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**1 - 5**of**5**### RESEARCH STATEMENT

, 2010

"... I am interested in studying the complexity of mathematical practice. In mathematics, as we all know, some structures are more complicated than others, some constructions more complicated than others, and some proofs more complicated than others. I am interested in understanding how to measure this c ..."

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I am interested in studying the complexity of mathematical practice. In mathematics, as we all know, some structures are more complicated than others, some constructions more complicated than others, and some proofs more complicated than others. I am interested in understanding how to measure this complexity and in measuring it. The motivations for this come from different areas. Form a foundational viewpoint, we want to know what assumptions we really need to do mathematics (ZF C is way much more than we usually use), and we are also interested in knowing what assumptions are used in the different areas of mathematics. Form a computational viewpoint, it is important to know what part of mathematics can be done by mechanical algorithms, and, even for the part that can’t be done mechanically, we want to know how constructive are the objects we deal with. Furthermore, it is sometimes the case that this computational analysis allows us to find connections between constructions in different areas of mathematics, and in many cases to obtain a deeper understanding of mathematical objects being analyzed. My work is quite diverse in terms of the techniques I have used, the approaches I have taken, and the areas of mathematics that I have analyzed. However, my background area is Computability Theory, and most of my work can be considered as part of this branch of Mathematical Logic.

### THERE IS NO ORDERING ON THE CLASSES IN THE GENERALIZED HIGH/LOW HIERARCHIES.

"... Abstract. We prove that the existential theory of the Turing degrees, in the language with Turing reduction, 0, and unary relations for the classes in the generalized high/low hierarchy, is decidable. 1. ..."

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Abstract. We prove that the existential theory of the Turing degrees, in the language with Turing reduction, 0, and unary relations for the classes in the generalized high/low hierarchy, is decidable. 1.

### EMBEDDINGS INTO THE TURING DEGREES.

, 2007

"... The structure of the Turing degrees was introduced by Kleene and Post in 1954 [KP54]. Since then, its study has been central in the area of Computability Theory. One approach for analyzing the shape of this structure has been looking at the structures that can be ..."

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The structure of the Turing degrees was introduced by Kleene and Post in 1954 [KP54]. Since then, its study has been central in the area of Computability Theory. One approach for analyzing the shape of this structure has been looking at the structures that can be

### A NOTE ON THE JOIN PROPERTY

"... Abstract. A Turing degree a satisfies the join property if, for every non-zero b < a, there exists c < a with b ∨ c = a. It was observed in [N4] that all degrees which are non-GL2 satisfy the join property. This, however, leaves open many questions. Do all a.n.r. degrees satisfy the join prope ..."

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Abstract. A Turing degree a satisfies the join property if, for every non-zero b < a, there exists c < a with b ∨ c = a. It was observed in [N4] that all degrees which are non-GL2 satisfy the join property. This, however, leaves open many questions. Do all a.n.r. degrees satisfy the join property? How about the PA degrees or the Martin-Löf random degrees? A degree b satisfies the cupping property if, for every a> b, there exists c < a with b ∨ c = a. Is satisfying the cupping property equivalent to all degrees above satisfying join? We answer all of these questions by showing that above every low degree there is a low degree which does not satisfy join. We show, in fact, that all low fixed point free degrees a fail to satisfy join, and moreover, that the non-zero degree below a without any joining partner can be chosen to be a c.e. degree. 1.

### JOINING UP TO THE GENERALIZED HIGH DEGREES

"... Abstract. We show that every generalized high Turing degree is the join of two minimal degrees, thereby settling a conjecture of Posner’s from the 70s. 1. ..."

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Abstract. We show that every generalized high Turing degree is the join of two minimal degrees, thereby settling a conjecture of Posner’s from the 70s. 1.