The "neural code" is the way the brain characterizes, stores, and processes information. Unraveling the neural code is a key goal of mathematical neuroscience. Topology, coding theory, and, recently, commutative algebra are some the mathematical areas that are involved in analyzing these codes. Neural rings and ideals are algebraic objects that create a bridge between mathematical neuroscience and commutative algebra. A neural ideal is an ideal in a polynomial ring that encodes the combinatorial firing data of a neural code. Using some algebraic techniques one hopes to understand more about the structure of a neural code via neural rings and ideals. In this paper, we introduce an operation, called "polarization," that allows us to relate neural ideals with squarefree monomial ideals, which are very well studied and known for their nice behavior in commutative algebra.2010 Mathematics Subject Classification. 13F20, 13P25, 13L99, 92B20.

We study the representation theory of the increasing monoid. Our results provide a fairly comprehensive picture of the representation category: for example, we describe the Grothendieck group (including the effective cone), classify injective objects, establish properties of injective and projective resolutions, construct a derived auto-duality, and so on. Our work is motivated by numerous connections of this theory to other areas, such as representation stability, commutative algebra, simplicial theory, and shuffle algebras.

The equivariant Hilbert series of an ideal generated by an orbit of a monomial under the action of the monoid Inc(N) of strictly increasing functions is determined. This is used to find the dimension and degree of such an ideal. The result also suggests that the description of the denominator of an equivariant Hilbert series of an arbitrary Inc(N)-invariant ideal as given by Nagel and Römer is rather efficient.2010 Mathematics Subject Classification. 13F20, 13A02, 13D40, 13A50.

The Eisenbud-Green-Harris (EGH) conjecture states that a homogeneous ideal in a polynomial ring K[x1, . . . , xn] over a field K that contains a regular sequence f1, . . . , fn with degrees ai, i = 1, . . . , n has the same Hilbert function as a lex-plus-powers ideal containing the powers x a i i , i = 1, . . . , n. In this paper, we discuss a case of the EGH conjecture for homogeneous ideals generated by n + 2 quadrics containing a regular sequence f1, . . . , fn and give a complete proof for EGH when n = 5 and a1 = • • • = a5 = 2.

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