## Nearly Finitary Matroids

In this thesis, we study nearly finitary matroids by introducing new definitions and prove various properties of nearly finitary matroids. In 2010, an axiom system for infinite matroids was proposed by Bruhn et al. We use this axiom system for this thesis. In Chapter 2, we summarize our main results after reviewing historical background and motivation. In Chapter 3, we define a notion of spectrum for matroids. Moreover, we show that the spectrum of a nearly finitary matroid can be larger than any fixed finite size. We also give an example of a matroid with infinitely large spectrum that is not nearly finitary. Assuming the existence of a single matroid that is nearly finitary but not *
k
*-nearly finitary, we construct classes of matroids that are nearly finitary but not

*-nearly finitary. We also show that finite rank matroids are unionable. In Chapter 4, we will introduce a notion of near finitarization. We also give an example of a nearly finitary independence system that is not*

**k***-nearly finitary. This independence system is not a matroid. In Chapter 5, we will talk about Psi-matroids and introduce a possible generalization. Moreover, we study these new matroids to search for an example of a nearly finitary matroid that is not*

**k***-nearly finitary. We have not yet found such an example. In Chapter 6, we will discuss thin sums matroids and consider our problem restricted to this class of matroids. Our results are motivated by the open problem concerning whether every nearly finitary matroid is*

**k***-nearly finitary for some*

**k***.*

**k**## Structure and Interpretation of Dual-Feasible Functions

We study two techniques to obtain new families of classical and general Dual-Feasible Functions: A conversion from minimal Gomory--Johnson functions; and computer-based search using polyhedral computation and an automatic maximality and extremality test.

## Cores with distinct parts and bigraded Fibonacci numbers

The notion of (a,b)-cores is closely related to rational (a,b) Dyck paths due to Anderson's bijection, and thus the number of (a,a+1)-cores is given by the Catalan number Ca. Recent research shows that (a,a+1) cores with distinct parts are enumerated by another important sequence- Fibonacci numbers Fa. In this paper, we consider the abacus description of (a,b)-cores to introduce the natural grading and generalize this result to (a,as+1)-cores. We also use the bijection with Dyck paths to count the number of (2k−1,2k+1)-cores with distinct parts. We give a second grading to Fibonacci numbers, induced by bigraded Catalan sequence Ca,b(q,t).

## Smooth polytopes with negative Ehrhart coefficients

In this note, we present examples of smooth lattice polytopes in dimensions 3 and higher whose Ehrhart polynomials have some negative coefficients. This answers negatively a question by Bruns. We also discuss Berline-Vergne valuations as a useful tool in proving Ehrhart positivity results and state several open questions.

## Spherical metrics with conical singularities on 2-spheres

Suppose that θ1,θ2,…,θn are positive numbers and n≥3. Does there exist a sphere with a spherical metric with n conical singularities of angles 2πθ1,2πθ2,…,2πθn? A sufficient condition was obtained by Gabriele Mondello and Dmitri Panov (arXiv:1505.01994 https://arxiv.org/abs/1505.01994). We show that it is also necessary when we assume that θ1,θ2,…,θn∉N.

## Crystal analysis of type C Stanley symmetric functions

Combining results of T.K. Lam and J. Stembridge, the type C Stanley symmetric function FCw(x), indexed by an element w in the type C Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.

## Solving deterministic and stochastic equilibrium problems via augmented
Walrasian

We described a method to solve deterministic and stochastic Walras equilibrium models based on associating with the given problem a bifunction whose maxinf-points turn out to be equilibrium points. The numerical procedure relies on an augmentation of this bifunction. Convergence of the proposed procedure is proved by relying on the relevant lopsided convergence. In the dynamic versions of our models, deterministic and stochastic, we are mostly concerned with models that equip the agents with a mechanism to transfer goods from one time period to the next, possibly simply savings, but also allows for the transformation of goods via production

## Representable Chow classes of a product of projective spaces

Inside a product of projective spaces, we try to understand which Chow classes come from irreducible subvarieties. The answer is closely related to the theory of integer polymatroids. The support of a representable class can be (partially) characterized as some integer point inside a particular polymatroid. If the class is multiplicity-free, we obtain a complete characterization in terms of representable polymatroids. We also generalize some of the results to the case of products of Grassmannians.

## On the notions of facets, weak facets, and extreme functions of the Gomory-Johnson
infinite group problem

We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.