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The local cohomology sheaf of a smooth complex variety along a closed subvariety comes endowed with a Hodge filtration, via Saito's theory of mixed Hodge modules. I will discuss some properties of this filtration, based on joint work with Mihnea Popa.

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What is the most singular possible singularity? What can we say about its geometric and algebraic properties? This seemingly naive question has a sensible answer in characteristic p. The ""F-pure threshold,"" which is an analog of the log canonical threshold, can be used to ""measure"" how bad a singularity is. The F-pure threshold is a numerical invariant of a point on (say) a hypersurface---a positive rational number that is 1 at any smooth point (or more generally, any F-pure point) but less than one in general, with ""more singular"" points having smaller F-pure thresholds. We explain a recently proved lower bound on the F-pure threshold in terms of the multiplicity of the singularity. We also show that there is a nice class of hypersurfaces--which we call ""Extremal hypersurfaces""---for which this bound is achieved. These have very nice (extreme!) geometric properties. For example, the affine cone over a non Frobenius split cubic surface of characteristic two is one example of an ""extremal singularity"". Geometrically, these are the only cubic surfaces with the property that *every* triple of coplanar lines on the surface meets in a single point (rather than a ""triangle"" as expected)--a very extreme property indeed.

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Let G be a linearly reductive group over a field K, with a linear action on a polynomial ring over K. Then the invariant ring is a pure subring of the polynomial ring; many key properties of classical invariant rings including finite generation and the Cohen-Macaulay property, as in the Hochster-Roberts theorem, follow from purity. Now let A denote either a field, or the ring of integers, or a ring of p-adic integers. When is a given finitely generated A-algebra a pure subring of a polynomial ring over A? We will discuss how this can be addressed via D-modules, Group Actions, and Frobenius! The recent Computing on Singularities is joint work with Jack Jeffries.

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We introduce differential primary decompositions for ideals in a commutative ring. Ideal membership is characterized by differential conditions. The minimal number of conditions needed is the arithmetic multiplicity. Minimal differential primary decompositions are unique up to change of bases. Our results generalize the construction of Noetherian operators for primary ideals in the analytic theory of Ehrenpreis- Palamodov, and they offer a concise method for representing affine schemes. The case of modules is also addressed. This is joint work with Bernd Sturmfels.

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As the constructible counterpart of the Fourier-Mukai transformation, the non-abelian Mellin transformation of a constructible complex can be considered as taking the hyper-cohomology of the complex twisted by all possible local systems simultaneously. We will explain a t-exactness result about the non-abelian Mellin transformation, generalizing a theorem of Gabber-Loeser on affine torus. We will also discuss some local vanishing properties of the Sabbah's specialization functor, which is a key step in the proof of the t-exactness result.

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We prove the Bernstein inequality and develop the theory of holonomic D-modules for rings of invariants of finite groups in characteristic zero, and for strongly F-regular finitely generated graded algebras with FFRT in prime characteristic. In each of these cases, the ring itself, its localizations, and its local cohomology modules are holonomic. We also show that holonomic D-modules, in this context, have finite length. This is based on joint work with Josep Àlvarez Montaner, Daniel J. Hernández, Luis Núñez-Betancourt, Pedro Teixeira, and Emily E. Witt.

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Suppose that some microphones are placed on a drone inside a room with planar walls/floors/ceilings. A loudspeaker emits a sound impulse and the microphones receive several delayed responses corresponding to the sound bouncing back from each planar surface. These are the first-order echoes. In this talks, we will discuss the problem of reconstructing the shape of the room from these echoes, an unlabeled distance geometry problem. The time delay for each echo determines the distance from the microphone to a mirror image of the source reflected across a wall. Since we do not know which echo corresponds to which wall, the distances are unlabeled. The problem is to figure out under which circumstances, and how, one can find out the correct distance-wall assignments and reconstruct the wall positions. This is joint work with Gregor Kemper.

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Using V-filtrations, Kashiwara and Malgrange constructed Riemann-Hilbert correspondence for nearby and vanishing cycles along a single holomorphic functions. Sabbah then constructed multi V-filtrations along a finite set of holomorphic functions and thus obtained the multivariate Bernstein-Sato polynomials. However, Sabbah's method also indicates that the method of Kashiwara and Malgrange can not be generalized to the multivariate case in general. In this talk, first I will explain the construction of Bernstein-Sato ideals and Alexander complexes by using Mellin transformations. Then, I will focus on the construction of Riemann-Hilbert correspondence for Alexander complexes (the multivariate generalization of nearby cycles) by using Bernstein-Sato ideals and relative holonomic D-modules.

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Most of the theory of D-modules has been developed only in characteristic zero. This includes holonomic modules. Some candidates for holonomic modules in characteristic p>0 have been proposed using definitions specific to characteristic p>0. The first characteristic-free definition of holonomicity was given in 2010 by the speaker, but only for modules over polynomial rings. In the talk I am going to describe an extension of this definition to arbitrary non-singular varieties. This is joint work with Wenliang Zhang.

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A number of the innovations used in studying singularities in commutative algebra have come from the study of tight closure and its test ideal in rings of equal characteristic. In replicating these results in rings of mixed characteristic, it has been useful to find closure operations that share key properties with tight closure. By studying the shared structure of common closure operations in commutative algebra, we show that many tight closure properties, in particular the structure of the test ideal, hold for a much larger set of closure operations, including (big Cohen-Macaulay) module closures and mixed characteristic closures. In this talk, I will describe the structures that these closure operations have in common and share some of the results on test ideals that have come out of this theory. Parts of this research are joint with subsets of Neil Epstein, Janet Vassilev, Felipe Pérez, and Zhan Jiang.

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A noncommutative analog of the Peskine-Szpiro Acyclicity Lemma.
Daniel Bath
Given a complex of finite R-modules (R commutative, Noetherian, local) satisfying a sliding projective dimension condition, the Peskine-Szpiro Acyclicity Lemma provides a criterion for verifying this complex is exact. We present a noncommutative analog for R an Auslander regular ring. The lemma is especially useful for D-modules and D[s]-modules as the requisite hypotheses are more easily verifiable. Potential use cases will be discussed, time permitting.

Minimal faithful permutation representations of finite groups
Neelima Borade
In this short talk we will introduce the notion of a minimal faithful permutation representation of a finite group and Cayley’s constant p(G) that measures its size. We’ll outline the history of the computation of p(G) for finite groups G and focus primarily on the case when G is a linear group. We'll end with some results for p(G) of a general linear group G based on my undergraduate paper with Dr. Takloo-Bighash.

Diamonds in Langlands Local Functoriality
Shanna Dobson
We use Scholze's "diamond" construct and Scholze and Fargue's geometrization of the local Langlands correspondence on the Fargues-Fontaine Curve to investigate Local Functoriality for p-adic groups and a concomitant diamond universal construction.

Tate's thesis and 3d mirror symmetry
Justin Hilburn
Geometric local Langlands for the multiplicative group G_m, as proved by Beilinson and Drinfeld, is the identification of the monoidal category of D-modules on the loop group G_m((t)) with with the category of coherent sheaves on the moduli space of G_m local systems on the punctured formal disc. This can be thought of as an infinite dimensional version of the Mellin transform. In joint work with Sam Raskin, I gave a coherent description of the D-mod(G_m((t)))-module category of D-modules on the loop space A*1((t)). This can be thought of as a geometric version of Tate's thesis. If time permits I will explain the statement of this result and how it is related to the physics of 3d mirror symmetry.

Singularities of a modification of the Grothendieck-Springer resolution
Mee Seong Im
I will introduce the extended Borel moment map, which is related to the Springer resolution and the Grothendieck-Springer resolution. The affine quotient of this map has interesting singular locus, which is intimately related to the Hilbert-Chow morphism. I will describe all the important objects mentioned in this abstract, and provide reasons why it is important to study the singular locus of this map.

Differential operators on singular rings
Devlin Mallory
At least since the work of Levasseur and Stafford in the 1980s, the question had been asked of whether one can characterize singularities of rings via certain properties of their rings of differential operators. In particular, one question is whether a ring with mild singularities is a simple module under the action of its ring of differential operators. While an answer in characteristic p had been provided by Karen Smith, no answer had been forthcoming in characteristic 0. We provide a counterexample showing that the expected connection does not exist, through the study of the global geometry of Fano varieties. More specifically, we show that certain del Pezzo surfaces do not have big tangent bundles, and thus their homogeneous coordinate rings are not simple under the action of their rings of differential operators, despite having “mild” singularities.

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In this talk, I will discuss some results and tools concerning equivariant D-modules, with a focus on representations of reductive groups having finitely many orbits. In particular, I provide explicit descriptions of: categories of equivariant D-modules as quivers, D-module structures of local cohomology modules supported in orbit closures, Lyubeznik numbers, Bernstein-Sato polynomials of holonomic functions, character formulas.

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In a polynomial ring over a perfect field, the symbolic powers of a radical ideal consist of the polynomials that vanish to order n on the corresponding variety, and can be described via differential operators. If we replace the field with a DVR, we need both differential operators and Joyal and Buium's notion of a p-derivation to give an analogous result. As an application, we will discuss an explicit Chevalley lemma for the symbolic powers of prime ideals in direct summands of polynomial rings. This is joint work with Alessandro De Stefani and Jack Jeffries.

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In this talk, I will survey some recent results on asymptotic vanishing of cohomology of lci varieties and explain an approach to extending these results to graded rings over a field. If time permits, I will explain an application of our approach to the study of rings of prime characteristic.

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Multiplicities of jumping numbers
Swaraj Pande
Multiplier ideals are important invariants of singularities of algebraic varieties. Jumping numbers and their multiplicities are numerical invariants arising from multiplier ideals, and are connected to other singularity invariants. In this talk, we'll present some new finiteness results for multiplier ideals of a point scheme. Namely, that the multiplicities of jumping numbers fit into a quasi-polynomial. Further, we'll see how the rate of growth of this quasi-polynomial is closely related to the valuations that compute jumping numbers.

On derived functors of graded local cohomology modules - II
Sudeshna Roy
This is a joint work with Puthenpurakal and Singh.

On Derived Functors of Graded Local Cohomology Modules
Jyoti Singh
This is the joint work with Prof. Tony J. Puthenpurakal, IIT Bombay.

Primary Decomposition of Binomial Modules
Afshan Sadiq
Let K be a field and R be a polynomial ring in n variables. A binomial module is a submodule of R^m generated by binomials. The aim of this paper is to prove that a binomial module has a primary decomposition into binomial primary modules and the associated primes are binomial ideals. The idea is to generalize the paper of Eisenbud and Strumfels. Eisenbud, D.; Sturmfels, B.: Binomial ideals. Duke Mathematical Journal 84 (No. 1), 1--45 (1996).

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In joint work with Kustin and Ulrich we compute the Kaehler different and the Dedekind different for several classes of ideals. Our techniques include residual intersections and linkage theory. In particular we obtain interesting formulas for determinantal ideals of generic matrices and perfect Gorenstein ideals of height three.

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GKZ hypergeometric systems were introduced by Gelfand, Kapranov and Zelevinsky as a generalization of Gauss hypergeometric differential equation. It can be shown that for certain parameters the GKZ-systems carry the structure of an irregular mixed Hodge module, a category recently defined by Claude Sabbah. We will discuss the Hodge and weight filtration of these D-modules.

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We calculate the Bernstein-Sato polynomial of the irreducible power series in two variables that have the simplest topology, via reduction to characteristic p. To do so, we determine uniform formulas for certain numerical invariants in prime characteristic by constructing, and solving, integer programs.
This is joint work with Daniel Hernández.

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