program

In this talk, we first introduce a new class of (commutative noetherian) local rings which we call uniformly dominant local rings, and show that uniform dominance is preserved under several basic operations. Next, we give certain conditions on syzygies of the residue field, and present some classes of local rings that satisfy one of those conditions. Finally, we prove that a local ring satisfying one of those conditions is uniformly dominant, and its singularity category has finite Orlov spectrum if it has an isolated singularity. If time permits, we apply our methods to direct summands of syzygies of the residue field which have finite G-dimension, and unify/refine the speaker's old results. 

Let $(S,m)$ be a local normal domain of dimension two with "nice" singularity. What kind of singularity one can expect for $R=S/xS$ for a general element $x$ in $m-m^2$? I will discuss some recent results in this direction, which offer new perspectives on singularity of curves and surfaces.

I will explain an attempt to describe the direct sum decomposition of the Frobenius direct images $F^e_*(L^n)$ on a smooth quintic del Pezzo surface $X$ in positive characteristic with respect to the anti-canonical polarization $L=-K_X$. It is shown by Mallory that there are only finitely many Frobenius summands (i.e., indecomposable vector bundles appearing in the decomposition) up to twists by $L$. Our approach is aiming at more detailed description of individual Frobenius summands. 

In this talk, we prove that if a 3-dimensional quasi-projective variety X over an algebraically closed field of characteristic p > 3 has only log canonical singularities, then so does a general hyperplane section H of X.

In the course of the proof, we give a sufficient condition for log canonical surface singularities over a field to be geometrically log canonical.

The coordinate ring of the variety of matrices with bounded rank at most can be realized in a natural way as an invariant ring of a polynomial ring; the same is true if one replaces "matrices" by "symmetric matrices" or "alternating matrices". This realization as a ring of invariants explains many of the nice algebraic properties enjoyed by determinantal rings, at least in characteristic zero, since is a retract of a polynomial ring via the Reynolds operator. Motivated by understanding the relationship between and in arbitrary characteristic, we consider the ideal generated by positive degree invariants inside of the ambient polynomial ring. For example, certain varieties of complexes as introduced by Buchsbaum and Eisenbud occur like so. We will discuss some aspects of the behavior of local cohomology with support in these ideals in different characteristics and some applications. This is based on work in progress with Pandey, Singh, and Walther, and earlier joint work with Hochster, Pandey, and Singh.

Searching for some finiteness in infinite free resolutions,  Serre asked whether the generating function for the ranks of the free modules in the minimal free resolution

 of the residue field of a local ring represents a rational function. Kaplansky even conjectured that this would be true (Anick finally showed that it is false in general). An early milestone in this research was a construction by Golod  giving a positive answer for a class of rings now called Golod rings. Recently I, with collaborators Cuong, Dao, Kobayashi, Polini and Ulrich, discovered that the resolution constructed by Golod has an unexpected internal structure, and experiments suggest much more than we can prove.

I will explain Golod's construction and our current results and conjectures.

The aim of this talk is to show the utility of cones of certain smooth projective varieties to construct graded rings over the Witt vectors with Frobenius lifts. Then I explain that the resulting rings have interesting singularities and connections with perfectoid towers and lim CM sequences.  This is a joint work with S. Ishiro, R. Ishizuka and K. Nakazato. 

Recently, Ein-Ha-Lazarsfeld proved that if I is a homogeneous ideal whose zero locus is a non-singular complex projective scheme, then the saturation degree sdeg I^n is bounded above by a linear function of n whose slope is less or equal the maximal generating degree of I. We show that for an arbitrary graded ideal I in an arbitrary graded ring, sdeg I^n is either a constant or a linear function for n large enough whose slope is one of the generating degrees of I.

In this talk, I will talk about some interaction between combinatorial commutative algebra of Stanley-Reisner rings and affine stresses of simplicial polytopes. I will explain a relation between affine stresses of polytopes and inverse systems of certain quotients of Stanley-Reisner rings, and show that this algebraic connection unable us to prove a partial affirmative answer to the conjecture of Gil Kalai telling that affine type of prime simplicial d-polytopes can be constructed from affine stresses when d is larger than or equals to 4.

The maximal minors of a matrix of indeterminates are a universal

Gröbner basis by a theorem of Bernstein, Sturmfels and Zelevinsky. On the

other hand it is known that they are not always a universal Sagbi basis. By an experimental

approach we discuss their behavior under varying monomial orders and their extensions to

Sagbi bases. These experiments motivated a new implementation of the Sagbi algorithm

which is organized in a Singular script and falls back on Normaliz for the combinatorial

computations. In comparison to packages in the current standard distributions of Macaulay 2 and

Singular it extends the range of computability by at least one order of magnitude.

This is joint work with Aldo Conca and appeared in J. Symb. Comp. 120 (2024).

This talk will discuss a somewhat surprising conjectured bound on the number of generators of a licci (in the linkage class of a complete intersection) ideal, namely that the number of generators of a homogeneous licci ideal is bounded above by the greatest last

twist in a minimal graded free resolution of the ideal. This is

continuing joint work with Claudia Polini and Bernd Ulrich.

This talk will discuss some recent work on using perfectoid algebras and big Cohen-Macaulay to measure singularities in mixed characteristic, analogous to work done by Watanabe and co-authors in the positive characteristic case. 

We explore Lech's inequality relating the colength and multiplicity of m-primary ideals in a Noetherian local ring. We introduce an invariant that measures the sharpness of Lech's inequality and show its connections with singularities of asymptotically semi-stable varieties and singularities arising from the minimal model program. We compute this invariant in various examples. This is a joint work in progress with Ilya Smirnov.

In the talk, I will discuss the vanishing of higher direct images of the structure sheaf on a normal Cohen-Macaulay variety that is proper over a smooth variety. 

We will talk about several ring-theoretic properties of normal

tangent cone; e.g. Gorensteinness, maximal embedding dimension,

maximal relative normal reduction number etc. For example,

1. We compute Hiibert series of some graded rings with

maximal relative normal reduction number. 

2. We prove that the normal tengent cone \barG(m)

of the maximal ideal of some hypersurface is Cohen-Macaulay and

characterize several properties in terms of basic invariants.

3. We can charcterize the Gorensteinness of \barG(I) for

integrally closed ieeal I of a 2-dimensional normal local

domain A in terms of anti-nef cycle Z corrsponding to I.

This is a joint work with M.Bhat, S.Dubey, S.K.Masuti. T.Okuma,  J.K.Verma

and K.-i.Watanabe.  

Let (A,m) be a complete local ring and G=gr_m(A) its associated graded ring. The problem of the descent of a property from G to A was extensively studied and the answers are predominantly positive. The problems arise passing from A to G, because we may lose many good geometric and algebraic properties. We present an overview up to a recent work with A. De Stefani and M. Varbaro.  We introduce a homogenization technique which allows to relate  G  to the special fiber and A to the generic fiber of a ``Groebner-like" deformation. Using this technique we prove sharp results concerning the connectedness of A and G. 

In the last 40 years, many researchers have studied the symbolic Rees rings of space monomial primes. For example, in 1994, Goto, Nishida, and Watanabe found the first example that is not finitely generated. This problem is closely related to the Nagata conjecture, and researches on them are still ongoing. In this talk, we will introduce some recent results.