Pure Mathematics
http://hdl.handle.net/10012/9932
Tue, 16 Jul 2019 06:20:46 GMT2019-07-16T06:20:46ZComputability Theory and Some Applications
http://hdl.handle.net/10012/14800
Computability Theory and Some Applications
Deveau, Michael
We explore various areas of computability theory, ranging from applications in computable structure theory primarily focused on problems about computing isomorphisms, to a number of new results regarding the degree-theoretic notion of the bounded Turing hierarchy.
In Chapter 2 (joint with Csima, Harrison-Trainor, Mahmoud), the set of degrees that are computably enumerable in and above $\mathbf{0}^{(\alpha)}$ are shown to be degrees of categoricity of a structure, where $\alpha$ is a computable limit ordinal. We construct such structures in a particularly useful way: by restricting the construction to a particular case (the limit ordinal $\omega$) and proving some additional facts about the widgets that make up the structure, we are able to produce a computable prime model with a degree of categoricity as high as is possible. This then shows that a particular upper bound on such degrees is exact.
In Chapter 3 (joint with Csima and Stephenson), a common trick in computable structure theory as it relates to degrees of categoricity is explored. In this trick, the degree of an isomorphism between computable copies of a rigid structure is often able to be witnessed by the clever choice of a computable set whose image or preimage through the isomorphism actually attains the degree of the isomorphism itself. We construct a pair of computable copies of $(\omega, <)$ where this trick will not work, examine some problems with decidability of the structures and work with $(\omega^2, <)$ to resolve them by proving a similar result.
In Chapter 4, the effectivization of Walker's Cancellation Theorem in group theory is discussed in the context of uniformity. That is, if we have an indexed collection of instances of sums of finitely generated abelian groups $A_i \join G_i \cong A_i \join H_i$ and the code for the isomorphism between them, then we wish to know to what extent we can give a single procedure that, given an index $i$, produces an isomorphism between $G_i$ and $H_i$.
Finally, in Chapter 5, several results pertaining to the bounded Turing degrees (also known as the weak truth-table degrees) and the bounded jump are investigated, with an eye toward jump inversion. We first resolve a potential ambiguity in the definition of sets used to characterize degrees in the bounded Turing hierarchy. Then we investigate some open problems related to lowness and highness as it appears in this realm, and then generalize a characterization about reductions to iterated bounded jumps of arbitrary sets. We use this result to prove the non-triviality of the hierarchy of successive applications of the bounded jump above any set, showing that the problem of jump inversion must be non-trivial if it is true in any relativized generality.
Mon, 15 Jul 2019 00:00:00 GMThttp://hdl.handle.net/10012/148002019-07-15T00:00:00ZUnitary Correlation Sets and their Applications
http://hdl.handle.net/10012/14793
Unitary Correlation Sets and their Applications
Harris, Samuel
We relate Connes' embedding problem in operator algebras to the Brown algebra Unc(n), which is defined as the universal unital C*-algebra generated by the entries of an n x n unitary matrix. In particular, we show that the embedding problem is equivalent to determining whether or not the algebra Unc(n) has the weak expectation property for any (equivalently, all) n at least 2. From this perspective, we develop a theory of what we call unitary correlation sets. These sets are analogous to the usual sets of bipartite probabilistic correlations arising from the typical models in quantum information theory. We show that the analogue of the weak Tsirelson problem for unitary correlation sets is again equivalent to Connes' embedding problem. Moreover, we show that as long as Alice and Bob's unitaries are of size at least 2, the set of spatial unitary correlations is never a closed set. This result is analogous to a recent theorem of Slofstra, which states that the set of quantum probabilistic correlations is not closed, so long as the input and output sets are large enough.
We also discuss several applications of the theory of unitary correlation sets. First, we show that the class of extended non-local games known as quantum XOR games is a rich enough class to detect the validity of Connes' embedding problem. That is, the embedding problem has a positive answer if and only if the value of every quantum XOR game in the commuting model agrees with the value of the game in the approximate finite-dimensional model. Second, we use a C*-algebraic analogue of the quantum teleportation and super-dense coding maps from quantum information theory to obtain separations between the tensor product model (or "quantum spatial" model) and the approximate finite-dimensional model (or "quantum approximate" model), for matrix-valued generalizations of the usual Tsirelson corrleation sets. We use some of the intermediate results to also obtain separations between the matrix versions of the finite-dimensional model (or "quantum model") and the tensor product model.
Thu, 04 Jul 2019 00:00:00 GMThttp://hdl.handle.net/10012/147932019-07-04T00:00:00ZApproximation Constants for Closed Subschemes of Projective Varieties
http://hdl.handle.net/10012/14764
Approximation Constants for Closed Subschemes of Projective Varieties
Rollick, Nickolas
Diophantine approximation is a branch of number theory with a long history, going back at least to the work of Dirichlet and Liouville in the 1840s. The innocent-looking question of how well an arbitrary real algebraic number can be approximated by rational numbers (relative to the size of the denominator of the approximating rational number) took more than 100 years to resolve, culminating in the definitive Fields Medal-winning work of Klaus Roth in 1955.
Much more recently, David McKinnon and Mike Roth have re-phrased and generalized this Diophantine approximation question to apply in the setting of approximating algebraic points on projective varieties defined over number fields. To do this, they defined an "approximation constant", depending on the point one wishes to approximate and a given line bundle. This constant measures the tradeoff between the closeness of the approximation and the arithmetic complexity of the point used to make the approximation, as measured by a height function associated to the line bundle.
In particular, McKinnon and Roth succeeded in proving lower bounds on the approximation constant in terms of the "Seshadri constant" associated to the given point and line bundle, measuring local positivity of the line bundle around the point. Appropriately interpreted, these results generalize the classical work of Liouville and Roth, and the corresponding McKinnon-Roth theorems are therefore labelled "Liouville-type" and "Roth-type" results.
Recent work of Grieve and of Ru-Wang have taken the Roth-type theorems even further; in contrast, we explore results of Liouville-type, which are more elementary in nature. In Chapter 2, we lay the groundwork necessary to define the approximation constant at a point, before generalizing the McKinnon-Roth definition to approximations of arbitrary closed subschemes. We also introduce the notion of an essential approximation constant, which ignores unusually good approximations along proper Zariski-closed subsets. After verifying that our new approximation constant truly does generalize the constant of McKinnon-Roth, Chapter 3 establishes a fundamental lower bound on the approximation constants of closed subschemes of projective space, depending only on the equations cutting out the subscheme.
In Chapter 4, we provide a series of explicit computations of approximation constants, both for subschemes satisfying suitable geometric conditions, and for curves of low degree in projective 3-space. We will encounter difficulties computing the approximation constant exactly for general cubic curves, and we spend some time showing why some of the more evident approaches do not succeed. To conclude the chapter, we take up the question of large gaps between the ordinary and essential approximation constants, by considering approximations to a certain rational point on a diagonal quartic surface. Finally, in Chapter 5, we generalize the Liouville-type results of McKinnon-Roth.
Wed, 19 Jun 2019 00:00:00 GMThttp://hdl.handle.net/10012/147642019-06-19T00:00:00ZDegrees of Categoricity and the Isomorphism Problem
http://hdl.handle.net/10012/14753
Degrees of Categoricity and the Isomorphism Problem
Mahmoud, Mohammad
In this thesis, we study notions of complexity related to computable structures.
We first study degrees of categoricity for computable tree structures. We show that, for any computable ordinal $\alpha$, there exists a computable tree of rank $\alpha+1$ with strong degree of categoricity ${\bf 0}^{(2\alpha)}$ if $\alpha$ is finite, and with strong degree of categoricity ${\bf 0}^{(2\alpha+1)}$ if $\alpha$ is infinite. For a computable limit ordinal $\alpha$, we show that there is a computable tree of rank $\alpha$ with strong degree of categoricity ${\bf 0}^{(\alpha)}$ (which equals ${\bf 0}^{(2\alpha)}$).
In general, it is not the case that every Turing degree is the degree of categoricity of some structure. However, it is known that every degree that is of a computably enumerable (c.e.) set\ in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a successor ordinal, is a degree of categoricity. In this thesis, we include joint work with Csima, Deveau and Harrison-Trainor which shows that every degree c.e.\ in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a limit ordinal, is a degree of categoricity. We also show that every degree c.e.\ in and above $\mathbf{0}^{(\omega)}$ is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk.
After that, we study the isomorphism problem for tree structures. It follows from our proofs regarding the degrees of categoricity for these structures that, for every computable ordinal $\alpha>0$, the isomorphism problem for trees of rank $\alpha$ is $\Pi_{2\alpha}$-complete.
We also discuss the isomorphism problem for pregeometries in which dependent elements are dense and the closure operator is relatively intrinsically computably enumerable. We show that, if $K$ is a class of such pregeometries, then the isomorphism problem for the class $K$ is $\Pi_3$-hard.
Finally, we study the Turing ordinal.
We observed that the definition of the Turing ordinal has two parts each of which alone can define a specific ordinal which we call the upper and lower Turing ordinals. The Turing ordinal exists if and only if these two ordinals exist and are equal.
We give examples of classes of computable structures such that the upper Turing ordinal is $\beta$ and the lower Turing ordinal is $\alpha$ for all computable ordinals $\alpha<\beta$.
Wed, 12 Jun 2019 00:00:00 GMThttp://hdl.handle.net/10012/147532019-06-12T00:00:00Z