ideals, maximal ideals => prime - Products of ideals (+ & x) - Properties: I|C => ∃! JI=C; UF

- The extension/completion: Γθ -> Ideals to enable UF. - Compare: V. Sp., extending from vectors to subspaces - Ex. p.661 (UF)

21.2 Solving algebraic equations - Central theme in 19th century - Cyclotomic extensions Q[a], an=1 (reduces to n prime) - Eq. solvable by radicals: reducible to xn=A - Constructing an n-gon ; n-1 must be 2k (otherwise deg. Eq>2) - Problems not solvable by quadratic extensions: - Trisecting an angle, doubling a cube - Squaring the circle - Transcendental numbers: - Liouville 1844 ∑ 10-(k!) - Charles Hermite 1873: e transcendental - Ferdinand Lindermann 1882: π transcendental - Cauchy: theory of permutations – degree, cycle - 1798 Paolo Ruffini: n=5 not solvable by √ (unreadable) - 1820 Niels H. Abel: solid proof Work of Galois (1811-1832) - Explained the idea of field extension - Introduced the group of the equation (Galois group) - Normal subgroups, tower of groups - Eq. solvable by radicals iff there is a maximal tower of cyclic extensions Jordan and Groups by Substitution

- Treatise of Algebra – including Galois work - Defines groups of permutations / transformations - Composition series : maximal tower of normal groups - Solvable group G <-> solvable equation <-> cyclic factors - Abelian eq. <-> G abelian => “resoluble” => solvable

21.3 Symbolic algebra - (England) Peacock studied basic properties of symbols representing numbers - De Morgan identifies Laws of Algebra - Hamilton studies C and discovers quaternions Q - Quaternions and the algebra of vectors - Bool and logi (Boolean algebra and functions) 21.4 Matrices and Systems of Linear Equations - Mid 19th century matrices were defined and theory developed - Eigenvalues, eigenvectors, characteristic equation, canonical forms, - Solutions of systems AX=B, rank, determinants, linear independents, solutions 21.5 Groups & fields – The Beginning (!) of Structure - Gauss discussed the theory of quadratic forms - Kronecker develops the F. Th. of Finite Abelian Groups - Cayley defines abstract groups in “On the Theory of Groups”

- Cayley table, isomorphism, free groups - H. Weber defines abstract fields

Ch. 22 Analysis in the 19th Century - Increased concern for rigor - Augustin-Louis Cauchy – most prolific math. of 19th c. - C: Established calculus based on limits (Convergence & Cauchy sequences; also by Bernhard Bolzano 1817) - C: defined integration in C, Th. known as Green’s Th. - Divergence Th.: M. Ostrogradsky & W. Thomson Progress in probability and statistics - Legendre: method of least squares - Gauss: derivation of the normal curve of errors - Laplace synthesis of the ideas of probability - Quetelet & English school: R&D basic tools 22.1 Rigor in analysis - Limits, continuity, intermediate value theorem, convergence, derivatives, integrals (Cauchy: sums rather than antiderivatives), Fundamental Th. of Calculus (Part I), Fourier series, Riemann integral, uniform convergence - What is a function? (p.724) 22.2 The Arithmetisation of Analysis - Dedekind: cuts (completion of Q), axioms of reals (∃! Com-plete ordered field) - Cantor: fundamental sequences (Cauchy sequences!), theory of sets: 1895 & 1897 (language; math code), continuum hypothesis.

22.3 Complex analysis - W. R. Hamilton developed C in abstract form as pairs of #s - Geometric interpretation of C: Casper Wessel 1797 (essay) - 1811 Gauss: defined complex integral for analytic functions & claimed independence of path - 1825 Cauchy’s Integral Theorem (independence of path) - 1777 Euler: analytic functions s.t. Z(x+iy)=Z(x-iy); by diff. => Cauchy-Riemann equations Mx=Ny & My=-Nx 1821 Cauchy Cours d’Analysis - Includes the study of C-functions (cont., conv. Etc.) - Residue of f at z1; (circular int.) ∫f=Res(f;z1) - 1846 Cauchy paper: C-fn. & line integrals - “Green’s Th.” ∫df=0 (+C,-C1,…-Cn) - In general ∫ω=∫∫dω - Cor.: Cauchy’s Integral Th. - 1828 George Green (1793-1841) – work on electricity and magnetism - Relation between ∫C=∑∫Ci & homology theory

Riemann and Complex Functions - Dissertation: Foundations for a general theory of fn. of 1-C variable - Defined C-functions as C-differentiable (∃ dw/dz) => - Preserve angles: conformal mapping => CR-equations => u,v are harmonic functions - Euler & Gauss new analytic fns. preserve angles - Riemann Mapping Theorem: D1, D2 simply connected => there is f:D1->D2 conformal transformation - Proves Green’s Th.: ∫∫divF dV=∫Fn dS - Introduced the idea of a Riemann surface (covering space & mapping); discovered analytic continuation (f determined by values on a “small domain”; Cauchy’s th.) - Beginnings of topology (compact surfaces, R.S. & string th.)

22. 4 Vector analysis - 1851 Riemsnn’s proof of “Green’s Th.” ∫∫dω=∫ω (in R2) - Line integrals and multiple connectivity; e.g. annulus, torus - Surface integrals and divergence th.: 1826 M. Ostrogradsky (1801-1861) Russia - George Stokes (1819-1903) - Stokes Th. ∫∫dω=∫ω (Green’s th. in R3) as Problem 8 at Smith’s Prize Exam at Cambridge U. 1854 (Maxwell did sit at the exam!) - Th. also stated by W. Thomson in a letter to Stokes 1850 - Higher “Clairaut-like” results: closed forms, as an integrability condition: they are exact in simple domains. - Gibbs: pure vector form of div. & Green’s th.:

∇σ=-div σ + ∇ x σ, ∫∫∫∇σ=∫∫σ - 1889 Vito Volterra (1890-1940): unified the results & generalized them as part of the study of hypersurfaces in n-dim space - H. Poincare (1854-1912): also generalized the results

- Closed forms - integrability condition (e.g. Clairaut Th.); - HPs Lemma (closed forms are exact in star-domains)

- Beginning of algebraic and differential topology

Ch. 23 Probability and Statistics in the 19th Century - Application of statistical methods in various fields => standard statistical techniques: least squares, normal curve interpretation etc. 23.1 The Method of Least Squares - Most important statistical method (regression: linear etc.); for N=1=> mean of m observations - Legendre applied the tool of Calculus (minimum of an objective fn. & linear algebra problem) - Gauss & derivation of the method - Priority dispute - Based on Cramer’s method => Gaussian elimination - Developments by Simpson, Bayes, Laplace: the error function : exp(-m|x|) - Gauss & normal distributions : C exp(-m x2) - First developed by De Moivre (work on probabilities) 23.2 Statistics and the Social Sciences - Quetelet & probable error - Galton & correlation coefficient - Other English statisticians: F. Edgeworth, Karl Pearson, George Udny Yule, Ronald Fisher 23.3 Statistical Graphs …

Ch. 24. Geometry in the 19th Century - Riemann’s 3rd topic for “Habilitationschrift”: “On the Hypothesis which lie at the foundation of geometry” (1954). Contained few details, but was packed with so many ideas about what geometry should be about … => Riemannian geometry studied for more than 100 years. - Applications of analysis led to new geometric ideas - Gauss considered ideas of differential geometry on surfaces, but clarified his ideas while completing a geodetic survey: - Intrinsic geometry & geometry of the embedding - Relation between curvature & sum of angles - Believed the parallel postulate had alternatives experiment has to decide; - 1820s N. Lobachevsky and Janos Bolyai published 1st full treatement of non-euclidean geometry; impact 40 years later! - B. Riemann & Herman von Helmholz 1868: studied the notion of general geometric manifold (n-dim Riemann manifold) - What geometry applies to our world? - A. Einstein’s Theory of General Relativity generalizes Gauss’s ideas & Newton’s approach: T=κG (Newton’s Law: F=m a)

(Enery-momentum tensor ~ Geometry/Ricci curvature) - Fundamental idea: “curve the space-time” NOT the motion! (Lines in curved S-T rather then curves in flat S-T)

- Advances in projective geometry: Poncelet, M. Charles, J. Plucker Felix Klein’s work - Connection between non-Euclidean geometry & projective geometry using the idea of metric (1871) - 1872: defines geometry in terms of transformations (Erlangen’s Program), connecting geometry with theory of groups - 1844 Hermann Grassmann: attempted a detailed study of n-dimensional vector spaces from a geometric point of view (algebraic tools rather then axiomatic); impact 40 years later! - Giuseppe Peano: set of axioms for a vector space - Ellie Cartan: applied Grassmann’s work to differential forms - Need for revising axiomatics of geometry (flaws in Euclid’s reasoning) => David Hilbert’s axioms.

24.1 Differential Geometry - Gauss’s work on surfaces - Curvature: lim n(A)/A when A->p & n:S->S2 Gauss map - Values in coordinates (see book!) 24.2 Non-euclidean geometry (elliptic on sphere, parabolic in plane, hyperbolic on ``saddle’’) - Alternatives to the parallel axiom; Lobachevsky & Bolyai; models: Poincare disk & sphere - Riemann’s hypothesis on geometry: manifolds, metric; agreeing with Gauss: experience must decide! (Big-band & Huble; universe oscillates?) 25.3 Projective Geometry - Cayley provides a metric for projective geometry - Klein’s Erlengen’s Programm 1872 - Unifying view for geometry: the study of properties of figures invariant under a group of transformations - Examples: Euclidean geom.. & rigid motions, similarities and reflections (Principal group)(i.e. Euclidean metric and isometries and conformal transformations) - Projective geometry: properties invariant by projections (projective transformations: colineations – “lines to lines”) - Principal group ⊂ Projective transformations => Th. in Proj. Geom. are true in Euclidean Geometry - Impact after translation in Italian, French, English; became central facet in geometry research till 20th C.

25.4 Graph Theory and the Four-Colour Problem … 25.4 Geometry in N-dimensions - H. Grassmann (1809-1877) - Def. “geometric multiplication”: geometric product of 2 and 3 vectors (parallelogram, parallelipiped) - Def. “Extensive quantities”: linear combinations of vectors and products of vectors (exterior algebra) - Vector space, exterior algebra (versus symmetric algebra) - Non-linear case (manifolds):

- Differential forms ; developed by E. Cartan (1869-1951) - Formal expressions, - Exterior derivative , - Integration on k-dim. manifolds & Stokes Theorem.

25.6 The Foundations of Geometry - Late 19th Ctry: appearance of axiom systems for various mathematical structures (groups, fields, V.sp., N, R etc.) - Model: Euclid’s axioms (incomplete though) - David Hilbert’s axioms (1862-1943) - 1899 Grundlagen der geometry - Idea: primary concepts (point, line, plane) + relations (5 sets of axioms: connection, order, parallels, congruence, continuity & completeness) - Axiomatic systems:

1) Consistency – typical proof: existence of a model within another system (!); ex.: geometry consistent if arithmetic is.

2) Independence – by constructing various models (A & B, A & nonB => A & B independent)

3) Completeness, i.e. every Statement may be proved or disproved (“connectedness” – existence of a “logical path”) - Role of Hilbert’s work: reinforced that any math. field must start with undefined terms and axioms specifying relationships (“logical category”)

Ch. 25 Aspects of the 20th Century and Beyond - Emmy Noether – mighty imagination (top-down design: idea/ project/ implementation) - 1976 K. Appel & W. Haken: proof of 4 Color Theorem (1852) with computer - Math output in 20th C.

- Far exceeds all other centuries’ output altogether - Undergraduate texts barely scratch the surface

=> Author selects 4 topics closer to U-grad. curriculum. I) Foundations of Math - Cantor’s work: - Trichotomy of cardinals: A < B, =, >; proof difficulties - Axiom of choice needed (1904 Zermelo; Zorn’s Lemma) - Axioms of set theory & Bertrand Russell’s paradox (“set of all sets”) - Kurt Godel’s Incompleteness Th. - Any axiomatic system containing arithmetic is incomplete => “failure” of axiomatic systems (End of the “chase” for a “complete system”; compare with GUTs) - 1900s Math: focus on axiomatic formalism; physics theoretical output rate: higher (focus on adequacy: correspondence with experiment) - 1940s Category Theory

- Introduces “classes”, not sets - Object (& relational) oriented language ) - It’s a “high level language” versus “math code”/ set th.

II) Growth of Topology - Point set topology (open, closed sets instead of a distance) - Combinatorial topology (counting components, holes etc.; Euler characteristic) - Algebraic topology & homology (language: category theory) Emmy Noether (1920 Gotingen - from Betti numbers to homology groups); III) Algebrization of topology (& geometry) - Language: Category Th. (1940s S. Eilenberg & S. Mac Lane) - Algebraization of analysis

Function analysis: Banach spaces and operator theory; C*-algebras & Quantum mechanics

IV) Development of Electronic Computer & their Applications - Algebra helped grow machine computations - Early attempts: C. Babbage & Ada Byron, Alan Turing & John von Newman - Math areas affected: error-correcting codes, linear programming, theory of graphs

25.1 Set Theory: Problems & Paradoxes - Trichotomy; related with well-ordering; one of Hilbert’s problems at 1900 ICM in Paris - Zermelo’s Th.: If M contains its subsets => contradiction (~ Russell’s paradox: “the set of all sets” => contradiction) - Axiom of choice: one may chose one element of each of the sets of a family of sets (kind of “obvious”!) - Zorn’s Lemma: If ascending chains are bounded => there is a maximal element (� Axiom of choice) - Axiomatization of Set Theory 25.2 Topology - Point-set topology grew out of Cantor’s study of real numbers (open & closed sets, connectedness etc.) - Frechet & function spaces: applying topology - Hausdorff and topological spaces: top. Space (family of neighborhoods), separation axioms - Combinatorial topology - E. Betti generalized idea of multiple connectivity to n-dim. Spaces; Betti numbers (dim homology spaces; ex.) - Homology: J. W. Alexander 1926: p-simplex, complex of simplices (ex. tetrahedron), chains, cycles & boundaries; homology groups & Euler characteristic

25.3 New Ideas in Algebra - Efforts to develop sets of independent axioms in algebra - New structures were introduced axiomatically - Kurt Hensel (1861-1941) discovered a new type of field: p-addic numbers (based on valuations) - Classification of fields 1910: prime fields, char. 0 or p; extensions: algebraic or transcendental; finite / infinite; separable/inseparable; algebraic closure - Banach spaces and linear operators 1922 The Theory of Rings 1870

- Algebras; nilpotent & idempotent elements; division algebras: R, Q, C, H, O.

- Noether: theory of Unique Factorization in more general rings: Noetherian rings (satisfy the ascending chain condition; ex. PIDs => UFD) Algebraic Topology - One of Noether’s suggestions (=> whole new area of study) - Contributions: L. Vietoris (homology group of a complex 1927), Walter Mayer (1887-1948) - Cohomology groups – de Rham (differential forms & exterior differential) capture the topological structure of a space; provides invariants, but … Poincare conjecture (finally proved recently; “simple problem” / difficult proof) - Homology is functorial (manifolds & continuous functions are mapped to groups and group homomorphisms)

Category Theory (1945) - Samuel Eilenberg (1903) & Saunders Mac Lane (1909) - Categories (objects & relations) and functors (correspondences between categories); ex. Sets, Top, Grp, k-Vect; - “Cousins”: graphs, networks, automata etc. - Generalizes in some sense Klein’s Erlengen program - The study of category and functors: building templates for “universal” mathematical structures (“Researcher’s Tool Box”) - Category Th. is on “object-oriented” math language, NOT developed for the sake of abstraction 25.4 The Statistical revolution …

25.5 Computers and Applications - Computers are used in mathematics: examples & proofs - Some branches of mathematics developed because of the need in Computer Science (CS) - Pre-history of computers (1600-1700) - Used to compute astronomers’ tables - Babbage’s difference engine 1821 - Turing 1936: computability, automaton (I,O & transitions) Claude Shannon - Algebra of switching circuits (Boolean algebra) - Ex. x,y…in Z2 & gates: And, Or etc. - Correspondence between circuits & Boolean functions - 40 Years later: quantum gates & unitary operators - Theory of information … - Von Newman’s Computer (1903-1957): 1951 Sections: arithmetic unit, memory, control, IO-device - Error detecting and correcting codes: R. Hamming 1950 (Hamming (k,n)-(linear)code), parity check etc. - Linear programming - Max/min linear functions w. constraints inequalities (difficulty: many (in)equations): simplex method - Applications in economics - Graph theory - Graphs; Euler’s 1736 bridges of Konigsburg - Hamilton’s problem: find a Hamiltonian circuit (& marketing: soled the game!)

- A. Cayley 1857: trees, rooted trees (ex.); applications to chemical isomers - Modern applications of graphs (100 years later): DE in NC spaces & physics (Feynman graphs, QFT & renormalization) - Another “toy problem” 4 Colors Theorem (1852 De Morgan’s letter to Hamilton; conjectured by a student: F. Guthrie) - Tools developed - Dual graphs – Whitney 1931 - Proof by computer: 1976 Debate 1: Is it a proof or not? (If not checked by a human mind) - Deep Blue – won against world chess champion (Kasparov?) - Classification of simple groups: “distributed proof” (many minds) Debate 2: Is “T/F” the only math main question? - Other: “Is it useful in applications?” (Ex. Feynman integrals: mathematically they DO NOT EXIST – until recently; but extremely useful in FAST computing probabilities for experiment outcomes) - Physics “hand waving” proofs as “blue prints” for mathematical proofs. 25.6 Old Questions Answered - The proof of Fermat’s Last Theorem - The classification of Finite Simple Groups - The Proof of the Four-Colour Theorem - The Poincare Conjecture (& Thorston’s geometrization programme).

“Ch. 26” Last Few Decades (1970s-2000) (History of Mathematics … and its Applications) - Categorification (1990s):

- From numbers to sets (& operations) - “Higher dimensional algebra” (J. Baez, L. Crane, L.I.) - From single object to multiple-objects & higher hierarchy

- Quantum algebra and quantum topology - QFT & Feynman Path Integral - Hopf algebra techniques in perturbative QFT - DE in the non-commutative world – Feynman Series - Quantum Computing - Feynman processor – Modeling the Quantum World - Why not the other way around? - Classical computing versus Quantum Computing - Quantum encryption based on entanglement

(Teleportation: “Beam me up Scotty!”) - AI (www.intellibuddy.org & Alice; Turing test)

I) Quantum Computing and Quantum Information Classical computing & information processing - Algorithm: precise recipe - key concept - Model for algorithm: Turing machine/ automaton (change program => change transition function) - Elementary components: gates (circuits / Boolean function); - Ex. 1-bit NOT gate – states & truth table (transition function) Q. Computing & Q. Information - Global perspectives - QC: the study of information processing tasks accomplished by quantum mechanical systems (bulky comp. => decoh.) - Fields contributing to QC: QM, CS, IT etc. - Math: “old” (operator theory etc.) with new applications - Future directions & implications: think physically about

computations. Quantum bits - bit: 2 states – fundamental concept of CC - qubit: 2 basic states & C-linear combinations - Changing coefficients: from Z2 to the (complex) Circle - Consequences of this “modest” change: - Superposition and interference - From probabilities to amplitudes - Enables “parallel computing” (exp versus poly input) - Multiple qubits – quantum memory

Quantum gates & Quantum circuits: Examples - Classical NOT gate & Q-NOT gate - Hadamard gate = “square root of NOT”; no classic analog! - Quantum gate = any unitary matrix! => Q-circuit: any U.mat. - Quantum computing = changes occurring to quantum states (so … unitary transformations … Operator Theory!) - 1st Problem: the hardware! (How to build circuits using atoms, photons etc.?!) - 2nd Problem: the software! (How to find quantum algorithms for our “classical problems”?) Examples - Q-algorithms: Peter Shor’s FT & factorization

(poly. time! but no hardware yet!) - Q-teleportation & cryptography

(“Beam me up, Scotty!” Done on “short” distances) - Q-parallelism & interference: Deutsche’s algorithm Math needed: linear algebra + bits & pieces …

Path Models - Models for systems composed of interacting subsystems (what else is there!?) - Linear problems: OK (geometry); non-linear: use perturbative methods (ex. Newton & power series; DE & p.s. solutions) - Models: global (DE in “spaces”) & relational (Path Models - appropriate language is Category Theory) - Applications of Path Models to: biology, chemistry, physics, engineering, probability & statistics, computer vision … and everything else! - Uses all kinds of math: graphs, measures, distributions etc. from algebra, analysis, geometry etc.) - Notable Path Models: Feynman-Kac formulae (Probability & statistics) and …

Feynman Path Integral methods in QFT - Elementary particles, Wilson chambers and … graphs! - Basic idea: solutions are expansions over graphs representing the amplitude of probability as a sum over possible histories (paths). - Markov chains: states and transition probabilities - “Feynman process”: states & multiple-transition amplitudes (multiple edges & complex numbers). - Dynamics: integral equations <-> DE & looking for graph series solutions! - Feynman Path Integrals & Quantum computing: deeply interrelated! (“Information is physical”, Rolf Landauer)

List of Questions for the Final Exam 1. What is analytic geometry? (More about it: Who? When? Why? if appropriate). 2. Number theory in 17th century: who, what, when and why? 3. What is Fermat’s Little Theorem? What is Fermat’s Last Theorem? 5. Explain “the congruence classes modulo a prime number” (briefly). 6. What is projective geometry? (More: WWW). 7. Describe the mathematical framework for probability. Use and example including the main “keywords” or the axiomatic approach (define a probability space as a special type of measure space). 8. Who are the founders of Calculus? 9. What are the Newton’s “fluxions”? What are Leibniz’s differentials? 10. What are indivisibles? Briefly explain the “progression” Galilei-Cavalieri leading to a certain integration method. 11. Explain the relation between integrals and differential equations. 12. What is, in modern terms, the Tangent Problem and the Area Problem? What is the Rectification of Curves? 13. Lagrange developed the Calculus of Variations. State and explain briefly on what principle it is based.

14. Newton’s Law F=ma is a special case of ______________ equations. (Fill the missing words). 15. Konigsberg bridges can be modeled as a graph (“unoriented” = not directed), as a special instance of a CW-complex of dimension 1. (a) What is the key ingredient used to define such a complex; (b) What part of algebra studies these objects and maps? 16. What is “Topology”? 17. What is a “Number System” (ring)? What is a field? 18. What is an irreducible element? What is a prime element? What is the relation between the two? 19. What is an ideal in a “number system” (ring)? What is a principal ideal? 20. What is an algebraic number? What is a transcendental number? 21. Who invented the quaternions? It was a progression of “doubling” number systems, starting from N …? 22. Where are pure quaternions (generated by i, j, k) used and what corresponds to their quaternion multiplication? 23. How did Hamilton construct negatives? 24. How did Hamilton construct complex numbers? 25. (a) Who is the founder of Set Theory? (b) He also introduced “fundamental sequences” now also known as “____________ sequences”. (Fill the space).

26. Name 2-3 theorem from analysis dating from 19th century (Think “Calculus”). 27. When is a complex function differentiable? 28. (a) What can you say about f=u(x,y)+iv(x,y) knowing that u and v satisfy Cauchy-Riemann equations? (b) To what part of analysis does the result belong? 29. What are the mathematicians and theorems (responsible for) generalizing the Fundamental Theorem of Calculus? 30. Stokes Theorem with differential forms and boundary operator is part of algebraic _________ and differential ________. (Fill the space). 31. What mathematical tools did Maxwell use to formulate Electromagnetism? 32. What is the framework of Riemannian geometry? 33. (a) What is the mathematical foundation of General Relativity? (b) In what sense is Gauss a precursor of Einstein? 34. What is “geometry” in the eyes of … Felix Klein? 35. How does exterior differentiation act on differential forms? 36. What is integration, really? State Stokes Theorem using differential forms, in view of the duality pairing between manifolds and differential forms. 37. What is an “axiomatic system”? Give an example.

38. What are the properties we require from an axiomatic system? 39. The “low-level” mathematics language founded by Cantor is ___________. 40. The “high-level, object-oriented” math language invented by Eilenberg and MacLane is _________. 41. (a) Is arithmetic, as an axiomatic system, complete? (b) Says who? 42. (a) What is a category, roughly? (b) What is a functor? 43. State two prominent names in the area of computability and theory of information. 44. What is an automaton, roughly? 45. What is the Turing Test for Artificial Inteligence? 46. How would you describe Feynman path integral approach to quantum field theory, roughly, when compared with Markov processes? 47. What “mathematics tools” are needed in Quantum Computing? 48. What part of this history course you liked more? … less? 49. Is mathematics Art or Science? (Brief answer please) 50. Name a couple of completions, extensions or generalizations you emphasized in your Project 2.