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Cohomology groups of complex algebraic varieties with coefficients in ℂ, can be considered more than just a vector space and they can equipped with various and rich linear algebra structures which are functorial with respect to morphisms between them. In this thesis we first review classic Hodge theory, Hodge decomposition and Lefschetz decomposition theorems which enable us to introduce the concept of pure Hodge structure on cohomology group H^n (X,C) of a compact Kähler manifold X. Then we define Frölicher spectral sequence for a complex manifold X and show that it will degenerate at E_1 when X is compact and Kähler. For generalization of pure Hodge structures to smooth non... 

Computational complexity of graph problems is an important branch in the-oretical computer science. We introduce to some of ideas for computing the complexity of graph problems with some kind and beautiful examples. Next, we show hardness and inapproximability of some problems. Representation number of graphs has been introduce by Pavel Erdos by Number theory. We prove n1−ϵ inapproximability of that. Lucky number η has been studied by Grytczuk et.al . We show for planar and 3-colorable graphs, it is NP-Complete whether η = 2. Note that since a conjecture, for those graphs, 2 ≤ η ≤ 3. Also for each k ≥ 2, we show NP-completeness of η ≤ k for the graphs. Proper orientation number −→ is a... 

In this thesis we review several arithmetical questions about the diophantine equation φ[x]=q where φ is a nondegenerate symmetric bilinear form on a finite dimensional vector space, φ[x] is the corresponding quadratic form and q is a nonzero element of the ground field. We obtain class number and mass formulae for the orthogonal group associated to φ in a number field utilizing the concept of quadratic forms, lattices and adelic language  

In this thesis, first we will have a thorough review on problems and models presented in the field of contagion in networks (spread of diseases, spread of computer viruses,etc.). Then we will investigate the basic model of virus inoculation game with purely selfish players and the modified version of it which takes into account a factor of friendship. We will discuss the results of these models including computing and comparing Nash Equilibria  

The Mordell–Weil theorem states that for an abelian variety A over a number field k, the group A(k) of k-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group. The case with A an elliptic curve E and k the rational number field Q is Mordell’s theorem, answering a question apparently posed by Poincare around 1908; it was proved by Louis Mordell in 1922. The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(Q)=2E(Q) which forms a major step in the... 

Let G be a simple graph. A harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number, h(G), is the least number of colors in such a coloring. In this project, first we discuss about complexity of harmonious coloring. Then we find some bounds for harmonious chromatic number  

We consider a network creation game in which, each player(vertex) has bounded budget to draw edges to other vertices. Each player wants to minimize its local diameter in MAX version, and its sum of distances to other vertices in SUM version.
In this thesis, we first prove that the problem of finding the best response for each player in this network creation game, is NP-Hard. But we show that always there exists a Pure Nash Equilibria for each of the versions. Then we focus on the diameter of equilibrium graphs. In Unit Budget case (when each player’s budget is one) we show that the diameter of each equilibrium graph in O(1), and always there exists a Mixed Nash Equilibirum. For each n,... 

” The story of the "Weil conjectures" is a marvelous example of mathematical imagination, and one of the most striking instances exhibiting the fundamental unity of mathematics.”In 1949,Andre Weil stated some conjectures on the zeta function of Algebraic varieties over finite fields .These conjectures were analogue of the properties of Riemann zeta function ,in particular Riemann hypothesis.In fact ,Weil built a bridge between Diophantine structure on varieties over finite fields (Counting of rational points on varieties) and cohomological structure of them over the field of complex numbers(topology of variety).In this thesis, first we state Weil’s motivations for these conjectures and state... 

This thesis is about one of the newest attractive theories in mathematics and theoretical physics; Topological Quantum Field Theory (TQFT) has some basic algebraic requirements called tensor categories, after introducing them and describing axioms of the theory which has suggested by Atiyah in 1988, we will see that the category of 2d-TQFT is equivalent to the category of commutative frobenius algebras over an arbitrary field .In the last part we focus on some topological aspects of theory in dimension 3 and some invariants of 3d manifolds , we will see these invariants are the main tool for construction a 3d-TQFT  

Let M be a closed manifold, oriented manifold of dimension -d. Let be the space of smooth loops in M. Chas and Sullivan defined a product on the homology H⋆(LM) of degree -d. They then investigated other structure that this product induces including a Batalin-Vilkovisky structure, and a lie algebra structure, and a lie algebra structure on the S1 equivariant homology HS1 ⋆ (LM). This algebraic structure as well as the others came under the general heading of the string topology of M  

Deligne Proved that cohomology groups of algebraic varieties over C have mixed Hodge structures.It is also proved that homotopy groups of these varieties have mixed Hodge structures in a proper sense.In this thesis using iterated integrals that introduced by Chen,we explain Hain’s approach for defining Hodge structures on truncated first homotopy groups(fundamental groups). Note that there is another approach for defining Hodge structures on homotopy groups by Morgan,that uses minimal models of Sullivan.For explaining Hain’s approach,some basic results on Hodge structures are also explained  

In this thesis, we study the f-vectors and h-vectors of simplicial complexes and we state and prove a theorem of Kruskal and Katona that characterizes the f-vectors of simplicial complexes. We then define the Stanley-Reisner ring A associated to a simplicial complex, and state certain connections between f and h-vectors of this simplicial complex with Hilbert function of A, and show that if A is a Cohen-Macaulay ring then the h-vector of the simplicial complex in an O-sequence. conversely any O-sequence, equivalently, the f-vector of any multicomplex is the h-vector of a simplicial complex. Finally it is shown that if a simplicial complex arises from a triangulation of the sphere and A is... 

Stochastic Algebraic Topology studies stochastic or less known spaces that depend on many random variables from the perspective of algebra. Examples of such spaces included random graphs of Erdos and Renyi and random simplicial spaces of higher dimension . In this thesis , we study algebraic aspects of these spaces such as homotopy groups and homology groups from probabilistic point of view , when the number of vertices of these spaces tend to infinity . The study of these spaces have a lot of applications, since the data for a given space can be incomplete . They are also used in big system structures. We have given a collection of new results in stochastic algebraic topology of two... 

A moduli spase consists in classifying geometric objects up tp equivalence relations and its point are in 1-1 corrospondence with equivalence classes. One of the most important moduli spaces in algebraic geometry is «moduli space of stable n-pointed curves of genus zero» which has many applications in mathematics and theoretical physics.In [10] Knudsen shows that for every n 3 there is a smooth projective variety Xn that is a moduli space for stable n-pointed curves of genus zero.In this thesis we study moduli spaces and introduce Xn and its relation with cross ratios. This way we show Xn is smooth projective variety and a fine moduli space. [5] After that we study a paper by Sean Keel. [9]... 

The primary goal of this thesis is to study the chromatic number of Kneser hypergraphs that made a connection between topology and combinatorics.We mention simplicial complexes, Fan’s and Tuckers lemma’s, nerve’s lemma,and topological Tverberg theorem as some tools for investigating the chromatic number of these graphs. Many of the concepts and propositions expressed in this paper, such as those mentioned above, are topological propositions about combinatorial objects, and vice versa. Using such connections, you can use the tools of each branch in another and use their combination  

The aim of this thesis is to introduce some algebraic topologies methods and apply them on fining the chromatic number of some famous graphs and also hypergraphs. In the first part, we will use a mixture of two well-known technics, Tucker lemma and Discrete Morse theory to find an upper bound for the chromatic number of s-stable Kneser for some specific vector s. to find the sharper upper bound, we will deviate our strategy and use another approach by finding an edge-labeling and apply some theorems in POSET algebraic topology. In this way, we also find a connection between Young diagrams and the numbers of spheres in the box complex related to Kneser graphs and hypergraph. Actually, we can... 

Topological combinatorics is an almost novel branch in mathematics whose dawn began to shine originally from Lovász studies at late 70's. The main purpose of the pursued subjects in this branch is to apply topological methods in order to conclude various combinatorial results based on the famous Borsuk-Ulam's theorem which itself is a very deep statement in algebraic topology and has many equivalent versions. For instance, one of these versions states that there is no continuous map f:S^n⟶S^(n-1) which behaves antipodally. In this thesis, we intend to take a considerably long journey in homology theory to pave the way toward a complete proof of this theorem and some of its generalizations.... 

In This thesis, we study first the analytic properties of Iterated integrals on complex line and discuss their shuffle relations and also regularization for divergent integrals, Next we turn in to Mixed Hodge theory and define a Hopf algebra of framed Mixed Hodge -Tate Structures. Associated to any iterated integral ∫ an+1 a0 dt ta1 dt ta2 ::: dttan
we define an n-framed Mixed Hodge-Tate structure denoted by IH(a0; a1; :::; an; an+1) . These structure come from the fundamental group of Cfa1; :::; ang and for this we recall the theory of Chen and Hain for these groups. At the end calculate a coproduct formula for these elements  

Class field theory studies abelian extensions of local and global fields. Classifying these abelian extensions was part of Hilbert’s 12th problem, which as of today is still open and is not completely solved, but that has been considerable progress towards solving it.One of these progresses is class field theory. In this theory, instead of concretely building theses extensions, we express their major properties, in terms of the arithmetic of the base field alone, which is encoded in a group called, the Idele class group (in global case) and the multiplicative group of the base field (in local case).In this thesis, we explain local and global class field theory, their relationship and the...