Jack Jeffries

What Makes a Neural Code Convex

Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley-Reisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in understanding the intrinsic combinatorial properties of convex codes.

Multiplicities of classical varieties

The j-multiplicity plays an important role in the intersec- tion theory of Stuckrad-Vogel cycles, while recent developments confirm the connections between the e-multiplicity and equisingularity theory. In this paper we establish, under some constraints, a relation ship between the j-multiplicity of an ideal and the degree of its fiber cone. As a conse- quence, we are able to compute the j-multiplicity of all the ideals defin- ing rational normal scrolls. By using the standard monomial theory, we can also compute the j- and e-multiplicity of ideals defining determinan- tal varieties: The found quantities are integrals which, qu ite surprisingly, are central in random matrix theory.

Non-simplicial decompositions of Betti diagrams of complete intersections

We investigate decompositions of Betti diagrams over a polyno- mial ring within the framework of Boij{Soderberg theory. That is, given a Betti diagram, we decompose it into pure diagrams. Relaxing the require- ment that the degree sequences in such pure diagrams be totally ordered, we are able to dene a multiplication law for Betti diagrams that respects the decomposition and allows us to write a simple expression the decomposition of the Betti diagram of any complete intersection in terms of the degrees of its minimal generators. In the more traditional sense, the decomposition of complete intersections of codimension at most 3 are also computed as given by the totally ordered decomposition algorithm obtained from (ES09). In higher codimension, obstructions arise that inspire our work on an alternative algo- rithm.

A Zariski--Nagata theorem for smooth ℤ-algebras

Abstract In a polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish up to order n on the corresponding variety. However, this description fails in mixed characteristic. In this paper, we use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic, and prove that this new object does coincide with the symbolic powers of prime ideals. This seems to be the first application of p-derivations to commutative algebra.