Extremal ideals are a class of square-free monomial ideals which dominate and determine many algebraic invariants of powers of all square-free monomial ideals. For example, the rth power Eqr of the extremal ideal on q generators has the maximum Betti numbers among the rth power of any square-free monomial ideal with q generators. In this paper we study the combinatorial and geometric structure of the (minimal) free resolutions of powers of square-free monomial ideals via the resolutions of powers of extremal ideals. Although the end results are algebraic, this problem has a natural interpretation in terms of polytopes and discrete geometry. Our guiding conjecture is that all powers Eqr of extremal ideals have resolutions supported on their Scarf simplicial complexes, and thus their resolutions are as small as possible. This conjecture is known to hold for r≤2 or q≤4. In this paper we prove the conjecture holds for r=3 and any q≥1 by giving a complete description of the Scarf complex of Eq3. This effectively gives us a sharp bound on the betti numbers and projective dimension of the third power of any square-free momomial ideal. For large i and q, our bounds on the ith betti numbers are an exponential improvement over previously known bounds. We also describe a large number of faces of the Scarf complex of Eqr for any r,q≥1.

Inspired by the Roller Coaster Theorem from graph theory, we prove the existence of artinian Gorenstein algebras with unconstrained Hilbert series, which we call Roller Coaster algebras. Our construction relies on Nagata idealization of quadratic monomial algebras defined by whiskered graphs. The monomial algebras are interesting in their own right, as our results suggest that artinian level algebras defined by quadratic monomial ideals rarely have the weak Lefschetz property. In addition, we discover a large family of G-quadratic Gorenstein algebras failing the weak Lefschetz property.

We introduce and study a new probabilistic variant of the classical parking protocol of Konheim and Weiss [29], which is closely related to Internal Diffusion Limited Aggregation, or IDLA, introduced in 1991 by Diaconis and Fulton [15]. In particular, we show that if one runs our parking protocol starting with a parking function whose outcome permutation (in the sense of the classical parking process of Konheim and Weiss) is the identity permutation, then we can compute the exact probability that all of the cars park. Furthermore, we compute the expected time it takes for the protocol to complete assuming all of the cars park, and prove that the parking process is negatively correlated. We also study statistics of uniformly random weakly increasing parking functions, a subset of parking functions whose outcome is the identity permutation. We give the distribution of the last entry, along with the probability that a specific set of cars is lucky, and the expected number of lucky cars.

Lefschetz properties and inverse systems have played key roles in understanding the h-vector of simplicial spheres. In 1996, Lee established connections between these two algebraic tools and rigidity theory, an area often used in the study of motions of geometric complexes. One of the key ideas, is to translate geometric information about a complex, coming from vertex coordinates, to the algebraic notion of a linear system of parameters. In this paper, we explore similar connections in the nonlinear case, by using recent results of Herzog and Moradi (2021) where they prove that a subset of the elementary symmetric polynomials is always a system of parameters for the Stanley-Reisner ideal of a complex. We investigate connections to the study of Lefschetz properties of monomial ideals. Using this perspective, we recover and extend the well known result of Migliore, Miró-Roig and Nagel on the failure of the WLP of monomial almost complete intersections, by showing that, with one simple exception, every homology sphere has a monomial artinian reduction failing the weak Lefschetz property. Finally, we state probabilistic consequences of our results under a model introduced by Linial and Meshulam. We prove that there exists an open interval for the probability parameter where failure of Lefschetz properties of monomial ideals should be expected.

The theory of Rees algebras of monomial ideals has been extensively studied, and as a consequence, many (sometimes partial) equivalences between algebraic properties of monomial ideals, and combinatorial properties of simplicial complexes and hypergraphs are known. In this paper we show how this theory can be used to find interesting examples in the theory of Lefschetz properties. We explore the consequences of known results from Lefschetz properties to the Rees algebras of squarefree monomial ideals, for example in the calculation of analytic spread. In particular, we show a connection between symbolic powers and f-vectors of simplicial complexes. This perspective leads us to a generalization of Postnikov’s ”mixed Eulerian numbers”. We prove the positivity of such numbers in our setting.

In 2022, Jinha Kim proved a conjecture by Engström that states the independence complex of a graph with no induced cycle of length divisible by 3 is either contractible or homotopy equivalent to a sphere. We give criteria for when the independence complex of a ternary graph is contractible, and describe the dimension of the sphere when it is not. We then apply our results to describe the multigraded betti numbers of the edge ideal of a ternary graph. In particular, we show that the regularity and depth of edge ideals of a ternary graph G are equal if and only if the independence complex of G is not contractible. Finally, we apply our results to partially recover and generalize recent results on the depth of edge ideals of some unicyclic graphs.

We consider Artinian level algebras arising from the whiskering of a graph. Employing a result by Dao-Nair we show that multiplication by a general linear form has maximal rank in degrees 1 and n−1 when the characteristic is not two, where n is the number of vertices in the graph. Moreover, the multiplication is injective in degrees < n/2 when the characteristic is zero, following a proof by Hausel. Our result in the characteristic zero case is optimal in the sense that there are whiskered graphs for which the multiplication maps in all intermediate degrees n/2,…,n−2 of the associated Artinian algebras fail to have maximal rank, and consequently, the weak Lefschetz property.

Recently, H. Dao and R. Nair gave a combinatorial description of simplicial complexes Δ such that the squarefree reduction of the Stanley-Reisner ideal of Δ has the WLP in degree 1 and characteristic zero. In this paper, we apply the connections between analytic spread of equigenerated monomial ideals, mixed multiplicities and birational monomial maps to give a sufficient and necessary condition for the squarefree reduction A(Δ) to satisfy the WLP in degree i and characteristic zero in terms of mixed multiplicities of monomial ideals that contain combinatorial information of Δ, we call them incidence ideals. As a consequence, we give an upper bound to the possible failures of the WLP of A(Δ) in degree i in positive characteristics in terms of mixed multiplicities. Moreover, we extend Dao and Nair's criterion to arbitrary monomial ideals in positive odd characteristics

This work is based on the article Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, by J. Huh, published in Journal of the American Mathematical Society (2012). The dissertation is devoted to the study of the methods used by Huh to prove a particular case of the Rota-Heron-Welsh conjecture. J. Huh used results from algebraic topology, combinatorics, algebraic geometry and commutative algebra to prove the log-concavity of the coefficients of the characteristic polynomial of a matroid representable over a field of characteristic zero.