Marcelo Mendes Disconzi, PhD student
Department of Mathmatics of SUNY - Stony Brook University

Research and Study

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General.
Here you find information about my academic work, my academic education, some notes I have taken and some links.
My interests are analysis, partial differential equation and mathematical physics. About the later I am primarily interested in the interactions between analysis and quantum field theory/string theory.

CV and papers.
You can get my CV here. Below is a list of my papers and preprints together with a short description of each of them.

Yamabe Solitons, Determinant of the Laplacian and the Uniformization Theorem for Riemann Surfaces, with L. F. di Cerbo. Letters in Mathematical Physics, Volume 83, Number 1, p. 13-18, January, 2008.
In this letter we prove that non-trivial compact Yamabe solitions or breathers do not exist. In particular our proof in the two dimensional case dependes only on properties of the determinant of the Laplacian and turns out to be independent of the classical uniformization theorem. Using this remarkable fact we are able to explain how the uniformization theorem for Riemann surfaces can be obtained using the Yamabe-Ricci flow.

Asymptotic states in coupled map lattices, with A. T. Baraviera. International Journal of Bifurcation and Chaos, Volume 18, Issue 2, p. 285-311, February 2008.
In this work we show that a certain family of Coupled Map Lattices presents different asymptotic behaviors when some parameters (including coupling) are changed. We proof that we can begin with a configuration with infinitely many different measures and, with a slight change in coupling, get an asymptotic state with only one measure describing the behavior of most orbits. Since our results are motivated by some results coming from physics, in order to establish a common language we also give a non-technical introduction to the theory of invariant measures and equilibrium in Dynamics.

Analysis of high-dimensional non-hyperbolic coupled systems through finite-time Lyapunov exponents, with L. G. Brunnet. Physica A, Volume 387, p. 425-431, 2008.
Using finite-time Lyapunov exponent we investigave the relations between Unstable Dimension Variability and the phase space dimension.

Dynamics at the interface dividing collective behavior and synchronized states in CML, with L. G. Brunnet. Physica A, Volume 360, p. 159-170, 2006.
A study is developed focusing on the lost of stability of the interface dividing two regions of different spatial patterns on a coupled map lattice.

Notes.
Here are some notes that I (and other people) have taken.

Mathematical Foundations of Classical and Quantum Field Theory - notes of two summer courses I took on the subject (pdf  file).

Elementary realization of of BRST symmetry and gauge fixing - notes of a series of lectures given by Martin Rocek. All ideas of BRST symmetry and BV formalism are developed at a very basic level using finite dimensional integrals instead of path integrals. Excellent for those interested in the general idea of the formalism (pdf file).

Some algebraic structures in physics - notes from a series of informal meetings that I and some other students organized with the goal of sharing our different background in physics and mathematics (pdf file).

Some ideas in Conformal Field Theory - (handwritten) notes from a talk I gave in the RTG Seminar in Geometry and Physics at Stony Brook (.zip file with a bunch of .jpg files, or click here to access each file separetely).

Topics in Differential Topology - notes by Somnath Basu of a course taught by Blaine Lawson (pdf file).

Some advanced techniques on PDE's - notes of (part of) a course taught by Marcus Khuri (pdf file).

Spontaneous symmetry breaking - introductory notes on the Higgs mechanism (pdf file).

Holographic renormalization - notes of a talk I gave in the RTG Seminar in Geometry and Physics at Stony Brook (pdf file).

Links and useful material.

The Comprehensive LaTex symbol list - excellent material by Scott Pakin (pdf file).

RTG Program in Geometry and Physics - this program is an attempt to reincorporate concepts of Theoretical Physics into the Geometry and Topology curricula at all levels, starting from the undergraduate to the post-doctoral.

Simons Center for Geometry and Physics - the intellectual focus of the Center is at the interface of Mathematics, in particular Geometry, and Theoretical Physics.