Variations on Invariance
last updated on February 25th, 2007
NEW MEETING TIME:
ANNOUNCEMENT: the first lecture on the 26th of February has been canceled. We will start on March 5th
* Welcome to the new seminar on algebraic and geometric invariants, and "related" topics (related is not a very mathematical concept, we know...).
* The seminar is targeted to first and second year students of the faculty of Mathematics and Physics (MFF) at Charles University. Everyone who is curious about topics in geometry, mathematical physics and so on can attend to this series of seminars.
* The lectures will be held by various members of the Department of Mathematical Sciences (MUUK) including young researchers, postdocs, and visiting guests. The official language of the seminar will be English.
* For any question, please do not hesitate to contact the organizer of the seminars Lukaš Krump or the administrator of this webpage Alberto Damiano
* Below you can find a list of the topics we plan to cover over the semester. Detailed abstracts and location of the seminars will be announced later.
* HOMEWORK: you are supposed to work on
all homework assignments given by the instructor for at least two of the
topics (=4 lectures) covered. Homework assigned are also posted on this
webpage for your convenience but only as a reminder of what has been said
in class. You should come to class to take notes and understand the
notation. If you don't understand something, please ask the instructor.
You should hand your assignments to the instructor of that particular
topic as soon as possible.
TOPICS (not necessarily in this order):
Quaternions --->
Roman Lavicka
Lobachevski-Poincare geometry -> Lukas Krump
Symmetries of differential equations ->Dalibor Smid
Ramanujan and his history --->
Petr Somberg
Riemann-Hurwitz theorem ---> Svatopluk Krysl
Galois theory --->
Peter Franek
Colouring graphs; knots --->
Alberto Damiano
Euler function --->
David Eelbode
Finite groups --->
Vladimir Soucek
LECTUREs 1&2 : David Eelbode Monday 9th and 16th, room K2 (Karlin 2nd floor), 8:10-9:50
On Euler’s function and its connection with Lie algebras
You can reach me by email at davideelbode"zavinac"gmail.com (replace the "zavinac" with the simbol @) or you can
come by my office, room K489 (Karlin, 4th floor, at the right end of the left wing). You can drop your homework in the
office or in A. Damiano's mailbox (Karlin, 3rd floor, next to J. Richter's office).
HOMEWORK:
HW1: Find all the possible root spaces in the plane. HINT: make a table of all possible cosine values
HW2: find the coefficients of the Euler series E(x)=Πk=1..∞ (1-xk) HINT: find the exponents that survive and divide into two parts
HW3: find SOME integers p such that the power Ep(x) is sparse, meaning that consecutive terms can be arbitrarily distant
HW4: Define ad(X)( - ):=[X, - ] and prove [ad(X),ad(Y)]=ad([X,Y]) for all X and Y in a Lie algebra
HW5: For every root diagram found in HW1, calculate the number of roots +2 and "verify" that this is one of the exponents of HW3
LECTUREs 3,4&5 : Vladimír Souček October 23rd and 30th, November 6th room K2, 8:10-9:40
Representation of finite groups
HOMEWORK:
HW1: Check that the pointwise product [i] * [j] := [ij] in Zn is well defined for every positive integer n
HW2: Prove that the map defined by [k]m --> ( [k]m1 , ... , [k]mn] ) for the Chinese Remainder Theorem is an isomorphism of rings
HW3: Write down all solutions to x2+1=0 in Zp for p prime <100 (or program a computer to do so...)
HW4: Calculate Leg(24,31) where Leg(_,_) is the Legendre symbol defined in class
HW5: Prove that the multiplicative group Zp* is cyclic if p is an odd prime
1) δa * δb = δa+b 2) (δa * f )(x) = f (x-a)
HW7: Write a matrix representing the Fourier transform for Zp using the basis {δa} and the orthonormal basis {ea/√n}
HW8: Calculate the Fourier transform for the functions 1 , ea , δa, 1/2 (δ1+δ-1) in Zp
HW9: non trivial : find a closed form for the Gauss sum g=F[h](-1) where h is the Legendre symbol and F[h] its Fourier transfrom
LECTUREs 6&7 : Alberto Damiano, November 13th and 20th, K2, 8:10-9:40
The Four Color Theorem, graphs, knots and the vector product in
R3Office: K432 (or 489 depending on the numbering... just on the 4th floor, at the end of the corridor, right wing)
email: damiano (at) karlin.mff.cuni.cz
HOMEWORK:
HW0: Go to Strahovský klášter and find an old map of Bohemia which is four-colored.
HW1: Generalize the definition of a graph in order to include also directed (=oriented) graphs and non simple graphs (=with multiple edges and loops). Define also the notion of realizations and embeddings for such graphs in R2
HW2: Give a precise definition of a 1) subgraph 2) cycle 3) tree 4) Kn complete graph in n vertices 5) bipartite graph Ka,b . Which of them are planar? How does an embedding look like? NOTE: be precise with your definitions using G=(V,E) and note that a subgraph is NOT just a graph (V',E') where V' is a subset of V and E' is a subset of E. Something more is required!
HW3: Define an alternative notion of coloring: an edge coloring. Define also a total coloring (find it online...).
HW4: Let Cn be a cycle with n vertices. Add a vertex and connect it with every vertex of Cn. Is it still planar? What is its coloring number?
HW5: Download the pdf file above with my abstract, and find a mistake in the graph of figure 2. Correct it and find the coloring number of EU. How did you proceed coloring the vertices? Can you generalize this procedure? Will it always work for planar graphs? Will it still hold after January 2007? What is a sufficient condition for a planar graph to be three-colored?
HW6: Calculate the Chromatic function P(G,t) for G= K3 and K4 . To do this, first calculate all the values of the function for t=0..5 (or more) and then see if you can make a prediction on the general value of the function for any t. Alternatively, calculate in how many ways you can color the first vertex, then how many colors you can use for the following one etc.
HW7: Prove that the Chromatic function satisfies P(G,t) = P( G-{e}, t) - P( G/e , t) for every choice of and edge e. Deduce that it is a polynomial in t.
HW8*: Prove that if you know the first n+1 values a0,a1,a2 ... an of the Chromatic polynomial, i.e. P(G, i) = ai for i=0..n then the Chromatic polynomial is the interpolating polynomial of the n+1 points (0,a0) , (1,a1) ... (n,an), where n is the number of vertices.
HW9: Calculate the Chromatic polynomial for a cycle Cn and a tree Tn using HW7. Can you also prove the formulas directly without induction?
HW10: Find a sharp solution for an equation L(x,y,z,t) = +/- R(x,y,z,t) of your choice (choose two different bracketings of the vector product in R3 but keep the order of the variables!) and then prove that the solution you found also respects the sign. If you are interested in knowing why, ask me for the paper of Kauffman.
LECTUREs 8,9&10 : Lukáš Krump, November 27th, December 4th and 11th K2, 8:10-9:40
Klein's approach to non-euclidean geometry
contact me: krump (at) karlin.mff.cuni.cz
HOMEWORK
HW1: Prove that the pair of isotropic points { <(0,1,i )> , <(0,1, -i)> } is invariant with respect to all euclidean motions
HW2: consider four points in RP2 as <(A,a)>,<(B,b)>,<(C,c)>,<(D,d)>. Prove that their cross ratio satisfies the formual given in class.
HW3: find the group preserving the following quadratic forms:
a) (1 0 b) (1 0
. 0 1) 0 -1)
LECTURE 11 : Dalibor Šmíd, December 18th, January 8th K2, 8:10-9:40
Symmetry solves differential equations (
or what Sophus Lie taught us and we almost forgot)
LECTUREs 12&13 : Roman Lávička, March 5th and 12th K8, 9:00-10:30
ANNOUNCEMENT: the first lecture on the 26th of February has been canceled.
Quaternions
Abstract: W.R. Hamilton discovered (or invented?)
quaternions in 1843. Ten years ago he realized that
the complex numbers had a deep relation to the geometry of the plane.
Indeed, complex numbers can be considered
just as pairs of real numbers. Then he was looking for a generalized complex
number system that would play
a similar role in the three dimensional geometry. However, such hypercomplex
numbers Hamilton tried to find do
not exist in three dimensions. Finally, in four dimensions he discovered the
quaternions by introducing a
non-commutative product of vectors similar to the product of
complex numbers.
We shall deal with basic properties of quaternions. We shall explain in
detail how to describe rotations in
three and four dimensions using quaternions. In conclusion, we shall discuss
conformal mappings, Möbius
transformations and, in particular, a description of Möbius transformations
in four dimensions using quaternions.
LECTUREs 14&15 : TBA, March 19th and 26th K8, 9:00-10:30
TBA
LECTUREs 16&17 : TBA, April 2nd and 16th K8, 9:00-10:30
TBA
LECTUREs 18&19 : TBA, April 23rd and 30th K8, 9:00-10:30
TBA
Abstract:
LECTUREs 20&21 : TBA, TBA K8, 9:00-10:30
TBA
Abstract:
LECTUREs 22&23 : TBA, TBA K8, 9:00-10:30
TBA
Abstract: