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# 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.

Do you wish to partner in this unique program? Let us know.

# Research – Geometric Group Theory

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 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 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 $$z \in C_j$$ then...

## 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 (** = open...

## 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.

# How it works

• Training bursts (3 to 4 weekly training sessions once every two months)
• Weekly presentation/problem solving (moderated)
• 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