I will discuss recent large-scale computations, which utilize numerical linear algebra and highly distributed, high-performance computing to generate data about the syzygies of various algebraic surfaces. Further, I will discuss how this data has led to several new conjectures.

A \(p^{-1}\)-linear map is a central object of study in the theory of singularities in prime characteristic \(p\) defined via Frobenius. For example, such maps have been used to create prime characteristic analogues of fundamental notions from characteristic \(0\) birational geometry such as log canonical singularities, klt singularities and multiplier ideals. In this talk we will survey the state of the art on \(p^{-1}\)-linear maps, with a focus on the relationship between the existence of such maps and Grothendieck's notion of excellent rings. The talk will be based on recent joint works with Takumi Murayama, and with Takumi Murayama and Karen E. Smith.

J. Hong and B. Ulrich proved that the integral closure of an ideal of height at least 2 is compatible with specialization by generic elements, allowing one to use induction on the height of the ideal to prove results about integrally closed ideals. In order to apply this theorem, one must extend the base ring to a polynomial ring or localization thereof in order to define generic elements. We have shown that the integral closure of an ideal of height at least 2 is compatible with specialization by general elements, allowing one to induct on the height of the ideal without extending the ring. This is based on joint work with Rachel Lynn.

The Frobenius complexity of a local ring \(R\) measures asymptotically the abundance of Frobenius actions of order \(e\) on the injective hull of the residue field of \(R\). It is known that, for Stanley–Reisner rings, the Frobenius complexity is either \(-\infty\) or \(0\). This invariant is determined by the complexity sequence \(\{c_e\}_e\) of the ring of Frobenius operators on the injective hull of the residue field. We will show that \(\{c_e\}_e\) is constant for \(e\geq 2,\) generalizing work of Àlvarez Montaner, Boix and Zarzuela. Our result settles an open question mentioned by Àlvarez Montaner in one of his papers.

When S is a ring of prime characteristic p > 0, the local cohomology of S carries a natural Frobenius structure. If S is regular, we have access to Lyubeznik's powerful theory of F-modules. We lose this if S is singular, but retain the notion of Frobenius actions. In this talk, we will present recent joint work with Eric Canton on some advantages to using a non-standard Frobenius action, defined when S is a complete intersection ring, and will discuss applications to questions about finiteness properties.

We introduce Lech–Mumford constant to study the relationship between colength and multiplicity of m-primary ideals in a Noetherian local ring. Basically this invariant measures the sharpness of Lech's inequality, and was considered by Mumford in his study of GIT. We discuss some connections with singularity theory and compute some examples in low dimensions. Work in progress with Ilya Smirnov.

Let \((R,\mathfrak{m},k)\) be a Cohen-Macaulay local ring, or positively graded over \(k\), and let \(M\) and \(N\) be finitely generated \(R\)-modules. We find sufficient conditions on \(R\) so that if \(\operatorname{Tor}_i^R(M,N)=0\) for \(i\gg0\), then either \(M\) or \(N\) has finite projective dimension. As a consequence, we show the Auslander-Reiten conjecture for rings of multiplicity at most 8. We also study this property in the case \(M=N\) and show that it holds for \(R\) whenever \(\mathfrak{m}^3\) vanishes. Finally, by specializing to the case of the canonical module, we settle Tachikawa's conjecture in the graded case. This is based on joint work with Justin Lyle and Sean Sather-Wagstaff.

The Hochschild cohomology ring of an associative algebra is a graded commutative ring with respect to the cup product. It has a Lie bracket (Gerstenhaber bracket) making it into a Gerstenhaber algebra. There are several equivalent definitions of the cup product, so it is well understood. The Gerstenhaber bracket on the other hand is not well understood. I will present the Gerstenhaber algebra structure on Hochschild cohomology for Koszul algebras defined by quivers and relations using the idea of homotopy liftings. This technique was first introduced by Yury Volkov.

Arithmetic and geometric properties of a normal domain are encoded in its divisor class group. For example, a normal domain is a unique factorization domain if and only if its divisor class group is the 0 group. As indicated by the title, we explore properties of the divisor class group of a strongly F-regular ring. Specifically, we will discuss why the torsion subgroup of the divisor class group of any local strongly F-regular ring is finite. We will also discuss a novel, elementary, and streamlined proof that the divisor class group of a 2-dimensional F-regular ring is finite.

Let \(S\) be an unramified regular local ring of mixed characteristic \(p>0\) and \(R\) the integral closure of \(S\) in a biradical extension of degree \(p^2\) of its quotient field, obtained by adjoining \(p\)-th roots of squarefree elements. We provide sufficient conditions for \(R\) to be Cohen Macaulay and show the existence of small Cohen Macaulay modules over a class of non Cohen Macaulay integral closures which are vector bundles on the punctured spectrum of \(S\). This is done by studying the properties of the conductor of \(R\) to the complete intersection ring \(S[\omega,\mu]\) where \(\omega\) and \(\mu\) are the aforementioned \(p\)-th roots.

The tight closure and integral closure of an ideal \(I\) are concepts which, in positive characteristic, can be described using the asymptotic behavior of either the ordinary powers \(I^n\) or Frobenius powers \(I^{[q]}\) of an ideal. Using a “mixed power” ideal which lies between these two powers and is parameterized by a real number \(s\), we may define a family of closures called \(s\)-closures which lie between tight and integral closure. These are built up from possibly non-idempotent “weak” \(s\)-closures.
In this talk we will define the (weak) \(s\)-closures and establish an important description of them in certain graded cases using rational powers of ideals. We will use this description to prove that the \(s\)-closure is equal to the \(s\)-closure in certain nice graded settings, including the setting of monomial ideals in toric rings. Finally, we will give a generalization of the Briançon-Skoda Thereom which compares \(s\)-closures for different values of \(s\).