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**1 - 2**of**2**### Spectra of Structures and Relations

, 2006

"... We consider embeddings of structures which preserve spectra: if g: M → S with S computable, then M should have the same Turing degree spectrum (as a structure) that g(M) has (as a relation on S). We show that the computable dense linear order L is universal for all countable linear orders under this ..."

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We consider embeddings of structures which preserve spectra: if g: M → S with S computable, then M should have the same Turing degree spectrum (as a structure) that g(M) has (as a relation on S). We show that the computable dense linear order L is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph G. Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on G. Finally, we consider the extent to which all spectra of unary relations on the structure L may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c ′ ≥T 0 ′ ′. 1

### 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.