Apr 15, 2024
Monday

09:45 AM  10:00 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



10:00 AM  11:00 AM


Congruence modules and criteria for detecting free summands
Srikanth Iyengar (University of Utah)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Wiles, in his work on modularity lifting, discovered a numerical criterion for a map R>T of noetherian complete commutative local rings over a fixed discrete valuation ring O, and of relative dimension zero, to be an isomorphism of complete intersections. The criterion is in terms of the "congruence module" of T attached to an augmentation T>O and the cotangent module of the composite map R>O. Diamond generalized this result to a numerical criteria for a module over R to be free, again involving suitable congruence modules and cotangent modules. In my talk, I will present extensions of these results to higher relative dimension. These have applications in number theory, but I will focus mostly on the commutative algebra aspects. This is based on joint work with Khare, Manning, and Urban.
 Supplements



11:00 AM  11:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Nilpotent Frobenius actions and computing Frobenius closures of parameter ideal
Lance Edward Miller (University of Arkansas)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Computing Frobenius closures of ideals is a classically difficult problem. Starting in 2006 with work of KatzmanSharp the problem is made tractable for CohenMacaulay rings. Since roughly 2019, starting from work of Quy and Maddox, there has been an explosion of activity in computing Frobenius closure of parameter ideals. This has come by studying Fnilpotent singularities. In this talk, we will review the recent aspects of this research, applications of Fnilpotence to computing Frobenius test exponents, and the significant recent advances in Fnilpotent singularities that arose from this.
 Supplements



12:30 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Differential operators of low order for an isolated hypersurface singularity
Claudia Miller (Syracuse University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
We study the ring D differential operators of low order for an isolated singularity hypersurface ring of the form R=k[x,y,z]/(f) for a field k of characteristic zero and homogeneous polynomial of of degree at least 3 (that is, the cone of a smooth projective plane curve). For such hypersurfaces, it was long known that there are no negative degree operators by results of Vigué building on the work of Bernstein, Gel’fand, and Gel’fand, the proofs going via investigations of certain sheaf cohomologies, but the explicit operators were not known as far as we know (and further recent results by Mallory provide similar behavior in 4 variables). We find the differential operators of order up to 3 for such hypersurfaces in 3 variables by completely new, homological methods, as well as minimal resolutions of the Rmodule that they form.
This result highlights further the contrasting and yet interactive behaviors in the smooth and nonsmooth cases. In the smooth case, such rings of differential operators tend to have many operators of negative degree, yielding that certain important Dmodules are simple. In addition, when R is a polynomial ring, for example, many important, but infinitely generated, Rmodules such as local cohomology become of finite length as Dmodules, which has been an important tool for studying them.
This is joint work with Rachel Diethorn, Jack Jeffries, Nick Packauskas, Josh Pollitz, Hamid Rahmati, and Sophia Vassiliadou.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Regularity functions of graded ideals
Ngo Trung (Institute of Mathematics)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
This talk concerns the behavior of the functions reg I^{n1}/I^n, reg R/I^n, reg I^n, where I is a graded ideal in a standard graded algebra R. If I is generated in a single degree, we can describe these functions in a satisfactory manner. We also study the behavior of the function of the saturation degree of I^n.
 Supplements



04:30 PM  05:30 PM


Ask Mike your Macaulay2 questions
Michael Stillman (Cornell University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
Zoom Link
 Supplements




Apr 16, 2024
Tuesday

10:00 AM  11:00 AM


Orlov's Theorem for dgalgebras
Michael Brown (Auburn University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
A landmark theorem of Orlov relates the singularity category of a graded Gorenstein algebra to the derived category of the associated noncommutative projective scheme. I will discuss a generalization of this theorem to the setting of differential graded algebras. This is joint work with Prashanth Sridhar.
 Supplements



11:00 AM  11:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Fregularity and finite generation of monoid algebras determined by convex cones
Rankeya Datta (University of Missouri)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
Conjectures about the finite generation of various classes of rings in function fields have been instrumental in the development of algebraic geometry and commutative algebra. In this talk we will introduce one such conjecture in prime characteristic via a variant of Fregularity, which is a prime characteristic analog of KLT singularities from characteristic 0 birational geometry. We will mention the connection of this problem to other longstanding conjectures in the theory of Fsingularities and tight closure. Our main goal is to discuss evidence in favor of the conjecture by considering the class of monoid algebras determined by lattice points inside convex cones of finite dimensional vector spaces. The talk is based on joint work with Karl Schwede and Kevin Tucker.
 Supplements


12:30 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Determinants, Pfaffians, symmetric quivers, and symmetric varieties
Jenna Rajchgot (McMaster University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Type A quiver loci are a class of generalized determinantal varieties. Special cases include classical determinantal varieties and varieties of complexes. Since the 1980s, mathematicians have found connections between these quiver loci and Schubert varieties in type A flag varieties. These connections were used to gain insights into commutativealgebraic properties of type A quiver loci (e.g., singularities, Hilbert series).
In this talk, I will motivate and recall some of this story. I will then discuss the related setting of H. Derksen and J. Weyman's symmetric quivers and their representation varieties. Special cases include varieties defined by minors (or Pfaffians) of symmetric and skewsymmetric matrices. I will show how one can unify the study of commutativealgebraic properties of finite type symmetric quiver representation varieties with corresponding properties for Borel orbit closures in symmetric varieties G/K (G = general linear group, K = orthogonal or symplectic subgroup). Finally, I will provide some commutativealgebraic consequences.
This is joint work with Ryan Kinser and Martina Lanini.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea and Poster Session

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Fpure threshold as a measure of singularities
Ilya Smirnov (BCAM  Basque Center for Applied Mathematics)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Takagi and Watanabe introduced Fpure threshold as a “Frobenius analogue” of log canonical threshold from birational geometry, showed that Fpure threshold detects whether a local ring is singular, and explored its connections with other invariants. This local study of Fpure threshold, was subsequently further expanded by other authors. I will report on a joint work with Alessandro de Stefani and Luís NúñezBetancourt, which instead takes the theory in a new direction by studying the geometric properties of Fpure threshold as a singularity measure.
 Supplements



04:30 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Apr 17, 2024
Wednesday

10:00 AM  11:00 AM


Constructing resolutions using Koszul homomorphisms
Benjamin Briggs (University of Copenhagen; University of Utah)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
This is a talk about a relative version of Koszul duality, and how it can tell us about the asymptotic homological algebra of (seemingly) nonKoszul local rings. It’s all joint work with James Cameron, Janina Letz, and Josh Pollitz.
A homomorphism f of commutative local rings, say S to R, has a derived fibre F (a differential graded algebra over the residue field k of R) and we say that f is Koszul if F is formal and its homology H(F) (the Tor algebra of R and k over S) is a Koszul algebra in the classical sense. I’ll explain why this is a good definition with interesting consequences, and how it is satisfied by many many examples. I might also talk about how to understand these through Ainfinity structures.
The main application is the construction of explicit free resolutions over R in the presence of certain Koszul homomorphisms. This construction simultaneously generalises the resolutions of Priddy over a Koszul algebra, the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring
 Supplements



11:00 AM  11:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Extremal Surfaces in Positive Characteristic
Janet Page (North Dakota State University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
What is the most singular possible point on any variety in positive characteristic? In joint work with coauthors, we gave a precise answer to this question for hypersurfaces using a measure of singularity called the Fpure threshold, and we called these “most singular” hypersurfaces extremal hypersurfaces. These special hypersurfaces only occur in degrees $d=p^e+1$, where $p$ is the characteristic. In each degree where they occur, there is one particular extremal hypersurface (up to a change of variables) which stands out—it is the cone over a smooth projective hypersurface. In this talk, we’ll focus on this special hypersurface in the 4variable case—that is, we’ll focus on an extremal hypersurface which is the cone over a smooth projective surface, and we’ll discuss some of its surprising geometric properties. In particular, we highlight a conjecture which further justifies the name “extremal,” which would answer a longstanding classical question about smooth projective surfaces of any degree.
 Supplements




Apr 18, 2024
Thursday

10:00 AM  11:00 AM


Classification of resolving subcategories and its applications
Ryo Takahashi (Nagoya University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Classifying specific subcategories of a given category has been studied actively in many areas of mathematics including ring theory, representation theory, homotopy theory and algebraic geometry. In this talk, we consider classifying resolving subcategories over a commutative noetherian ring. We start with explaining a motivation by using a concrete examples of modules over a polynomial ring.
 Supplements



11:00 AM  11:05 AM


Group Photo

 Location
 SLMath: Front Courtyard
 Video


 Abstract
 
 Supplements



11:05 AM  11:30 AM


Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Closure operations and rational singularities
Rebecca R.G. (George Mason University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
This talk will focus on the ways closure operations can be used to detect rational singularities and its analogues. I will give characteristicfree properties a closure operation needs to have in order to generate something like Frational singularities (joint with Zhan Jiang). Then I will define a closure operation in equal characteristic 0 that can be used to detect singularities there without reduction to characteristic p (joint with Neil Epstein, Peter McDonald, and Karl Schwede).
 Supplements



12:30 PM  01:30 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:30 PM


Multigraded syzygies of toric embeddings
Christine Berkesch (University of Minnesota)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
A smooth normal toric variety X is determined by a multigraded polynomial ring S and a monomial ideal encoded by the fan of X. When a normal toric variety Y is embedded into X, recent results show that the multigraded free Sresolution of the ideal for Y has an abundance of rich combinatorial structure. We will explain the important geometric implications provided by this large new suite of explicit cellular resolutions over multigraded rings, which includes a generalization of Beilinson's resolution of the diagonal for projective space. This is ongoing joint work with Lauren Cranton Heller, Mahrud Sayrafi, Greg Smith, and Jay Yang.
 Supplements



02:30 PM  02:45 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




Apr 19, 2024
Friday

10:00 AM  11:30 AM


Poster Session and Morning Break

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:30 AM  12:30 PM


Density functions for some algebraic invariants
Vijaylaxmi Trivedi (Tata Institute of Fundamental Research)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
A density function for an algebraic invariant is a measurable function on R which measures the invariant on an Rscale, that is, a density function for a given algebraic invariant, say , is an integrable function f: R −→ R, which gives a new measure μf on R such that the integration R E f on a subset E ⊂ R is ‘the measure of the invariant on E’.
This function carries a lot more information related to the invariant without seeking extra data (than needed to study the invariant itself). The HK density function, which was introduced by the speaker to study HilbertKunz multiplicity, turned out to be a useful tool in answering some long standing open questions. In this talk we discuss the existence of density function for some well known and some elusive algebraic invariants and its applications. A part of the talk is based on some work with K.I. Watanabe, and some with S.Das and S.Roy.
 Supplements



12:30 PM  02:00 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Polarization and Gorenstein liaison
Patricia Klein (Texas A & M University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The major open question in the theory of Gorenstein liaison is whether or not every arithmetically CohenMacaulay subscheme of $\mathbb{P}^n$ can be Glinked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then indeed it can be after reembedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$. Motivated by this result, we consider techniques for constructing Glinks on a scheme from Glinks on a closelyrelated reduced scheme.
Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. After reviewing the key aspects of liaison and polarization, we will describe conditions that allow for the lifting of a basic double Glink induced from a vertex decomposition of the StanleyReisner complex of the polarization of an ideal $I$ to a basic double Glink of $I$ itself. We will introduce a notion of polarization of a Gr\"obner basis of an arbitrary homogeneous ideal and give a result for geometric vertex decomposition and elementary Gbiliaison that is analogous to our result on vertex decomposition and basic double Glinkage. Time permitting, we will give an application to vertex decompositions of StanleyReisner complexes of polarizations of stable CohenMacaulay monomial ideals and of artinian monomial ideals, extending a recent result of Fl{\o}ystad and Mafi.
The work described in this talk is joint with Sara Faridi, Jenna Rajchgot and Alexandra Seceleanu.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Commutative Algebra up to Symmetry
Uwe Nagel (University of Kentucky)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
In various contexts, one encounters sequences of ideals or modules over polynomial rings with a rich algebraic structure, indicated, for example, by invariance under the action of a symmetric group. It is natural to expect that the properties of related symmetric modules eventually become predictable even when the number of variables increases as one moves along the sequence. We discuss a framework for studying all but finitely many objects in such a sequence simultaneously. It has been used to establish asymptotic results for invariants of the objects such as the Hilbert function and multiplicity as well as graded Betti numbers and CastelnuovoMumford regularity. However, the boundaries for these stabilization results are not clear, and there are many open questions.
 Supplements



