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Parameter spaces of holomorphic dynamical system.
Parameter spaces of holomorphic dynamical system.
Speaker: Sabyasachi Mukherjee
Abstract: In this mini-course, we will discuss some fundamental properties of parameter spaces of holomorphic dynamical systems generated by rational maps on the Riemann sphere. We will introduce the notion of structural stability and show that structurally stable parameters form a dense open subset of the parameter space. We will then focus on the parameter space of quadratic polynomials, and describe dynamically natural uniformizations of its structurally stable components. Finally, we will explicate the construction of a natural measure on the bifurcation locus of quadratic polynomials (complement of structurally stable maps) using potential-theoretic tools, and mention some equidistribution results for this measure.
Berkovich Projective line: potential theory and dynamics.
Berkovich Projective line: potential theory and dynamics.
Speaker: Niladri Sekhar Patra
An introduction to Teichmüller spaces and Teichmüller's theorems on extremal mappings.
An introduction to Teichmüller spaces and Teichmüller's theorems on extremal mappings.
Speaker: Kingshook Biswas
Abstract: We give an introduction to the theory of Teichmüller spaces. We start by introducing the notion of quasiconformal mappings, describing their basic properties, and giving a sketch of the proof of the Measurable Riemann Mapping Theorem. We then define Teichmüller spaces, and discuss how pants decompositions and Fenchel-Nielsen coordinates can be used to prove that the Teichmuller space of a compact surface of genus g is homeomorphic to a ball of dimension 6g-6. We then discuss holomorphic quadratic differentials on Riemann surfaces, and prove Teichmüller's theorems on the existence and uniqueness of Teichmüller mappings (maps affine with respect to flat structures induced by holomorphic quadratic differentials) within any homotopy class of maps between compact Riemann surfaces.
A proof of the Howe-Moore theorem on the decay of matrix coefficients.
A proof of the Howe-Moore theorem on the decay of matrix coefficients.
Speaker: T. N. Venkataramana
Abstract: We give a proof of the Howe-Moore theorem on the decay of matrix coefficients at infinity for a unitary representation of a non-compact simple linear Lie group G (restricting our attention mainly to SL(n,R)). We will deduce from this the ergodicity of the action of a non-compact subgroup of G on the quotient of the Lie group G by a lattice.
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