Probabilistic and analytic aspects of Boolean functions
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hdl:2117/361113
Tipus de documentTreball Final de Grau
Data2022-01
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Reconeixement-NoComercial-SenseObraDerivada 3.0 Espanya
Abstract
This thesis will focus on the study of Boolean functions. In point of fact, they can be represented with a Fourier expansion and many of the definitions and results for these functions can be rewritten in terms of the Fourier coefficients. The definition of Boolean functions is simple, this implies that they have a natural interpretation and hence have applications in many areas of scientific research. Specifically, in this thesis we will see applications in Social Choice Theory, Theoretical Computer Science and Combinatorics. For the first area, we will see and prove with Fourier analysis Arrow's theorem and KKL theorem in order to show that it is not possible to define a perfect voting election system from an ethical standpoint. Additionally, we will translate the proof presented by Arrow for his own theorem in terms of mathematical language. The work for the second application will follow the steps to prove Sensitivity Conjecture which, although it has remained unsolved for 30 years, Huang has presented a brilliant short proof in a paper published at Annals of Mathematics very recently (2019). For the last area we will present the strange phenomena of thresholds in Random Graph properties and we will show Margulis-Russo Formula to study this event in terms of Boolean functions Fourier analysis.
TitulacióGRAU EN MATEMÀTIQUES (Pla 2009)
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memoria.pdf | 676,7Kb | Visualitza/Obre |