Note: seminars at CUNY are now all virtual

**The seminar will meet via ZOOM SESSIONS on selected Fridays
4:00-5:00 PM**

**To obtain the zoom link for our seminar for the Spring ****2021****
Semester (do not use the one from last semester!), please register
here and you will be provided with a personal link and password
to join the seminar on all the scheduled dates
**

Organizers:

Laura Ghezzi, New York City College of Technology (CUNY), lghezzi@citytech.cuny.edu

Hans Schoutens, New York City College of Technology and the Graduate Center (CUNY), hschoutens@citytech.cuny.edu

Fei, Ye, Queensborough Community College (CUNY), Feye@qcc.cuny.edu

**Abstract**: Lifting property of
modules is a fundamental question in commutative ring theory, which is
tightly connected to deformation theory of modules and has important
applications in the theory of maximal Cohen-Macaulay approximations. This
notion was studied by Auslander, Ding, and Solberg for modules and by
Yoshino for complexes. In this talk, I will survey recent developments on
the lifting theory for differential graded (DG) modules over strongly
commutative and non-negatively graded DG algebras and describe the
relation between this notion and the long-standing Auslander-Reiten
conjecture. This investigation requires some new ideas using homotopy
limits and diagonal ideals.

Abstract: Let $R$ be a local ring of positive characteristic and $X$ a complex with nonzero finitely generated homology and finite injective

dimension. We prove that if derived base change of $X$ via the Frobenius (or more generally, via a contracting) endomorphism has finite injective

dimension then $R$ is Gorenstein.

**Abstract**: Let S be a regular ring. By
work of Hochster-Roberts, Boutot, Smith, Hochster-Huneke, Schoutens, and
Heitmann-Ma, we know that every pure subring of S is pseudo-rational,
hence Cohen-Macaulay. This applies in particular to rings of invariants of
linearly reductive groups. For pure maps R -> S of rings essentially of
finite type over fields of characteristic zero, Boutot's result is even
stronger: if S is pseudo-rational, then R is pseudo-rational. We show that
Boutot's theorem holds more generally if R and S are quasi-excellent
Q-algebras, which we use to give new proofs of the theorems of
Hochster-Huneke and Schoutens. This solves a conjecture of Boutot and
answers a question of Schoutens in the quasi-excellent case. To prove
these results, we show that the Grauert-Riemenschneider vanishing theorem
holds for Q-schemes, solving a conjecture of Boutot.

**Abstract**: We generalize the theory of globally F-regular
pairs to mixed characteristic, which we call +-regularity, and introduce
certain stable sections of adjoint line bundles. This is inspired by
recent work of Bhatt on the Cohen-Macaulayness of the absolute integral
closure. We will discuss some applications of these results to birational
geometry in mixed characteristic. Joint work with Bhargav Bhatt, Zsolt
Patakfalvi, Karl Schwede, Kevin Tucker, Joe Waldron, and Jakub Witaszek.

**Abstract**: As weve seen in two previous talks, the construction
given in Smoothness, Semistability, and Alterations by Aise Johan de Jong
(1996) of an alteration of any variety by a regular variety begins by
reducing immediately to the case of a family of curves whose generic fiber
is smooth of genus g with at least 3 marked points on each irreducible
component of each fiber. Consequently, there is a morphism from an open
subset of the base of the family to the moduli stack of stable curves of
genus g with n marked points. Morally, projection onto the domain factor
from the closure of the graph of this morphism yields an alteration of the
base over which our old family extends to a family of stable curves. Now,
the old family whose fibers could have terrible singularities has been
improved to a new family whose fibers have at-worst nodal singularities.
There is a little issue, however: there is not necessarily a morphism from
the new family onto the old family, only a not-everywhere-defined
birational map!

To show that this birational map from the new to the old family is indeed
a morphism, we study the projection onto the domain factor from the
closure of the graph of the map. We may arrange that the new family is
normal, and so the projection is a birational morphism onto a normal
variety. If we can show that this morphism is finite, i.e. has finite
fibers, Zariskis Main Theorem will guarantee that the morphism is an
isomorphism, and so we will have a morphism from the new family to the old
family after all. While trying to show that the fibers of this morphism
are finite essentially, that there are no contractions the 3 points
weve marked on each irreducible component of each fiber will come to the
rescue!

**Abstract**: Heisuke Hironaka solved the problem of resolution of
singularities for a variety over a field of characteristic zero in 1964,
although the story goes that his proof was so complicated as to stump
Alexander Grothendieck! Efforts have been made since then to simplify the
proof and see if it can be made to work for positive characteristic, but
to no avail.

Regardless of characteristic:

The singularities of a curve are resolved in one fell swoop by the
normalization, a powerful tool hailing from algebra made to work for us in
geometry, by Jean-Pierre Serres Criterion for Normality, and we owe much
to Oscar Zariski for teaching us about the normalization. Joseph
Lipman dealt with surfaces in as much generality as one could hope
for. Threefolds are rumored to have been handled by Vincent Cossart
and Olivier Piltant, although their proof is quite long. Fourfolds and up
are uncharted territory for the most part.

In his 1996 paper Smoothness, Semi-Stability, and Alterations, Johan de
Jong introduces a paradigm shift for solving the problem of resolution of
singularities by relaxing a resolution of singularities to what he calls
an alteration. The distinction is that a resolution is a birational
morphism (generically one-to-one), whereas an alteration is a generically
finite morphism (generically finite-to-one). Every variety can be altered
to a nonsingular variety regardless of the characteristic. In fact, de
Jongs technique paired up with Dan Abramovichs geometric insight yields
a proof that every variety over a field of characteristic zero admits a
resolution of its singularities in a paper consisting of only twelve
pages, see Smoothness, Semi-Stability, and Toroidal Geometry. de Jongs
technique relies on (at least) two ingenious ideas:

The first is the statement that there is a simple blowup of any variety
that admits a morphism to a projective space of one less dimension whose
fibers are curves ~ intuitively, any variety can be modified to be a
family of curves. Taking the intermediate variety in what is known as the
Stein factorization of the morphism instead of the target projective space
ensures the curves in the family are connected.

The second is that the curves in the family of curves that results can be
marked so as to become stable (in the sense of Pierre Deligne and David
Mumford) so that it gives rise to morphism into the moduli space of stable
curves, where we may take advantage of established facts about the moduli
space of stable curves, including the fact that its compactification has
curves with at worst nodal singularities.

My talk has the goal of introducing Johan de Jongs paradigm shift, and
indicating, to the extent that I can, the idea of the proof that every
variety may be altered to a nonsingular variety. An emphasis will be
placed on the picture that demonstrates that there is a simple blowup of
any variety that admits a morphism to a projective space of one less
dimension whose fibers are curves.

**Friday, Oct 16**

**Abstract**:See above.

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**Friday, Oct 30**

**Speaker**: **Miodrag Iovanov (Univ. of Iowa)**

**Title**: Diamond Rings - a class of rings close to Noetherian (click
here for a recording
of the talk)

Abstract. A ring is said to have property D(=diamond) if the injective
hull E of any simple module is locally Artinian, that is, a finitely
generated submodule of such an E is Artinian. Commutative Noetherian rings
are known to have property D. This property is of interest in
noncommutative rings, as Noetherian rings don't always have it, and it is
closely related to outstanding questions in algebra such as Jacobson's
conjecture (which states that the intersections of the powers of the
Jacobson radical of a noncommutative Noetherian ring is zero).

Perhaps somewhat surprisingly, this has not been studied much for the
commutative case. We present several motivating examples and questions
from the non-commutative case, and then describe such recent results for
the commutative case. It turns out rings with this property D are exactly
rings which have "enough" modules of finite length. This simple
observation leads to generalizations of several classical results of
commutative Noetherian algebra - such that the Krull intersection theorem
- to the generality of rings with property D, and which recover also
results of Jans, Matlis, Vamos. On the other hand, we show that for
complete rings (with a suitable natural topology consisting of ideals of
co-finite length), property D is equivalent to the ring being Noetherian,
which further points to property D being quite "close" to Noetherian.

**Friday, Nov 13**

**Speaker**: Xiaolei Zhao (University of California, Santa Barbara)

**Title**: Title: Elliptic quintics on cubic fourfolds, O'Grady 10, and
Lagrangian fiberations (click here for a recording
of the talk).

Abstract: We will show how to construct hyperkδhler manifolds of O'Grady
10 type out of smooth cubic fourfolds. Applications to classical algebraic
geometry questions on intermediate jacobian fibrations of families of
cubic threefolds, and quintic elliptic curves on cubic fourfolds will be
explained. This is based on joint work with Chunyi Li and Laura Pertusi.

**Friday, Dec 4**

**Speaker**: Federico Caucci (University of Florence)

**Title**: The basepoint-freeness threshold of a polarized abelian
variety (click here for a recording
of the talk).

Abstract: We will introduce the basepoint-freeness threshold of a
polarized abelian variety, as defined by Jiang and Pareschi, and explain
its relation with the syzygies of the section algebra of the polarization.
After an overview on equations defining abelian varieties and higher
syzygies, we will mainly focus on a conjectural upper bound for this
threshold which has been very recently studied by Ito and Jiang. If true,
it would have a direct application to syzygies of (primitive) polarized
abelian varieties, that can be seen as a revised and extended version of
the generalized Fujita's freeness conjecture, in the case of abelian
varieties.

**Friday, Feb 14**

**Speaker**: Daniele Bartoli (Universitΰ Degli Studi di Perugia)

**Title**: Applications of curves over finite fields to polynomial
problems

**Abstract**: Algebraic curves over finite fields are not only
interesting objects from a theoretical point of view, but they also have
deep connections with different areas of mathematics and combinatorics. In
fact, they are important tools when dealing with, for instance,
permutation polynomials, APN functions, planar functions, exceptional
polynomials, scattered polynomials, Moore-like matrices. In this talk I
will present some applications of algebraic curves to the above mentioned
objects.

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**Friday, Feb 28**

**Speaker**: Jai Laxmi (University of Connecticut)

**Title**: Spinor structures on free resolutions of codimension four
Gorenstein ideals

**Abstract**: The problem of classifying Gorenstein ideals
of codimension four was around ever since Buchsbaum-Eisenbud classified
Gorenstein ideals of codimension three. The first result related to the
structure of codimension four case were obtained by Kustin and Miller. We
analyze the structure of spinor coordinates on resolutions of Gorenstein
ideals of codimension four. As an application we produce a family of such
ideals with seven generators which are not specializations of
Kustin-Miller model.

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**Friday, Mar 13**(CANCELED)

**Speaker: **Malgorzata Marciniak (Laguardia)

**Title: **Analyzing the Game of Cycles

**Abstract: **Playing the game of cycles (as introduced in Su's 2020
book Mathematics for Human Flourishing) can be interpreted as converting a
planar non-directed graph into a planar directed graph. Thus, every stage
of the game can be represented as a graph of mixed type with a suitable
adjacency matrix. The rules of the game and the winning moves have
interpretations in terms of that matrix, which suggests another way to
analyze the game and its strategies.

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**Friday, Mar 27**(CANCELED)

**Speaker**: Wim Veys (University of Leuven)

**Title**: Zeta functions and the monodromy conjecture

**Abstract**: The monodromy conjecture is a mysterious open problem in
singularity theory. It relates arithmetic and topological/geometric
properties of a multivariate polynomial over the integers. The case of
interest is when the zero set of the polynomial has singular points. We
will present some history, motivation, and related results.

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**Friday, Apr 24**

**Speaker**: James Meyer (GC)

**Title**: TBA

**Abstract**:

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**Friday, Sep 6**

**Speaker**: Bart Van Steirteghem (FAU Erlangen-Nόrnberg and CUNY)

**Title**: Momentum polytopes of projective spherical varieties

**Abstract**: Spherical varieties are a natural generalization of
toric varieties, where the acting torus is replaced by a general connected
reductive group. They also include flag varieties and symmetric varieties.
In this talk, I will explain how one can use their rich combinatorics to
characterize the momentum polytopes of projective spherical varieties. I
will also discuss an application to the Kδhlerizability problem for
compact multiplicity free Hamiltonian manifolds. This is joint work with
S. Cupit-Foutou and G. Pezzini.

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**Friday, Sep 20****, 4PM**

**Speaker**: Hans Schoutens (CityTech and the CUNY Graudate Center)

**Titl**e: Generalized Hasse-Schmidt derivations

**Abstract**: Hasse-Schmidt derivations are certain types of higher
order differential operators (introduced by Hasse and Schmidt and studied,
among others, by Matsumura). In characteristic zero, they are ubiquitous,
but their presence in positive characteristic is more subtle: they are no
longer classical differential operators. Vojta has used them to construct
arc schemes. In this talk, I will discuss a generalization of
Hasse-Schmidt derivations and show the corresponding connection with
generalized arc schemes (which I introduced a decade ago, and are used,
among others, by Mustata and Stout). The recipe is quite simple: replace
in the definitions $k[t]/t^nk[t]$ by an arbitrary local Artinian
k-algebra.

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**Friday, Oct 4****, 4PM**

**Speaker**: Ray Hoobler (CUNY)

**Title: **Bases of Azumaya algebras

**Abstract**: It is well known that any central separable algebra over
a field is a matrix ring over a division algebra. If X is a regular scheme
I will show that there is an essentially unique reflexive (semi-stable if
X is projective), maximal order in the division algebra component of the
generic point of any Azumaya algebra on X. As an application I shall show
that for a projective smooth scheme X over a field of characteristic 0, an
Azumaya algebra admits a flat connection if and only if its second Chern
class vanishes.

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**Friday, Oct 18****, 4:15PM**

**Speaker**: Hans Schoutens (CityTech and the CUNY Graudate Center)

**Title**: Using Hasse-Schmidt derivations to construct small MCMs.

**Abstract**: This talk complemens my talk from last month, where I
introduced (general) Hasse-Schmidt derivations. Today I will sketch the
argument how the existence of some `nice' Hasse-Schmidt derivation over a
complete three-dimensional local ring of positive characteristic yields
the existence of a small MCM (i.e., a finitely generated module of depth
three), resolving Hochster's conjecture for this special case.

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**Friday, Nov 1****, 4:15PM**

**Speaker**: Ben Blum-Smith (Fordham and New School)

**Title**: Square-free Grφbner degenerations and cohomologically full
rings

**Abstract**: Let I be a homogeneous ideal in a polynomial
ring S := K[x_1,
,x_n]. A choice of monomial order on S specifies an
initial ideal in(I) for I. The ring S/in(I) is the special fiber in a flat
family whose generic fiber is S/I, and it has long been understood that
S/I is at least as good" as S/in(I) according to various notions of good
behavior" for rings. For example, the depth of S/I is at least the depth
of S/in(I); thus S/I is Cohen-Macaulay if S/in(I) is Cohen-Macaulay.
Because of this principle, S/in(I) is called a Grφbner degeneration of
S/I. In full generality, the converse of this principle is false, as
implied by the term degeneration good behavior of S/I need not pass to
S/in(I). However, in 1987, Bayer and Stillman showed that in an important
special situation, namely when I is in generic coordinates, the
relationship between S/I and S/in(I) is quite tight. In 2018, Aldo Conca
and Matteo Varbaro announced a theorem that if the monomial ideal in(I)
happens to be generated by squarefree monomials, then the relationship is
tighter still. The result established and sharpened a conjecture of
Herzog, and allowed the authors, as a corollary, to resolve a
long-standing mystery in the theory of algebras with straightening law
(ASLs). The proof is a first major application of the notion of
cohomologically full rings, introduced very recently by Hailong Dao,
Alessandro De Stefani, and Linquan Ma. In this talk, we will discuss the
results of Conca and Varbaro, the proof methods, and speculate about
possible applications.

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**Friday, Nov 8****, 4:15PM**

**Speaker**: Roger Wiegand (University of Nebraska)

**Titl**e: Vanishing of Tor over fiber products

**Abstract**: Let (S,m,k) and (T,n,k) be local
rings, and let R denote their fiber product over
their common residue field k. (Thus R = {(s,t) | p(s) = q(t),
where p:S >k and q:T>k are the reductions modulo the maximal
ideals.) Inspired by work of Nasseh and Sather-Wagstaff, we explore
consequences of vanishing of Tor^R_m(M,N) for various values
of m, where M and N are finitely generated
R-modules. For instance, assume that neither S nor
T is a discrete valuation domain and that Tor^R_m(M,N) = 0 for
some m > 5. Then at least one of M, N has
projective dimension at most one. This is joint work with
Thiago Freitas, Victor Hugo Jorge Pιrez, and Sylvia Wiegand.

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**Friday, Nov 15****, 4:15PM**

**Speaker**: Ben Blum-Smith (Fordham and New School)

**Title**: Square-free Grφbner degenerations and cohomologically full
rings (Part II)

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**Friday, Feb 15**

**Speaker**: Giovan Battista Pignatti (GC)

**Title**: Reinterpreting birational geometry through the theory of
valuation rings.

**Abstract:** The Zariski-Riemann space ZR(K|k) of a field extension
K|k is the set of all valuation rings between the two fields. It is
naturally endowed with a topology. To every collection of valuation rings
it is possible to associate a normal k-variety. Using a special class of
open coverings of ZR(K|k), we will build a category which turns out to be
equivalent to the category of all proper, normal k-varieties with
birational morphisms. This allows to revisit many geometric notions only
using the purely algebraic language of valuation rings. (The talk is based
on a set of notes by Hans Schoutens which is still work in progress.
Participation of the audience is welcomed and encouraged).

**Friday, Mar 1**

**Speaker**: Ben Blum-Smith (Eugene Lang College / The New School)

**Title**: Bounding the degree of generation of invariant fields

**Abstract**: It has long been of concern to invariant theorists to
know something about the degrees in which invariant rings are generated.
For example, a classical result of E. Noether states that if G is a finite
group acting linearly on a polynomial ring over a field k, then the
invariant ring is generated by polynomials of degree at most the order of
G (subject to a restriction on the characteristic of k). In this talk, we
consider the analogous question for invariant fields. This question is
relevant to applications, for example in biological imaging. We show that
various results from linear algebra on tensor decomposition have
interpretations in terms of the degree of generation of invariant fields,
and provide some notes toward a general theory in the case that G is
abelian and k has characteristic 0.

**Friday, Mar 15**

**Speaker**: Fei Ye, QCC (CUNY)

**Title**: Introduction to D-modules

**Friday, Mar 29, 4:00-5:00PM**

**Speaker**: Mingyi Zhang (Northwestern University)

**Title**: Hodge ideals and Hodge filtration of a left D-module
associated to Q-divisor with certain isolated singularities

**Abstract**: Given a singular subvariety Z on a smooth
complex variety X, Saito studied a special D-module O_X(*Z) which
underlines a mixed Hodge module. It carries a canonical Hodge filtration.
In general, it is hard to describe this filtration. However, when Z has an
isolated singularity given by a weighted homogeneous polynomial, Saito
gave an explicit formula of the Hodge filtration. On the other hand, this
D-module corresponds to a series of ideal sheafs, called Hodge ideals,
associated wit the integral divisor Z. I will give a crash introduction of
Hodge ideals and Hodge filtration of a D-module associated with Q-divisor,
and how to generalize Saito's formula to the Q-divisor case.

**Friday, April 12, 4:00-5:00PM**

**Speaker**: Uma Iyer (BMCC and the Graduate Center)

**Title**: A question on path algebras

**Abstract**: Lunts and Rosenberg defined the algebra of
differential operators on noncommutative algebras. One major source of
noncommutative algebras are the path algebras of quivers. We introduce a
study of the algebra of differential operators on the path algebra of an
arbitrary finite quiver.

**Tuesday, April 30, 5:00-6:00PM in room 5212 (note
special time, place, and day)**

**Speaker**: Nero Budur (University of Leuven)

**Title**: Jets, quivers, and SL_n(Z)

**Abstract**: Using jet schemes and quivers, we show that
the space of representations of the fundamental group of a compact Riemann
surface of genus at least two has rational singularities. We apply this to
show that the number of irreducible complex representations of SL_n(Z) of
dimension at most m grows at most as the square of m, for a fixed n>2.

**Friday, Sep 28** (oral defense)

**Speaker**: Giovan-Battista Pignatti (GC)

**Title**: Infinite directed unions of local quadratic transforms (abstract).

**Friday, October 5**

**Speaker**: Jai Laxmi (Indian Institute of Technology, Bombay)

**Title**: Tate resolutions and deviations of graded algebras (abstract).

**Friday, October 19**

**Speaker**: Ethan Cotterill (Boston College)

**Title**: Real inflection points of real linear series on real
hyperelliptic curves (joint with I. Biswas and C. Garay Lσpez)

Abstract: According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.

**Friday, November 2**

**Speaker**: Ben Blum-Smith (Eugene Lang College / The New School)

**Title**: Cohen-Macaulayness of invariant rings is determined by
inertia groups

Abstract: If a finite group G acts on a Cohen-Macaulay ring A, and the order of G is a unit in A, then the invariant ring A^G is Cohen-Macaulay as well, by the Hochster-Eagon theorem. On the other hand, if the order of G is not a unit in A, then the Cohen-Macaulayness of A^G is a delicate question that has attracted research attention over the last several decades, with answers in several special cases but little general theory. In this talk we show that the statement that A^G is Cohen-Macaulay is equivalent to a statement quantified over the inertia groups for the action of G on A, acting on strict henselizations of appropriate localizations of A. In a case of long-standing interest a permutation group acting on a polynomial ring we show how this can be applied to find an obstruction to Cohen-Macaulayness that allows us to completely characterize the permutation groups whose invariant ring is Cohen-Macaulay regardless of the ground field. This is joint work with Sophie Marques.

**Friday, November 16**

**Speaker**: Hans Schoutens (CUNY)

**Title**: Total blow-up ring along a valuation.

Abstract: Given a smooth variety $X$, any valuation on its function field has a unique center $x\in X$. This then determines a unique local blowing up at $x$ along the valuation. Abhyankar showed that when $X$ has dimension two, iterating this construction recovers the valuation ring; later, his student Shannon gave counterexamples in dimension three. So, two natural questions emerge: when do we get the valuation ring, and if we don't, what do we get? I will concentrate on the latter question, continuing the discussion of Giovan-Battista from last month. (This is joint work with Loper, Heinzer, Olberding and Toeniskotter).

**Friday, November 30**

**Speaker**: Chuanhao Wei (Stonybrook University)

**Title**: Zeros of log-one-forms and families of log-varieties

Abstract: I will introduce the result about the relation between the zeros of holomorphic log-one-forms and the log-Kodaira dimension, which is a natural generalization of Popa and Schnell's result on zeros of one-forms. Some geometric corollaries will be stated, e.g. algebraic Brody hyperbolicity of log-smooth family of log-pairs of log-general type, which also serves as a motivation to the main theorem.

**Friday, February 2**

**Speaker**: Jai Laxmi (Indian Institute of Technology, Bombay)

**Title**: Decomposing Gorenstein Rings as Connected Sums

Abstract: In 2012, Ananthnarayan, Avramov and Moore gave a new
construction of Goren-

stein rings from two Gorenstein local rings, called their connected sum.
Given a Gorenstein

Artin ring, we investigate conditions on the ring which force it to be
indecomposable as

a connected sum. We will see characterizations for Gorenstein Artin local
rings to be de-

composable. Finally, we will show that the indecomposable components
appearing in the

connected sum decomposition are unique up to isomorphism.

This is joint work with H. Ananthnarayan, E. Celikbas and Z. Yang.

**Friday, February 16**

**Speaker**: Andrew Stout (BMCC)

**Title**: An Introduction to Algebraic Spaces

**Abstract**: Grothendieck's celebrated results on the
existence of Hilbert spaces as schemes required projectivity assumptions
-- a fact that one can detect proved frustrating as one reads through the
details in FGA. Inspired by ideas from deformation theory, M. Artin found
that we may remove projectivity if we expand the category of schemes to
the new category of algebraic spaces. Roughly speaking, a quasi-separted
algebraic space is a sheaf in the etale topology which contains a dense
open scheme. The density assumption and the fact that the etale topology
is invariant under infinitesimal thickenings makes it possible to extend
geometric properties of schemes to these functors. Although these functors
are not geometric spaces, they behave well enough to allow for an
extension of most major results from scheme theory to algebra spaces
(e.g., Stein factorization, Chow's lemma, Finiteness of Cohomology, GAGA).
Furthermore, every algebraic space may also be viewed as a quotient of a
scheme modulo an etale equivalence relation, which immediately leads to
numerous examples. This material also serves as motivation to the larger
theory of algebraic stacks. In this talk, I will offer an accesible
overview of the material above as time permits.

**Friday, March 2
Speaker**: Laura Ghezzi (CityTech)

**Abstract**: This talk is a sequel to a previous one on
Invariants of Cohen Macaulay rings associated to their canonical ideals.
The big picture is to refine our understanding of Cohen-Macaulay rings
(which are not Gorenstein). Let R be a Cohen-Macaulay local ring of
dimension one with a canonical ideal. First we review the definition of
canonical degree and the necessary background. We then show that almost
Gorenstein rings are exactly those rings that have minimal canonical
degree. We also discuss generalizations of this result and open questions.

**Friday, April 13
Speaker**: Shizhuo Zhang (Indiana University)

**Friday, April 27
Speaker**:

**Abstract**: A classical question in algebraic geometry is
to determine whether or not a variety is rational. In this talk, I will
focus on its opposite side: how "irrational" a variety can be? In
particular, I will discuss my work on certain measures of irrationality of
hypersurfaces of large degree in projective spaces.

** **

**Friday, October 6**

**Speaker**: Malgorzata Marciniak, La Guardia Community College, CUNY

**Title**: Hessian Dynamics on the Hesse pencil

Abstract: Our ultimate goal is to answer the question of whether the group
of rational points on an elliptic curve $C$ is related to the group of
rational points on its Hessian $H(C)$. Prior to attacking this problem, we
considered the Hessian dynamics on the pencil of cubic curves (called the
Hesse pencil). Here we describe and analyze the Hessian dynamics in terms
of projective coordinates on the space of parameters on the Hesse pencil.
The presentation will be based on research activities conducted for the
Beginners Explorations in Algebraic Geometry (BEAG) during the academic
year 2016/17.

**Friday, October 20**

**Speaker**: Fei Ye

**Title**: Multiplier Ideals and Some Applications to Adjoint Line
Bundles

Abstract: Multiplier ideals are associated with ideals on algebraic variety and satisfy certain vanishing theorems. In this talk, I will explain how we can associate a maximal ideal with a multiplier ideal. As applications, I will present two results on adjoint line bundles on smooth projective varieties of dimension 3.

**Friday, November 3**

**Speaker**: Hans Schoutens (NYCCT and GC)

**Title**: From families of Weyl algebras to Bernstein algebras

Abstract:

Example 1: let $x$ and $y$ be non-commuting variables over the affine line $\text{Spec}(k[t])$ and consider the `variety' with equation $xy-yx=t$ and coordinate ring $B_1$; its fibers are all Weyl algebras, except above the origin, which is a polynomial ring;

Example 2: the forgetfull functor from the category of (non-commutative) algebras to the category of modules with an alternating form (given by the commutator bracket) admits an adjoint, associating to a module with an alternating form its `universal alternating algebra' $B_2$;

Example 2bis: the forgetfull functor from the category of (commutative) algebras with a Poisson bracket to the category of modules with an alternating form admits an adjoint, associating to a module with an alternating form its `universal Poisson algebra' $B_3$.

I want to describe a framework that (i) will encompass the algebras $B_i$ described in these examples; and (ii) allows for a general structure theory as in the first example. This is what I call a Bernstein algebra: a filtered (non-commutative) algebra whose degree zero elements are central, generated by finitely many linear elements, whose commutators have degree zero. The algebras $B_1$ and $B_2$ are both Bernstein, whereas $B_3$ is realized as the associated graded ring of a Bernstein algebra. To obtain a good structure theory, we require that the Bernstein algebra be smooth, which will in particular imply that its associated graded ring is a polynomial ring (the above examples are of this type).

**Friday, November 17**

**Speaker**: Bart van Steirteghem

**Title**: Smooth affine spherical varieties: classification and
application

Abstract: A smooth affine complex variety, equipped with an action of a reductive group G, is called spherical if its coordinate ring is multiplicity free as a representation of G. I will give an overview of the classification(s) of these varieties, including joint work with G. Pezzini and F. Knop. I will also show how, thanks to work of Knop's, the combinatorics of such varieties can be used to classify the (real) multiplicity free Hamiltonian manifolds of symplectic geometry. As an illustration, I will explain (following joint work with Pezzini and K. Paulus) how one can recover C. Woodward's result that every reflective Delzant polytope is the momentum polytope of such a manifold.

**Friday, December 1**

**Speaker**: Jai Laxmi (Indian Institute of Technology, Bombay)

**Title**: Embeddings of Canonical Modules

Abstract: It is well-known that, for a Cohen-Macaulay local ring $S$ with
a canonical module $\omega_S$, if $S$ is generically Gorenstein, then
$\omega_S$ can be identified with an ideal of $S$, that is, $\omega_S$
embeds into $S$.

In this talk, we are concerned with a specific embedding of a canonical
module of $R/I_{m,n}$ to itself, where $I_{m,n}$ is an ideal
generated by all square-free monomials of degree $m$ in a polynomial ring
$R$ with $n$ variables. We discuss how to construct such an embedding
using a minimal generating set of $\text{Hom}_R(R/I_{m,n}, R/I_{m,n})$.

This talk is based on a recent joint work with Ela Celikbas and Jerzy
Weyman.