In this work we study K-theoretic Donaldson-Thomas theory. We derive an explicit formula for the capped vertex with two legs in a certain gauge. Using this result we obtain an explicit formula for the operator corresponding to relative geometry of the resolved conifold with two nontrivial legs. As a consequence, we prove polynomiality in the Kahler variable of the operator for the corresponding absolute geometry.

Proefschrift--Utrecht.

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface frequently arise in enumerative problems. We use the K-theoretic Donaldson-Thomas theory of certain toric Calabi-Yau threefolds to study K-theoretic variants of such expressions. We study limits of the K-theoretic Donaldson-Thomas partition function of a toric Calabi-Yau threefold under certain one-parameter subgroups called slopes, and formulate a condition under which two such limits coincide. We then explicitly compute the limits of components of the partition function under so-called preferred slopes, obtaining explicit combinatorial expressions related to the refined topological vertex of Iqbal, Kos\c{c}az and Vafa. Applying these results to specific Calabi-Yau threefolds, we deduce dualities satisfied by a generating function built from tautological bundles on the Hilbert scheme of points on $\C^2$. We then use this duality to study holomorphic Euler characteristics of exterior and symmetric powers of tautological bundles on the Hilbert scheme of points on a general surface.

Thesis--Utrecht.

Proefschrift--Utrecht.

In this thesis we discuss various classical problems in enumerative geometry. We are focused on ideas and methods which can be used explicitly for practical computations. Our approach is based on studying the limits of elliptic stable envelopes with shifted equivariant or Kahler variables from elliptic cohomology to K-theory. We prove that for a variety X we can obtain K-theoretic stable envelopes for the variety of the G-fixed points of X, where G is a cyclic group acting on X preserving the symplectic form. We formalize the notion of symplectic duality, also known as 3-dimensional mirror symmetry. We obtain a factorization theorem about the limit of elliptic stable envelopes to a wall, which generalizes the result of M. Aganagic and A. Okounkov. This approach allows us to extend the action of quantum groups, quantum Weyl groups, R-matrices etc., to actions on the K-theory of the symplectic dual variety. In the case of X = Hilb, our results imply the conjectures of E. Gorsky and A. Negut. We propose a new approach to K-theoretic quantum difference equations.

Probability theory is a branch of mathematics concerned with determining the long run frequency or chance that a given event will occur. This chance is determined by dividing the number of selected events by the number of total events possible, assuming these events are equally likely. Probability theory is simply enumerative combinatorial analysis when applied to finite sets. For a given finite sample space, probability questions are usually "just" a lot of counting. The purpose of this thesis is to provide some in depth analysis of several combinatorial methods, including basic principles of counting, permutations and combinations, by specifically exploring one type of probability problem: C ordered possible elements that are equally likely s independent sampled subjects r distinct elements, where r = 1, 2, 3, …, min (C, s) we want to know P(s subjects utilize exactly r distinct elements). This thesis gives a detailed step by step analysis on techniques used to ultimately finding a general formula to solve the above problem.

Esta dissertação descreve um jogo de baralho com caráter pedagógico, Resto Zero, o qual apresenta forte ligação com probabilidade, divisibilidade, análise combinatória e operações aritméticas elementares. Especificamente calculamos a probabilidade de alguns eventos principais que ocorrem no desenvolvimento do jogo. Apresentamos também uma relação do uso do Resto Zero aos anos/séries em que pode ser trabalhado. / In this dissertation we present and develop a simple game based upon a deck of cards which we call Residue Zero. We study and describe some characteristics of this game by observing its strong connections with probability, combinatorics and basic arithmetic operations. In particular, we compute the probability of several events that occur during the development of this game. We finally provide a relation of the scholar grades in which some features of this game could be worked out.

Let Y be a smooth complex projective Calabi{Yau threefold. Donaldson-Thomas invariants [Tho00] are integer invariants that virtually enumerate curves on Y. They are organised in a generating series DT(Y) that is interesting from a variety of perspectives. For example, well-known series in mathematics and physics appear in explicit computations. Furthermore, closer to the topic of this thesis, the generating series of birational Calabi-Yau threefolds determine one another [Cal16a]. The crepant resolution conjecture for Donaldson-Thomas invariants [BCY12] conjectures another such comparison result. It relates the Donaldson{Thomas generating series of a certain type of three-dimensional Calabi-Yau orbifold to that of a particular resolution of singularities of its coarse moduli space. The conjectured relation is an equality of generating series. In this thesis, I first provide a counterexample showing that this conjecture cannot hold as an equality of generating series. I then verify that both generating series are the Laurent expansion about different points of the same rational function. This suggests a reinterpretation of the crepant resolution conjecture as an equality of rational functions. Second, following a strategy of Bridgeland [Bri11] and Toda [Tod10a, Tod13, Tod16a], I prove a wall-crossing formula in a motivic Hall algebra relating the Hilbert scheme of curves on the orbifold to that on the resolution. I introduce the notion of pair object associated to a torsion pair, putting ideal sheaves and stable pairs on the same footing, and generalise the wall-crossing formula to this setting, essentially breaking the former in many pieces. Pairs, and their wall-crossing formula, are fundamentally objects of the bounded derived category of the Calabi-Yau orbifold. Finally, I present joint work with J. Calabrese and J. Rennemo [BCR] in which we use the wall-crossing formula and Joyce's integration map to prove the crepant resolution conjecture for Donaldson-Thomas invariants as an equality of rational functions. A crucial ingredient is a result of J. Rennemo that detects when two generating functions related by a wall-crossing are expansions of the same rational function.

Esta dissertação descreve um jogo de baralho com caráter pedagógico, Resto Zero, o qual apresenta forte ligação com probabilidade, divisibilidade, análise combinatória e operações aritméticas elementares. Especificamente calculamos a probabilidade de alguns eventos principais que ocorrem no desenvolvimento do jogo. Apresentamos também uma relação do uso do Resto Zero aos anos/séries em que pode ser trabalhado. / In this dissertation we present and develop a simple game based upon a deck of cards which we call Residue Zero. We study and describe some characteristics of this game by observing its strong connections with probability, combinatorics and basic arithmetic operations. In particular, we compute the probability of several events that occur during the development of this game. We finally provide a relation of the scholar grades in which some features of this game could be worked out.