Jeffrey Giansiracusa

Equations of tropical varieties

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as

In vitro kinetic studies on the mechanism of oxygen-dependent cellular uptake of copper radiopharmaceuticals

\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)

A Grassmann algebra for matroids

by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric variety. Upon passing to the set of

The framed little 2-discs operad and diffeomorphisms of handlebodies

\mathbb{T}

THE STABLE MAPPING CLASS GROUP OF SIMPLY CONNECTED 4-MANIFOLDS

-points this reduces to Kajiwara-Paynes extended tropicalization, and in the case of a projective hypersurface we show that the scheme structure determines the multiplicities attached to the top-dimensional cells. By varying the valuation, these tropicalizations form algebraic families of

On the relation between hyperrings and fuzzy rings

\mathbb{T}

Cyclic operad formality for compactified moduli spaces of genus zero surfaces

-schemes parameterized by a moduli space of valuations on R that we construct. For projective subschemes, the Hilbert polynomial is preserved by tropicalization, regardless of the valuation. We conclude with some examples and a discussion of tropical bases in the scheme-theoretic setting.

The diffeomorphism group of a K3 surface and Nielsen realization

The development of hypoxia-selective radiopharmaceuticals for use as therapeutic and/or imaging agents is of vital importance for both early identification and treatment of cancer and in the design of new drugs. Radiotracers based on copper for use in positron emission tomography have received great attention due to the successful application of copper(II) bis(thiosemicarbazonato) complexes, such as [(60/62/64)Cu(II)ATSM] and [(60/62/64)Cu(II)PTSM], as markers for tumour hypoxia and blood perfusion, respectively. Recent work has led to the proposal of a revised mechanism of hypoxia-selective cellular uptake and retention of [Cu(II)ATSM]. The work presented here describes non-steady-state kinetic simulations in which the reported pO(2)-dependent in vitro cellular uptake and retention of [(64)Cu(II)ATSM] in EMT6 murine carcinoma cells has been modelled by using the revised mechanistic scheme. Non-steady-state (NSS) kinetic analysis reveals that the model is in very good agreement with the reported experimental data with a root-mean-squared error of less than 6% between the simulated and experimental cellular uptake profiles. Estimated rate constants are derived for the cellular uptake and washout (k(1) = 9.8 +/- 0.59 x 10(-4) s(-1) and k(2) = 2.9 +/- 0.17 x 10(-3) s(-1)), intracellular reduction (k(3) = 5.2 +/- 0.31 x 10(-2) s(-1)), reoxidation (k(4) = 2.2 +/- 0.13 mol(-1) dm(3) s(-1)) and proton-mediated ligand dissociation (k(5) = 9.0 +/- 0.54 x 10(-5) s(-1)). Previous mechanisms focused on the reduction and reoxidation steps. However, the data suggest that the origins of hypoxia-selective retention may reside with the stability of the copper(I) anion with respect to protonation and ligand dissociation. In vitro kinetic studies using the nicotimamide adenine dinucleotide (NADH)-dependent ferredoxin reductase enzyme PuR isolated from the bacterium Rhodopseudomonas palustris have also been conducted. NADH turnover frequencies are found to be dependent on the structure of the ligand and the results confirm that the proposed reduction step in the mechanism of hypoxia selectivity is likely to be mediated by NADH-dependent enzymes. Further understanding of the mechanism of hypoxia selectivity may facilitate the development of new imaging and radiotherapeutic agents with increased specificity for tumour hypoxia.

On the homotopy type of the Deligne–Mumford compactification

We introduce an idempotent analogue of the exterior algebra for which the theory of tropical linear spaces (and valuated matroids) can be seen in close analogy with the classical Grassmann algebra formalism for linear spaces. The top wedge power of a tropical linear space is its Plücker vector, which we view as a tensor, and a tropical linear space is recovered from its Plücker vector as the kernel of the corresponding wedge multiplication map. We prove that an arbitrary d-tensor satisfies the tropical Plücker relations (valuated exchange axiom) if and only if the dth wedge power of the kernel of wedge-multiplication is free of rank one. This provides a new cryptomorphism for valuated matroids, including ordinary matroids as a special case.

ON THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS

The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (that is, the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of Costellos theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ‘open string’ modular operad made from moduli spaces of Riemann surfaces with marked intervals on the boundary. Our technique also recovers a theorem of Braun that the derived modular envelope of the cyclic operad that describes associative algebras with involution is homotopy equivalent to the modular operad made from moduli spaces of unoriented Klein surfaces with open string gluing.