PM 467/667 Algebraic Topology, Course Outline, Winter 2008
Course: MWF 10:30-11:20, room MC 4041. Instructor: Stephen New, oce MC 5163, extension 5554, oce hours MTW 11:40-1:20 Text: A Basic Course in Algebraic Topology, by William Massey. References: Top
Pure Math 354, Midterm Topics Midterm Exam: February 26, 7:00-8:30PM in MC 2035. The midterm examination will be on a selection of topics from the measure theory section of this course. The exam will contain definitions, statements of named theorems (some
PMATH 467/667 Algebraic Topology, Assignment 1
Due: ?, 2009
1: Determine (with proof) which of the following statements are true for all topological spaces X, Y and Z. (a) If X is connected and is an equivalence relation on X, then X/ is connected. (b) If
PMATH 467/667 Algebraic Topology, Assignment 3
1: (a) Let G = a, b a3 = 1, b9 = 1, a = bab . Show that G Z3 . =
Due: Wed Mar 25, 2009
(b) Let Q be the quaternion group; Q = a, b a2 = b2 , a = bab . Show that |Q| = 8. (c) Let G = a, b, c abcbac = 1 and let
PMATH 467/667 Algebraic Topology, Assignment 4
Not to hand in.
1: (a) Let : I2 X. Define : I2 X by (s, t) = (t, s). Find ( - - ) in terms of and , where , , : I3 X are defined by (u + w, v) , if u + w 1 (u, v, w) = (1, v) (u, v, w) = (1, 0) (u, v, w) = (1
PMATH 467/667 Algebraic Topology, Assignment 2
1: Let (t) = eit + 2ei4t for 0 t 3. (a) Sketch (the image of) the path in C . dz (b) Evaluate each of the path integrals , z
Due: Fri Feb 13, 2009
dz and z+2
dz . + 2z
2: Find 1 (X, a) for each of the foll
Skills Summary _
Good analytical skills
Execellent stamina for long
Flexible and adaptable
Fork lift truck
Great attention to detail
Course Outline - PMath 766 -Introduction to Knot Theory - Fall
Instructor - Louis H. Kauman
Dept of Combinatorics and Optimization
University of Waterloo
Waterloo, On N2L 3G1
(519) 888-4567 x5596
Oce MC 5132
Let V be a vector space over the eld K, and let W be a vector space of V . That is, W
is a non-empty subset of V for which x, y W and k W implies that kx + y W . Given
v V , dene the coset of W with representative v by
v + W := cfw_v + w :
Due Wednesday March 30.
(a) Let be a convex region. Suppose that f (z ) is analytic on and Re f (z ) > 0 for all
z . Prove that f is one to one.
Hint: nd an expression for f (zz2 f1 z1 ) as an line integral.
(b) Show by examp
PM 352 Assignment 1 Solutions
1. Write f (x + iy ) = u(x, y ) + iv (x, y ). Then
g (x + iy ) = f (x iy ) = u(x, y ) iv (x, y ) =: u(x, y ) + iv (x, y ).
Thus u(x, y ) = u(x, y ) and v (x, y ) = v (x, y ). Therefore
vx (x, y ) =
x (u(x, y )
x (v (x, y
Due Friday February 12.
1. Suppose that T is a Mbius map which takes R to itself and sends to 0.
(a) What is the image of the family of lines parallel to R?
(b) What is the image of the family of lines perpendicular to R?
Due Friday January 29.
1. Let f (z ) = e1/z for z = 0. Let r > 0, and set Ar = cfw_z C : 0 < |z | < r. Determine the
range f (Ar ). Hint: Solve e1/z = a.
2. Dene sin z =
and cos z =
for z C.
(a) Show that sin(w +
Due Monday March 22.
Evaluate the following integrals using the residue theorem. Be careful to specify the
curves that you integrate over, and explain why extraneous terms go to 0.
a + sin2 x
for a > 0.
PM 352 Assignment 2 Solutions
1. If a = 0, there are no solutions. If a = rei = 0, then z = log r + i( +2k ) for some k Z. Hence
z = log r+i(+2k) for k Z. Thus this set of solutions contains complex numbers of arbitrarily
small modulus, so there are s
Due Wednesday March 10.
1. Show that every convex region is simply connected.
2. Let U be a simply connected open set in C, and suppose that f (z ) is analytic on U and
never vanishes. Show that there is an analytic function g (z ) on
PM 352 Assignment 6 Solutions
1. (a) Let z1 , z2 . Then (t) = z1 + (z2 z1 )t for 0 t 1 be the straight line from z1
to z2 , which lies in because is convex. Observe that
f (z2 ) f (z1 )
z 2 z1
f (z ) dz =
f ( (t) (z2 z1 ) dt.
PM 352 Assignment 4 Solutions
1. Pick a point p in the region R. If (t) for 0 t 1 is a closed curve in R, then the line
segments from p to each point on lies in R. So we can dene a homotopy by
(s, t) = (1 s) (t) + sp
0 s, t 1.
Check that this is conti
PM 352 Assignment 5 Solutions
1. (Type I) Let (x) = eix for 0 x 2 and set z = eix and dz/iz = dx
a + sin2 x
a + sin2 x
z 2 1 2
2(2a + 1)z 2 + 1
|w |< 1
2(2a + 1)z 2 + 1
Solve z 4 2(2a +
1.1 Denition: Let X be a set. A topology on X is a set T of subsets of X such that (1) T and X T . (2) If U T and V T then U V T . (3) If U T for all A, where A is some index set, then U T .
A topological spa
PMATH 955: Assignment 1
Due: Wednesday, 10 February, 2010
1. Tangent and normal bundles of submanifolds of Rn . Recall that a smooth
k -dimensional submanifold of Rn is a subset M Rn with the following
property: given any point p M , there exists an open
PMATH 955 - Topics in Geometry
Instructor: R. Moraru
Course title: Topics in Geometry: Gauge Theory
Lectures: MW 9:00-10:20, MC5046
Office hours (MC 5170): W 11:00 12:00 and Th 15:00-17:00 (subject to change) or by
PMATH 955: Assignment 1
Due: Friday, 20 January, 2012
In the this assignment, V is an n-dimensional vector space over a eld k .
1. Let : V V k be a bilinear form and W V .
(a) Show that is non-degenerate if and only if the induced linear map
: V V , X (Y
PMATH 955: Assignment 2
Due: Tuesday, 7 February, 2012
In this assignment, V is an n-dimensional real vector space.
1. Induced Dirac structures. Consider the linear Dirac structure on V given
the pair (E, E ), where E V and E : E E is a skew-symmetric
PMATH 955: Assignment 3
Due: Thursday, 1 March, 2012
1. Let M be a smooth manifold and let E and E be smooth real or complex
vector bundles bundles on M . Suppose that E has rank r and E has rank
r . Prove the following:
(a) If E is a real smooth vector b
AMATH 361 COURSE SUMMARY
- Forcing and Response:
r (t ) = () f (t )d
- Kernel & Fading Memory: () 0 as
+ q 0 r = p 2 2 + p1
+ p0 f
drdp (t )
f dp (t ) =
f sp (t ) = Grsp (t )
- Spring & Dashpot
AMATH 473 COURSE SUMMARY
History of Quantum Theory
- Planck and Thermal [Blackbody] Radiation
- Einstein and the Photoelectric Effect:
- Bohr and Atomic Spectral Line Emissions
E = h = h
- Brief Intro to Relativistic Quantum Mechanics and Q
AMATH 331 COURSE SUMMARY
A metric space X is a non-empty set on which a distance function, or metric, is defined.
A function d : X X R is a distance function, or metric, if is satisfies the following:
d ( x, y ) 0 x, y X
ii) d ( x,