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Dream

[/et_pb_text][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" _builder_version="3.25.4"][et_pb_row _builder_version="3.25.4"][et_pb_column type="4_4" _builder_version="3.25.4"][et_pb_text _builder_version="3.25.4"]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. 

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Research - Arithmetic Dynamics

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Arnab Dey Sarkar Pursuing Ph.D. at St. Louis on Arithmetic Dynamics Publication:
A Dichotomy for the Gelfand–Kirillov Dimensions of Simple Modules over Simple Differential Rings  Other Papers

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Understand the problem

Arithmetic dynamics is a field that amalgamates two areas of mathematics, discrete dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function. A fundamental goal is to describe arithmetic properties in terms of underlying geometric structures.
Given an endomorphism \( f \) on a set \( X \) ; \( f:X\to X \)
a point x in X is called preperiodic point if it has ``finite forward orbit under iteration of $f$ with mathematical notation if there exist distinct \( n \) and \( m \) such that
\( f^{n}(x)=f^{m}(x) \) (i.e it is eventually periodic'').
We are trying to classify for which maps are there not as many preperiodic points as their should be in dimension 1. It is a more precise version of I.N. Baker's theorem which states ``Let P be a polynomial of degree at least two and suppose that P has \( \textbf{\underline{no}} \) periodic points of period n. Then n=2 and P is conjugate to \( z^2-z \).''.

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Useful information

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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
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Some possible questions

Existence of (pre)Periodic Points. These are the topics expanding on I.N. Baker’s (1) Is there a lower bound on the number of minimal periodic points of period n in terms
of the degree of the map?
(2) What about Morton’s criteria for using generalized dynatomic polynomials (i.e.,
preperiodic points)? i.e., find a polynomial which vanishes at the c values where
there is collapse of (m, n) periodic points.
(3) What existence of periodic points for fields with characteristic p > 0? (such as finite
fields, or p-adic fields)
(4) In higher dimensions, are there always periodic points of every period? See Ivashkovich.
What about morphisms versus rational maps? 

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Papers

  (1) Silverman Arithmetic of Dynamical Systems: p 165. Bifurcation polynomials. Exercise 4.12 outlines some known properties and open questions (** = open question).

(2) Patrick Morton: Arithmetic properties of periodic points of quadratic maps, II –
article describes where there is collapse for the family of quadratic polynomials
(3) Vivaldi and Morton: Bifurcations and discriminants for polynomial maps
(4) Hagihara - Quadratic rational maps lacking period 2 orbits
(5) John Doyle https://arxiv.org/abs/1501.06821 – a related topic but not quite the
same
(6) https://arxiv.org/abs/1703.04172 – perhaps related to the question in characteristic p > 0.
(7) On fixed points of rational self-maps of complex projective plane - Ivashkovich http:
//arxiv.org/abs/0911.5084 Its the only goal is to provide examples of rational selfmaps of complex projective plane of any given degree without (holomorphic) fixed
points. This makes a contrast with the situation in one dimension.

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Books

  • Abstract Algebra (Dummit Foote)
  • Commutative Algebra (Atiyah)
  • An Invitation to Algebraic Geometry (Smith Kahanpaa)
  • The Arithmetic of Dynamical Systems (Silverman)
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