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Através do Método da Transversal Completa apresentamos neste trabalho a classificação dos germes simples de Rn em R, a classificação dos germes do plano no plano de corank 1 e A-codimensão no máximo 4 e uma breve classificação de bigermes de R em R2. / Applying the Complete Transversal Method we obtain, in this work, a classification of simple germs of smooth function from Rn to R, a classification of germs of maps from the plane to the plane with A-codimension up to 4 of corank 1 and an introduction to the classification of bigerms of maps from R to R2.

Através do Método da Transversal Completa apresentamos neste trabalho a classificação dos germes simples de Rn em R, a classificação dos germes do plano no plano de corank 1 e A-codimensão no máximo 4 e uma breve classificação de bigermes de R em R2. / Applying the Complete Transversal Method we obtain, in this work, a classification of simple germs of smooth function from Rn to R, a classification of germs of maps from the plane to the plane with A-codimension up to 4 of corank 1 and an introduction to the classification of bigerms of maps from R to R2.

This thesis concerns the study of chiral algebras over schemes of arbitrary dimension n. In Chapter I, we construct a chiral algebra over each smooth variety X of dimension n. We do this via the Hilbert scheme of points of X, which we use to build a factorisation space over X. Linearising this space produces a factorisation algebra over X, and hence, by Koszul duality, the desired chiral algebra. We begin the chapter with an overview of the theory of factorisation and chiral algebras, before introducing our main constructions. We compute the chiral homology of our factorisation algebra, and show that the D-modules underlying the corresponding chiral algebras form a universal D-module of dimension n. In Chapter II, we discuss the theory of universal D-modules and OO- modules more generally. We show that universal modules are equivalent to sheaves on certain stacks of étale germs of n-dimensional varieties. Furthermore, we identify these stacks with the classifying stacks of groups of automorphisms of the n-dimensional disc, and hence obtain an equivalence between the categories of universal modules and the representation categories of these groups. We also define categories of convergent universal modules and study them from the perspectives of the stacks of étale germs and the representation theory of the automorphism groups.

The present thesis is dedicated to two topics in Dynamics of Holomorphic maps. The first topic is dynamics of simple parabolic germs at the origin. The second topic is Polynomial-time Computability of Julia sets.\\ Dynamics of simple parabolic germs. Let $F$ be a germ with a simple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ It is convenient to apply the change of coordinates $z=-1/w$ and consider the germ at infinity $$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$$ The dynamics of a germ $f$ can be described using Fatou coordinates. Fatou coordinates are analytic solutions of the equation $\phi(f(z))=\phi(z)+1.$ This equation has a formal solution \[\tilde\phi(z)=\text{const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},\] where $\sum b_jz^{-j}$ is a divergent power series. Using \'Ecalle's Resurgence Theory we show that $\tilde$ can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from $\tilde \phi$ using Borel-Laplace summation. J.~\'Ecalle and S.~Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs with a parabolic fixed point. We give a new proof of validity of \'Ecalle's construction. \\ Computability of Julia sets. Informally, a compact subset of the complex plane is called \emph if it can be visualized on a computer screen with an arbitrarily high precision. One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be computed in a polynomial time. The main result of Chapter II is the following: Theorem. Let $f$ be a rational function of degree $d\ge 2$. Assume that for each critical point $c\in J_f$ the $\omega$-limit set $\omega(c)$ does not contain either a critical point or a parabolic periodic point of $f$. Then the Julia set $J_f$ is computable in a polynomial time.

This dissertation examines how the relationship between microbes and the human body has been reconfigured over the course of the twentieth century and into the first decades of the twenty-first century. It presents a counter-narrative to the ways in which we have tended to view microbe-human relations to make sense of the emergence of twenty-first century microbial selves by focusing on the normal microbiota. / History of Science

The present thesis is dedicated to two topics in Dynamics of Holomorphic maps. The first topic is dynamics of simple parabolic germs at the origin. The second topic is Polynomial-time Computability of Julia sets.\\ Dynamics of simple parabolic germs. Let $F$ be a germ with a simple parabolic fixed point at the origin: $F(w)=w+w^2+O(w^3).$ It is convenient to apply the change of coordinates $z=-1/w$ and consider the germ at infinity $$f(z)=-1/F(-1/z)=z+1+O(z^{-1}).$$ The dynamics of a germ $f$ can be described using Fatou coordinates. Fatou coordinates are analytic solutions of the equation $\phi(f(z))=\phi(z)+1.$ This equation has a formal solution \[\tilde\phi(z)=\text{const}+z+A\log z+\sum_{j=1}^\infty b_jz^{-j},\] where $\sum b_jz^{-j}$ is a divergent power series. Using \'Ecalle's Resurgence Theory we show that $\tilde$ can be interpreted as the asymptotic expansion of the Fatou coordinates at infinity. Moreover, the Fatou coordinates can be obtained from $\tilde \phi$ using Borel-Laplace summation. J.~\'Ecalle and S.~Voronin independently constructed a complete set of invariants of analytic conjugacy classes of germs with a parabolic fixed point. We give a new proof of validity of \'Ecalle's construction. \\ Computability of Julia sets. Informally, a compact subset of the complex plane is called \emph if it can be visualized on a computer screen with an arbitrarily high precision. One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be computed in a polynomial time. The main result of Chapter II is the following: Theorem. Let $f$ be a rational function of degree $d\ge 2$. Assume that for each critical point $c\in J_f$ the $\omega$-limit set $\omega(c)$ does not contain either a critical point or a parabolic periodic point of $f$. Then the Julia set $J_f$ is computable in a polynomial time.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste trabalho, mostramos que equivalÃncia HÃlder de germes de funÃÃes analÃticas (C2, 0) → (C, 0) admite moduli contÃnuo. Mais precisamente, construimos um invariante da equivalÃncia HÃlder de tais germes que varia continuamente numa famÃlia ft : (C2, 0) → (C, 0). Para um Ãnico germe ft o invariante de ft à dado em termos dos coeficientes principais das expansÃes assintÃticas de ft ao longo dos ramos da curva polar genÃrica de ft. / In this work, we show that HÃlder equivalence of analytic functions germs (C2, 0) → (C, 0)admits continuous moduli. More precisely, we constructed an invariant of the HÃlder equivalence of such germs that varies continuously in a family ft : (C2, 0) → (C, 0). For a single germ ft the invariant of ft is given in terms of the leading coefficients of the asymptotic expansion of ft along the branches of generic polar curve of ft .

Nos trabalhos iniciais sobre o nÃmero de Milnor, ele à definido como a dimensÃo do n-Ãsimo grupo de homologia de uma fibra de Milnor. Esse trabalho irà verificar algumas outras equivalÃncias, com o objetivo de mostrar que o nÃmero de Milnor pode ser escrito como a dimenÃÃo de um C-espaÃo vetorial que vem do quociente entre o anel de germes de funÃÃes holomorfas e de seu jacobiano. / In early work on the number of Milnor, it is defined as the dimension of the nth homology group of a Milnor fiber. This work will check some other equivalences, with the aim of showing that the number of Milnor can be written as the demension of a vector C-space that comes from the ratio of the germ ring of holomorphic functions and its Jacobian.

Magistro darbe tiriama mikrobangų poveikis įvairių rūšių medžiams. Darbo objektas – Paprastosios pušies ( Pinus sylvestris L. ) sėklų gemalai, paprastosios eglės (Picea abies (L.) Karst.) gemalai ir augimo kūgeliai, paprastojo uosio ( Fraxinus excelsior L.) gemalai, karpotojo beržo (Betula pendula L.) žiedadulkės. Darbo tikslas – Įvertinti mikrobangų poveikį skirtingų medžių rūšių subrendusių gemalų ir žiedadulkių gyvybingumui. Darbo metodika – mikrobangomis paveikiami subrendę gemalai, augimo kūgeliai žiedadulkės ir distiliuotas vanduo. Įvertinama mikrobangų įtaka gemalų, augimo kūgelių, žiedadulkių gyvybingumui. Rezultatai – Mikrobangos 600 W rėžime neigiamai veikia eglės, pušies ir uosio subrendusių gemalų gyvybingumą. Iš tirtų 480 subrendusių gemalų gyvybingi liko 183 eksplantai ( 38,1 %). Paveikus mikrobangomis beržo žiedadulkes, bangos stabdė žiedadulkių vystymąsi. Paveiktas mikrobangomis distiliuotas vanduo teigiamai veikė karpotojo beržo žiedadulkes, skatindamas žiedadulkių dygimą. Paprastosios eglės augimo kūgeliai paveikti mikrobangomis žuvo visuose eksperimento variantuose, išskyrus nepaveikti mikrobangomis augimo kūgeliai. / In the postgraduate thesis studied effect of microwaves on different trees species. Study object. Mature germ of Common Scots pine (Pinus sylvestris L.), Common Ash (Fraxinus excelsior L.), mature germ and growth cones of Norway spruce (Picea abies (L.) Karst.), pollen of Silver Birch (Betula pendula L.). Purpose of the study – to estimate the effect of microwaves on different trees species of mature germ and pollen viability. Study methods. Affected by microwaves mature germs, growth cones, pollen and distilled water. Rated microwave power of germs, growth cones and pollen viability. Results. Microwave 600 W mode affects spruce, pine and ash mature germs viability. Of investigated 480 mature germs remained viable explants 183 (38.1%). Birch pollen exposed of microwave, the waves slowed the development of pollen. Affected by microwave distilled water acted positively Silver birch pollen and stimulating germination. Growth cones of Norway spruce were killed of microwave affect all versions on the experiment, except do not affect the growth cone of microwaves.