Author: Ashani Dasgupta
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Ends
Motivation Start with a locally finite simplicial complex X. Locally finite: Each vertex is attached to only finitely many simplices. Why locally finite: To make sure it is a CW complex. Notice that the closure-finite criteria require each cell of a CW complex to meet only finitely many other cells. Hence we do not have a situation like…
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Which manifold is this?
This is the first exercise from Thurston’s Three Dimensional Geometry and Topology Vol. 1. Which manifold is this? It is like an old trick. Try following the lines. There are actually 6 loops (circles) in this maze. Here is a color coded picture of it.
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Round robin tournament
Problem : Suppose there are teams playing a round robin tournament; that is, each team plays against all the other teams and no game ends in a draw.Suppose the team loses games and wins games.Show that = Solution : Each team plays exactly one match against each other team. Consider the expression Since each team…
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Orthogonality
Let ABC be a triangle and D be the midpoint of BC. Suppose the angle bisector of \[\angle ADC \] is tangent to the circumcircle of triangle ABD at D. Prove that \[\angle A = 90^o \] . (Regional Mathematics Olympiad, India, 2016)
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Construction of polynomials
The polynomial P(x) has the property that P(1), P(2), P(3), P(4), and P(5) are equal to 1, 2, 3, 4, 5 in some order. How many possibilities are there for the polynomial P, given that the degree of P is strictly less than 4? (Duke Math Meet 2013 Tiebreaker round) Discussion: Let \[P(x) = a…
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A mathematician’s bookshelf
A mathematician’s bookshelf is probably more informative than his resume. The idea of ‘book’ has been recently challenged by the advent of technology. Outstanding authors such as Hatcher (of ‘Algebraic Topology’ fame) prefers to keep an electronic copy of his book. This electronic copy is updated from time to time.
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Some Beautiful Books
Straight Lines and Curves by Vasiliyev N. B. Vasilyev was the chief architect of Mathematical Olympiads in Soviet Union. This gem from erstwhile Soviet Union’s publication, explores loci of points in plane and space. The entire discussion is aided by geometric intuition. The authors occasionally use algebraic tools to augment the ideas. The holistic nature…
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A SAGE experiment on Nested Approximate Subgroups
(this is a continuing write-up mostly for personal records) Spoiler Before you read on, here is what I have found curious so far (this section is constantly changing): A little computation reveals that though group action conjugation preserves most elements of a subgroups in majority of cases, there exists some subgroups of \[S_4 \] which are ill…
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Number Theory in Math Olympiad – Beginner’s Toolbox
This article is aimed at entry level Math Olympiad (AMC and AIME in U.S. , SMO Junior in Singapore, RMO in India). We have complied some of the most useful results and tricks in elementary number theory that helps in problem solving at this level. Note that only with a lot of practice and conceptual…
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Lifting the exponent and math olympiad number theory
In math olympiads around the world, number theory problems have many recurring themes. One such theme is the ‘LTE’ or lifting the exponent.