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Basics on persistent homology
Novák, Jakub ; Šťovíček, Jan (advisor) ; Hrbek, Michal (referee)
Abstract:In this work, the reader is introduced to the theory of persistent ho- mology and its applications. In the first chapter we will show the basics of sim- plicial and singular homology and we will prove the basic relations, especially the independence of simplicial homological groups on the chosen △-complex and isomorphism between homological groups of homotopic spaces. In the second chapter, we explain the motivation behind persistent homology, describe its al- gebraic structure and how it can be visually represented. We describe and prove the corectness of the algorithm for its calculation. We then illustrate the theory on an example. 1
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Modules over string algebras
Löwit, Jakub ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
The aim of this thesis is to investigate the categories of modules over the so called string algebras. In particular, we try to understand the cotorsion pairs in these categories, which boils down to understanding the decompositions of extensions of such modules. For string algebras with some oriented tree for the underlying quiver, we describe some classes given by these cotorsion pairs in terms of purely combinatorial closure properties. For any string algebras, the combinatorics appears to be similar, althought more complicated.
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Counting the points on elliptic curves over finite fields
Eržiak, Igor ; Šťovíček, Jan (advisor) ; Stanovský, David (referee)
The goal of this thesis is to explain and implement Schoof's algorithm for counting points on elliptic curves over finite fields. We start by defining elliptic curve as a set of points satisfying certain equation and then proceeding to define an operation on this set. Theoretical background needed for the algorithm is presented in the second chapter. Finally, the Schoof's algorithm is introduced in the third chapter, supplemented by an implementation in SageMath open-source software.
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Elliptic curves over finite fields
Beran, Adam ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
In this thesis, we study the theory of elliptic curves, with the main focus on elliptic curves over finite fields. We present basic theory, taking several technical aspects into consideration (singularity of the curve, effect of field characteristic on the form of the equation of elliptic curve). We algebraically deduce and formulate the group law, that is the definition of addition on a set of points on elliptic curve). We prove a known result saying that the set of points on elliptic curve under addition forms a group. We present an elementary proof, some of the calculations will be carried out in computer program Mathematica due to their complexity. Finally, we study endomorphisms of elliptic curves over finite fields (homomorphisms on the set of points on elliptic curve that are defined by rational functions). Using obtained results, we prove the Hasse's theorem, which provides an estimate of the order of the group of points on elliptic curve over finite field. 1
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Quotients in algebraic geometry
Kopřiva, Jakub ; Šťovíček, Jan (advisor) ; Příhoda, Pavel (referee)
This thesis is concerned with the existence of pushouts in two different settings of algebraic geometry. At first, we study the pushouts in the cat- egory of affine algebraic sets over an infinite field. We show that this can be regarded as an instance of much general problem whether the pullback of finitely generated algebras over a commutative Noetherian ring is finitely generated. We give a partial solution to this problem and study some ex- amples. Secondly, we examine the existence of pushouts in the category of schemes with an emphasis on diagrams of affine schemes. We use the methods of Ferrand [2003] and Schwede [2004] and generalise some of their results. We conclude by giving some examples and suggest another approach to the problem.
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Jones polynomial
Gajdová, Anna ; Stanovský, David (advisor) ; Šťovíček, Jan (referee)
The topic of this thesis is the Jones polynomial of a given knot and its com- putation. First we define the Jones polynomial in two ways: using skein relations and using the bracket polynomial and we prove that these definitions are equi- valent. Next we derive an algorithm for computation of the Jones polynomial based on its relation with the bracket polynomial. We prove that the time com- plexity of the algorithm is O 20.823n , where n denotes number of crossings in a link diagram. Lastly we present the results of running the algorithm and its variants on data. We test the algorithm among others on small table knots, bigger random knots and on torus knots. We estimate that the fastest vari- ant of the algorithm runs on random knots with the average time complexity O 20.487n+o(n) . 1
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Modules and localization
Lysoněk, Tomáš ; Trlifaj, Jan (advisor) ; Šťovíček, Jan (referee)
In the Thesis, we define the notion of a localization and explore its connections to properties of modules - local properties and AD-properties, especially the projectivity. We present a proof of the fact that projectivity is an AD propety of modules due to Raynaud and Gruson in the corrected version by Perry from 2010. The proof is presented in full detail and accompanied by examples, and by the role of the investigated notions in the context of algebraic geometry. 1
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Invariant theory for finite groups
Žurav, Martin ; Šťovíček, Jan (advisor) ; Šaroch, Jan (referee)
The prime goal of this thesis is to give a decent introduction to the theory of invariants for finite groups. We begin our characterisation with symmetric polynomials and their fundamental properties. In particular, we study the ring of symmetric polynomials and we prove that it is finitely generated by elementary symmetric functions. Then we deal with some of the criteria for a polynomial to be symmetric. In the second part, we generalise these ideas for any finite subgroup of GL(n,k). We define an action of a finite linear group on k[x_1,...,x_n] and consider polynomials that are invariant under such action. We show that they form a ring which is always finitely generated, as follows from the Noether's bound theorem. At the end, we describe the ring of invariants and relations among its generators more profoundly.
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Plane geometry problems solved by algebraic geometry
Trummová, Ivana ; Šťovíček, Jan (advisor) ; Růžička, Pavel (referee)
In this thesis I focus on a certain part of algebraic geometry which studies plane curves and their intersection points. The main part is a proof of Bézout's theorem and an overview of its corollaries, which have an interesting geometric visualization. The most important corollary is the proof of associativity of adding points on elliptic curves. This fact is widely used in modern cryptography. 21
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Solving systems of polynomial equations
Kubej, Lukáš ; Šťovíček, Jan (advisor) ; Holub, Štěpán (referee)
This work is about theory of systems of polynomial equations. Its main purpose is to prove the Elimination theorem and the Extension theorem, where the Elimination theorem helps us to solve a given systems of polynomial equations and the Extension theorem tells us, which partial solutions can be extended into a complete solutions. To formulate and prove those theorems, we will explain theory including monomial orders, division algorithm for polynomial and mainly the key concept of Groebner basis. At the end are solved examples showing application of explained theory.
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