Date
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What we discussed/How we spent our time
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Aug 28
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Syllabus. Text.
We discussed CE/BCE notation for dating. (No year zero.)
We discussed a coarse timeline of development for Homo sapiens.
We discussed some of the the earliest mathematical objects/writings,
namely:
The Ishango bone:
Plimpton 322:
The Rhind papyrus:
The Moscow papyrus:
We discussed an
ancient algorithm for how to solve the
simultaneous system $x+y=s, xy=p$,
and explained why this was equivalent to solving
a general quadratic equation.
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Aug 30
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We gave several proofs of the Pythagorean Theorem,
then discussed the chord and tangent method
for producing Pythagorean triples.
Here are 118 proofs of the Pythagorean Theorem.
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Sep 1
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We showed how to derive the
formula for Pythagorean triples
from the parametrization of
rational points on the circle.
We defined segments $a$ and $b$
to be commensurable if there is a
segment $c$ and positive whole numbers
$m$ and $n$ such that $|a| = m*|c|$,
$|b| = n*|c|$.
We mentioned that $a$ and $b$ are
commensurable iff the ratio of their
lengths is a rational number.
We proved $\sqrt{2}$ is irrational by reductio ad absurdum.
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Sep 6
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We introduced and justified the
Euclidean algorithm for finding
the gcd of two positive whole numbers.
We gave a geometric version of the
algorithm and explained how it could be
used to determine the commensurability
of two lengths. (Construct an $a\times b$
rectangle and repeatedly delete maximal square
subregions. The algorithm terminates iff
$a$ and $b$ are commensurable.) We used this
algorithm to show that neither $\sqrt{2}$
nor the Golden Ratio is rational.
We also used the algorithm to solve Bézout's identity:
given integers $m, n$ such that $\gcd(m,n)$ divides
$p$, find integers $x, y$ so that $mx+ny=p$.
Quiz 1.
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Sep 8
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We worked on this handout.
We discussed the quiz. We
began a discussion of Euclid's Elements.
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Sep 11
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We discussed Euclid's five postulates for plane geometry.
We described the Poincare model, which satisfies
postulates 1-4, but not 5 (the parallel postulate).
Quiz 2.
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Sep 13
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We discussed ruler and compass
constructions, and the reduction
of geometry to algebra.
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Sep 15
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We discussed more ruler and compass
constructions, along with Thales' Theorem,
the Inscribed Angle Theorem,
the construction of a regular $5$-gon, and Platonic solids.
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Sep 18
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We discussed four constructions problems,
and started to explain how to algebraize them.
Quiz 3.
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Sep 20
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Today we identified which numbers are constructible,
but have yet to give a good way to recognize these numbers.
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Sep 22
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We defined algebraic and transcendental numbers.
We asserted that the collection of algebraic numbers forms a field,
so, for example, the sum or product of two algebraic numbers is algebraic.
We discussed field extensions. We sketched the explanation
of why it is that if
$\theta$ is a constructible number, then $\theta$
is algebraic and $[\mathbb Q[\theta]:\mathbb Q]$
is a power of $2$. We also sketched the explanation
of why it is that the degree of the minimal
polynomial of $\theta$ must be a power of $2$.
We then stated Lindemann's Theorem, and used it to show that $\sqrt{\pi}$ is
not algebraic, so a circle of radius $1$ cannot be squared with
straightedge and compass. We also noted that $\sqrt[3]{2}$
is algebraic, but its minimal polynomial $x^3-2$
has degree $3$, which is not a power of $2$, so the cube
of sidelength $1$ cannot be doubled with straightedge and compass.
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Sep 25
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Today we discussed the theorem which says that the following
are equivalent about a number $n$:
a regular $n$-gon is constructible with straightedge and compass.
$\cos(2\pi/n)$ is a constructible number.
$\cos(2\pi/n)$ has minimal polynomial whose degree
is a power of $2$.
$\varphi(n)$ is a power of $2$.
$n=2^r\cdot p_1\cdots p_s$ for some $r\geq 0$ and some
Fermat primes $p_1,\ldots,p_s$.
We defined the Euler phi-function, $\varphi(n)$,
and Fermat primes. We listed the known Fermat primes,
$3, 5, 17, 257, 65537$.
Quiz 4.
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Sep 27
|
Read pages 468-578.
Today we discussed Descartes' Total Angular Defect Theorem,
Euler's Formula, Euler characteristic, the classification
of compact, connected, 2-dimensional surfaces, Gaussian curvature,
and the Gauss-Bonnet Theorem. We concluded that
the angular defect at a vertex of a polyhedron
is a measure of curvature.
We computed some things, too, namely the Euler characteristic of
a sphere and a torus and how to predict the number of
vertices, edges, and faces of a regular polyhedron
from a knowledge of the angular defect at one vertex.
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Sep 29
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Read Sections 3.1-3.2.
We used Euler's Formula to show that it
is impossible to draw more than 3 circles
in the plane in general position. Then we discussed
polygonal numbers, the infinitude of primes, and
perfect numbers.
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Oct 2
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Read Sections 3.3-3.4.
We discussed using continued fractions to
to solve Pell's equation.
Quiz 5.
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Oct 4
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We discussed Brahmagupta's Identity,
and his method for solving Pell's equation.
(We also mentioned Brahmagupta's Formula
for finding the area of a cyclic quadrilateral.)
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Oct 6
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Read Sections 4.3-4.4.
We discussed the Method of Exhaustion, including
the Archimedes' Quadrature of the Parabolic Segment.
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Oct 9
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Read Section 5.2.
We defined congruences on $\mathbb Z$,
and discussed the Chinese Remainder Theorem.
Quiz 6.
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Oct 11
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I circulated a
review sheet.
Then we discussed existence and uniqueness
of solutions to systems of congruences,
following this handout.
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Oct 13
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Midterm. (So, no quiz or HW due next week.)
Midterm solutions.
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Oct 16
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Read Sections 6.1-6.5.
We discussed the cubic formula.
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Oct 18
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We worked on this handout.
We began discussing how to extract roots of complex numbers.
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Oct 20
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We discussed extraction of roots in the complex plane.
We then explained how to choose the two cube roots
of the Cardano formula consistently.
Finally, we discussed how to solve quartic equations.
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Oct 23
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We discussed the Quartic Formula.
We discussed the role of Bring radicals
in solving the quintic.
Quiz 7.
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Oct 25
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We discussed the problem of finding the roots
of higher order polynomial equations. We discussed the contributions
of Bring, Lagrange, Ruffini, Abel, Galois and Arnold.
We defined automorphisms of $\mathbb C$, and described
how commutators $[\alpha,\beta]=\alpha^{-1}\beta^{-1}\alpha\beta$
act on the roots of polynomials, when the roots are
expressible with one layer of radicals.
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Oct 27
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We discussed Vieta's Formulas,
the Fundamental Theorem of Symmetric
Polynomials, and the Newton-Girard formulas.
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Oct 30
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Read Chapter 7 and Sections 8.1-8.5.
We began discussing new developments
in geometry between 1630-1900.
Some of the things we discussed today were:
incidence geometry versus metric geometry;
Euclidean, Elliptic, Hyperbolic and Projective geometry.
We began discussing the classification of conics
up to affine transformations.
Quiz 8.
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Nov 1
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We talked about Klein's Erlangen Program, which asserts that
each geometrical language had its own appropriate concepts,
and these are the concepts invariant under a particular
group of transformations.
We talked about transformation groups
of $\mathbb R^2$, specifically (i) the group
of translations, (ii) the orthogonal group,
(iii) the Euclidean group, (iv) the general linear
group, and (v) the affine group.
We discussed the classification of conics
up to the action of a given group.
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Nov 3
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We discussed the classification of parabolas
up to the action of the translation group,
the orthogonal group, and the affine group.
We then asked: how many intersection points between two conics
should one expect? We stated Bezout's Theorem which
answers this question. The theorem requires
us to consider intersection multiplicity,
complex intersections, and intersections at infinity.
We began discussing the third of these by
introducing the projective plane.
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Nov 6
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We defined how to represent the points of the projective
plane in homogeneous coordinates, and showed that
the projectivization of the curve $y=x^2$ is $yz=x^2$,
which has one point at infinity.
Quiz 9.
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Nov 8
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We discussed the homogenization of curves
and how to find those points at infinity
that lie on a curve. We began discussing
projective transformations.
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Nov 10
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We discussed the effect on the equation of a curve
when we change coordinates in the Cartesian or projective plane.
After examining what happens to the equation $y=x^2$
when we translate or rotate the axes in $\mathbb R^2$,
we began discussing the transformation groups
$\textrm{PGL}_3(\mathbb R)$ and $\textrm{PGL}_3(\mathbb C)$.
We proved that, given two $4$-points $\Delta=(p_1,p_2,p_3,p_4)$
and $\Gamma=(q_1,q_2,q_3,q_4)$,
there is a unique projective transformation that maps
$\Delta$ to $\Gamma$.
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Nov 13
|
By using the fact that
$\textrm{PGL}_3(\mathbb C)$ acts transitively
on $4$-points in $\mathbb C\mathbb P^2$, we were able
to explain how to transform an arbitrary irreducible conic
in $\mathbb C\mathbb P^2$ to one whose equation
has the form $x^2+Bxy+y^2=z^2$.
We did not have time to show how to eliminate the $B$ coefficient.
Quiz 10.
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Nov 15
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We completed the proof that
an arbitrary irreducible conic
in $\mathbb C\mathbb P^2$
is projectively equivalent
to one whose equation
has the form $x^2+y^2=z^2$.
Then we discussed
intersection multiplicity.
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Nov 17
|
We began a discussion of Hilbert's Problems.
We said a few words about the life and work of
David Hilbert,
especially about the book
Grundlagen der Geometrie.
We then began discussing the influence of
Hilbert's
Problems.
We started discussing the first problem in detail:
the Continuum
Hypothesis (CH).
We described the
intuition
behind sets,
described the nature of the
ZFC
axioms, defined
cardinality,
discussed
countable versus
uncountable
sets,
stated the
Cantor-Schroder-Bernstein Theorem,
discussed power sets
and
Cantor's
Theorem, and defined
the
aleph
and
beth
numbers. We ended the lecture with the reformulation
of CH, namely $\aleph_1=\beth_1$, and also of the
GCH, namely
$\aleph_{\kappa}=\beth_{\kappa}$ for all $\kappa$.
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Nov 27
|
We reviewed some topics from the Nov 17 lecture,
especially the statement of CH and the
aleph and beth numbers. Then we
discussed an analogy:
On the one hand we have (1) the axioms
of ZFC, (2) the question
of the existence of a 1-1 function $f\colon \aleph_2\to \beth_1$,
(3) Godel's proof that $f$
does not exist in a minimal model of set theory,
and (4) Cohen's proof that any model of set theory
can be enlarged to a model containing $f$.
This can be compared
to a simpler situation: We have (1) the axioms
of fields of characteristic zero, (2) the question
of the existence of a root $i$ of $x^2+1=0$,
(3) an argument that $i$ does not exist
in the minimal model $\mathbb Q$,
and (4) an argument that any field
of characteristic zero can be enlarged to
a model containing $i$. The approach of Godel and Cohen
to the CH problem is similar in spirit to the fields problem.
Quiz 11.
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Nov 29
|
We discussed Hilbert's second problem,
about the consistency of number theory.
We discussed the axioms of PA, the notions
of provability and consistency.
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Dec 1
|
We started to discuss Dehn's solution to Hilbert's third
problem. We began by discussing equidecomposability
in the plane. We proved the Wallace-Bolyai-Gerwien Theorem.
We then defined the Dehn invariant, which takes
values in $\mathbb R\otimes [0,\pi)$.
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Dec 4
|
We defined the tensor product of two abelian groups,
and calculated some Dehn invariants.
No quiz!
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Dec 6
|
We discussed Hilbert's 7th problem, which was resolved
by the Gelfond-Schneider Theorem.
We noted that $1, \sqrt{2}$ and $i$ are algebraic,
but the GS Theorem implies that $2^{\sqrt{2}}$ and $e^{\pi}$
are not.
We explained why the collection of
algebraic numbers
forms a field, and reformulated the GS Theorem as:
If $\alpha$ and $\gamma$ are nonzero algebraic numbers
and the set $\{\log(\alpha),\log(\gamma)\}$ is independent
over the rational numbers, then it is independent
over the algebraic numbers.
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Dec 8
|
We discussed Hilbert's 10th problem, which
asks for an algorithm to determine
if a polynomial Diophantine equation
has an integer solution. We gave several
examples of Diophantine equations:
Bezout's Identity, Pell's equation,
Fermat's equation, the Erdos-Strauss Conjecture.
We discussed the possible meaning of the word algorithm,
with references to
Thue systems (string rewriting),
recursive functions, and Turing machines.
We defined Diophantine,
recursive, and recursively enumerable
subsets of $\mathbb N$.
We ended
with a statement of the
Matiyasevich-Robinson-Davis-Putnam Theorem,
which states that a subset of $\mathbb N$
is Diophantine iff it is recursively enumerable.
Conclusion: There is no algorithm to determine
if a polynomial Diophantine equation
has an integer solution.
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Dec 11
|
Ramsey's Theorem.
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Dec 13
|
Review.
Practice problems.
FCQ's.
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