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Representation theory of gentle algebras
 Mlezivová, Anna ; Šťovíček, Jan (advisor) ; Chan, Aaron (referee)
The object of study of this thesis is a special class of quiver algebras called gentle algebras. To study modules over them, we can use a combinatorial or geometric view. Thanks to Theorem 6.1. in the article Chan and Demonet [2020], we can find the lattice of torsion classes of modules over gentle algebras using string combinatorics. In the thesis, we apply this theorem for a few examples. Especially we derive the lattice of torsion classes of Kronecker algebra, and we do the first steps for finding the lattice for Markov algebra. The emphasis is placed on understanding the relationship with the geometric view. 1
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Poncelet's porism
 Mlezivová, Anna ; Šťovíček, Jan (advisor) ; Žemlička, Jan (referee)
The thesis presents a complete proof of Poncelet's porism, which states that if we have two conic sections and for given n exists an n-gon, which is circumscribed by one and inscribed in the other, there are infinitely many such n-gons. In the first chapter we introduce the necessary theory from the field of algebraic geometry. The second chapter deals with the proof. We show that it is enough to prove the porism only for a concentric circle and ellipse. We also use a series of isomorphisms between projective varieties to transform the problem into a form of an elliptic curve with a known group structure. 1
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