Dream

1000 research tracks. 300 mathematics cafes. 5000 researchers and students.

We dream of rejuvenating the research atmosphere in India. Cheenta Research Track specifically caters to that dream.

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Research – Geometric Group Theory

Research Track leader

Ashani Dasgupta

Pursuing Ph.D. at University of Wisconsin Milwaukee on Geometric Group Theory

Understand the problem

Suppose a relatively finitely presented group G is acting stably and minimally on a real tree T. Create a graph of groups decomposition of G using this group action.

Rips theory is used to analyze the action of finitely presented groups on real trees. We propose to extend the theory to relatively finitely presented case.

Arithmetical Dynamics: Part 6

Consider fix point of \( R(z) = z^2 - z \) . Which is the solution of $$ R(z) = z \\ \Rightarrow z^2 - z =z \\ \Rightarrow z^2 - 2z =0 \\ \Rightarrow z(z -2) =0 $$ Now , consider the fix point of \( R^2(z) \) . \( \\ \) $$ i.e. R^2(z) = R . R(z) = R(z^2 -z)\\...

Arithmetical Dynamics: Part 5

And suppose that R has no periodic points of period n . Then (d, n) is one of the pairs \( (2,2) ,(2,3) ,(3,2) ,(4 ,2) \) , each such pair does arise from some R in this way . The example of such pair is $$ 1. R(z) = z +\frac {(w-1)(z^2 -1)}{2z} ; it \ has \ no \...

Arithmetical Dynamics: Part 0

Rational function \( R(z)= \frac {P(z)}{Q(z)} \) ; where P and Q are polynimials . There are some theory about fixed points . Theorem: Let \( \rho \) be the fixed point of the maps R and g be the Mobius map . Then \( gRg^{-1} \) has the same number of fixed points at...

Arithmetical Dynamics: Part 4

\( P^m(z) = z \ and \ P^N(z)=z \ where \ m|N \Rightarrow (P^m(z) - z) | (P^N(z)-z) \) The proof of the theorem in Part 0 : Let , P be the polynomials satisfying the hypothesis of theorem 6.2.1 . Let , \( K = \{ z \in C | P^N(z) =z \} \\ \) and let \( M =\{ m \in Z :...

Arithmetical Dynamics: Part 3

Theory: Let \( \{ \zeta_1 , ......., \zeta_m \} \) be a ratinally indifferent cycle for R and let the multiplier of \( R^m \) at each point of the cycle be \( exp \frac {2 \pi i r}{q} \) where \( (r,q) =1 \) . Then \( \exists \ k \in Z \) and\( mkq \) distinct...

Arithmetical Dynamics: Part 2

The lower bound calculation is easy . But for the upper bound , observe that each \( z \in K \) lies in some cycle of length m(z) and we these cycles by \( C_1 , C_2 .....,C_q \) . Further , we denote the length of the cycle by \( m_j \) , so , if \( z \in C_j \)...

Arithmetical Dynamics: Part 1

Definition: Suppose that \( \zeta \in C \) is a fixed point of an analytic function \( f \) . Then \( \zeta \) is : a) Super attracting if \( f^{'} (\zeta) =0 \rightarrow \) critical point of \( f \) b) Attractting if \( 0 < |f^{'}( \zeta )|< 1 \ \rightarrow \)...

Research for School

Research projects for school students, in mathematics and data science. For advanced learners who are in love with mathematical science.

Arithmetical Dynamics: Two possible problems

1.1. Existence of (pre)Periodic Points. These are the topics expanding on I.N. Baker’s theorem. Related reading: (1) Silverman Arithmetic of Dynamical Systems: p 165. Bifurcation polynomials. Exercise 4.12 outlines some known properties and open questions (** =...

Arithmetical Dynamics: An intro:

Here I gave some introduction of Arithmetical Dynamics. What is it and why do we study it. Also there is one motivational question upraised.

Useful information

How it works

  • Training bursts (3 to 4 weekly training sessions once every two months)
  • Weekly presentation/problem solving (moderated)
  • Bi-weekly advisor meeting
  • Students are required to present at least once per month
  • Students are required to submit an expository article once every 6 months

Papers

Henry Wilton – Rips Machine

Topological Methods in Group Theory by Scott and Wall

Books

  • Contemporary Abstract Algebra (Gallian)
  • Bridson Haeflager
  • Scott and Wall
  • Hatcher