Weil’s Rosetta stone (or Conjecture):
Number Theory (1) | Curves over Finite Fields (2) | Riemann Surfaces (3)
Weil wanted to link up these 3 distinct Maths, as in the Langands Program.
Langrands’ original idea on the Left Column (1) Number Theory & the Middle Column (2):
1. He related :
representations of the Galois groups of number fields (objects studied in number theory)
to:
automorphic functions (objects in harmonic analysis).
2. The middle column (2):
Galois group relevant to curves over finite fields.
Also there exists a branch of harmonic analysis for automorphic functions.
3. How to translate column (3) Riemann Surfaces ?
We have to find geometric analogues of the Galois groups and automorphic functions in the theory of Riemann surfaces.
Next we have to find suitable analogues of the automorphic functions ?
It was a mystery until 1980 solved by the Russian Vladimir Drinfeld (Fields medalist for…
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and measures should be additive over disjoint sets in X.
defined on a 
-algebra X of subsets of X such that
for all 
is any disjoint sequence (
) of sets in X, then
.
, we say it is finite. More generally, if there exists a sequence 
and such that 
for all n, then we say that 
be definied on X by 
, for all 
. We can see that 
be defined by 
, 
if 
. 
defined on B which coincides with length on open intervals. I.e. if E is the nonempty interval (a,b), then 
. This measure is usually called Lebesgue measure (or sometimes Borel measure). It is not a finite measure since 
. But it is 
) such that 
for all n.
.
if f and f’ are path homotopic.
and 
are equivalence relations.
. The map F(x,t) is the required homotopy.
. Let F be a homotopy between f and f’. We can then verify that G(x,t) = F(x, 1-t) is a homotopy between f’ and f.
, we show that 
. Let F be a homotopy between f and f’, and let F’ be a homotopy between f’ and f”. This time, we need to define a slightly more complicated homotopy G: X x I -> Y by the equation![G(x,t) = \begin{cases} F(x,2t) &\text{for }t\in [0,\frac{1}{2}],\\ F'(x, 2t-1) &\text{for } t\in [\frac{1}{2}, 1].\end{cases}](https://s0.wp.com/latex.php?latex=G%28x%2Ct%29+%3D+%5Cbegin%7Bcases%7D+F%28x%2C2t%29+%26%5Ctext%7Bfor+%7Dt%5Cin+%5B0%2C%5Cfrac%7B1%7D%7B2%7D%5D%2C%5C%5C+F%27%28x%2C+2t-1%29+%26%5Ctext%7Bfor+%7D+t%5Cin+%5B%5Cfrac%7B1%7D%7B2%7D%2C+1%5D.%5Cend%7Bcases%7D&bg=ffffff&fg=1a1a1a&s=0&c=20201002)
are also in M(X,X).
, let 
be the “truncation of f” defined by 
be defined similarly. We will work out the proof that 
to 
.
is measurable.
, and using an earlier corollary that says that if a sequence 
is in M(X,X) converges to f on X, then f is also in M(X,X), we have that 
belongs to M(X,X).
, and hence fg also belongs to M(X,X).
implies 
.
consists of a half-line which is either of the form 
or the form 
. (We will show later that both cases can occur.) Thus, the set will belong to the Borel algebra B which is the 
, then when 
, the set will be 
.
, 
belong to X and the real-valued function 
defined by 
is measurable.
and 
, then we have that 
which is in X since it is the complement of the union of A and 
.
, then 
, which is a union of two sets in X and hence also in X.
and 
when 
, and 
when 
, due to a similar reason as above. Therefore f is measurable!
by Stephen Covey (Highly recommended to read), there are four types of activities:
.
, suitable for students who really need some encouragement and motivation to overcome fear of math! Albert Einstein once said, “You never fail until you stop trying.” Hence, even if you have not done well in math for the past years, there is still hope, don’t give up!
is an interesting book by Professor Diaconis, featuring Magic Tricks that have a mathematical background! This book is a great idea for a gift for students, teachers, or friends!
.
, Daniel A. Fleisch. Using children’s blocks and laymen language, he explains what is a tensor clearly and succinctly in a way that is unbelievably crystal clear.
, how do we find the (shortest) distance of a plane to the origin?
must be perpendicular to the plane, i.e. parallel to the normal vector 
.
, 
is either 0 or 180 degrees, hence 
is either 1 or -1.
.
would be suitable for children training for GEP Screening Test and Math Olympiad simultaneously at the same time.
of the tank can be filled in 1 hour.
, words cannot describe the huge advantage he/she has over someone who has not seen a logic pattern puzzle before.
is a good book to start learning from. An intuitive, yet precise introduction to probability theory, stochastic processes, and probabilistic models used in science, engineering, economics, and related fields. You may also wish to refer to our comprehensive list of 
