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**1 - 2**of**2**### RESEARCH STATEMENT

"... In mathematics, the most elegant results emerge only once one has found the right setting for them to do so. For example, the famous Bezout’s theorem in algebraic geometry, which says that any two plane curves of degree d and e intersect in exactly de points, only holds if one uses the complex numbe ..."

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In mathematics, the most elegant results emerge only once one has found the right setting for them to do so. For example, the famous Bezout’s theorem in algebraic geometry, which says that any two plane curves of degree d and e intersect in exactly de points, only holds if one uses the complex numbers instead of the reals, uses the projective plane instead of the affine plane, counts with multiplicity rather than naively, and works in the derived setting rather than the discrete one (in order to deal with cases of self-intersection). The appropriate structure is necessary to produce the most perfect result. My research in both pure and applied mathematics has tended to revolve around this principle. On one end of the spectrum, my Ph.D. thesis is a generalization of the above Bezout’s theorem, except that it takes place in the category of manifolds rather than schemes, and hence it takes place in a similarly-structured category. On the other end of the spectrum, my work in applied mathematics has been focused on finding categories which best express the dynamics of a given subject in computer science, such as realizing databases as sheaves on simplicial sets. In the following sections, I will discuss my current work and future goals in both the pure and the applied side of my research.

### WALDHAUSEN ADDITIVITY: CLASSICAL AND QUASICATEGORICAL

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"... Abstract. We give a short proof of classical Waldhausen Additivity, and then prove Waldhausen Additivity for an ∞-version of Waldhausen K-theory. Namely, we prove that Waldhausen K-theory sends a split-exact sequence of Waldausen quasicategories A → E → B to a stable equivalence of spectra K(E) → K( ..."

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Abstract. We give a short proof of classical Waldhausen Additivity, and then prove Waldhausen Additivity for an ∞-version of Waldhausen K-theory. Namely, we prove that Waldhausen K-theory sends a split-exact sequence of Waldausen quasicategories A → E → B to a stable equivalence of spectra K(E) → K(A) ∨ K(B) under a few mild hypotheses. For example, each cofiber sequence in A of the form A0 → A1 → ∗ is required to have the first map an equivalence. Model structures, presentability, and stability are not needed. In an effort to make the article self-contained, we provide many details in our proofs, recall all the prerequisites from the theory of quasicategories, and prove some of those as well. For instance, we develop the expected facts about (weak) adjunctions between quasicategories and (weak) adjunctions