BEGIN:VCALENDAR
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BEGIN:VEVENT
SUMMARY:Mike Stillman (Cornell University)
DTSTART:20200423T203000Z
DTEND:20200423T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/1
DESCRIPTION:Title: Quadratic Gorenstein rings and the Koszul property\nby Mike Stillm
an (Cornell University) as part of Fellowship of the Ring\n\n\nAbstract\nA
graded ring R = S/I is Gorenstein (S = polynomial ring\, I =\nhomogeneous
ideal) if the length of its free resolution over S is its\ncodimension in
S\, and the top betti number is one. R is called Koszul\nif the free reso
lution of k = R/(maximal homogeneous ideal) over R is\nlinear. Any Koszul
algebra is defined by quadratic relations\, but the\nconverse is false\, a
nd no one knows a finitely computable criterion.\nBoth types of rings have
duality properties\, and occur in many\nsituations in algebraic geometry
and commutative algebra\, and in many\ncases\, a Gorenstein quadratic alge
bra coming from geometry is often\nKoszul (e.g. homogeneous coordinate rin
gs of most canonical curves).\n\nIn 2001\, Conca\, Rossi\, and Valla asked
the question: must a (graded)\nquadratic Gorenstein algebra of regularity
3 be Koszul?\n\nIn the first 45 minutes\, we will define these notions\,
and give\nexamples of quadratic Gorenstein algebras and Koszul algebras. W
e\nwill give methods for their construction\, e.g. via inverse systems.\nA
fter a short break\, we will use these techniques to answer negatively\nth
e above question\, as well as see how to construct many other\nexamples of
quadratic Gorenstein algebras which are not Koszul.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christine Berkesch (University of Minnesota)
DTSTART:20200430T203000Z
DTEND:20200430T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/2
DESCRIPTION:Title: The geometry of toric syzygies\nby Christine Berkesch (University
of Minnesota) as part of Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Jeffries (CIMAT)
DTSTART:20200507T203000Z
DTEND:20200507T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/3
DESCRIPTION:Title: Two applications of $p$-derivations to commutative algebra\nby Jac
k Jeffries (CIMAT) as part of Fellowship of the Ring\n\n\nAbstract\nThe no
tions of derivations and modules of differentials have been central in com
mutative algebra for much of its history. A somewhat more exotic notion is
that of $p$-derivations: these are maps that satisfy functional equations
similarly to derivations\, but are not even additive. Nonetheless\, $$-de
rivations and related constructions have found applications in arithmetic
geometry. In this talk\, we will give a basic introduction to p-derivation
s\, and discuss two applications to commutative algebra\, based on project
s with Melvin Hochster and Anurag K. Singh (each).\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Erman (University of Wisconsin)
DTSTART:20200514T203000Z
DTEND:20200514T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/4
DESCRIPTION:Title: Boundedness questions for polynomials in many variables\nby Dan Er
man (University of Wisconsin) as part of Fellowship of the Ring\n\n\nAbstr
act\nI will discuss questions and results related to polynomials in a larg
e number of variables\, starting with classical results of Hilbert and mov
ing to Stillman's conjecture and its proof by Ananyan and Hochster. Then I
will describe ways to think about the limit of polynomial rings as the nu
mber of variables goes to infinity\, and how this can be applied to obtain
new finiteness results. The original work covered in this talk is all joi
nt with Steven V Sam and Andrew Snowden.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudiu Raicu (University of Notre Dame)
DTSTART:20200521T203000Z
DTEND:20200521T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/5
DESCRIPTION:Title: Commutative algebra with $S_n$-invariant monomial ideals\nby Claud
iu Raicu (University of Notre Dame) as part of Fellowship of the Ring\n\n\
nAbstract\nConsider a polynomial ring in $n$ variables\, together with the
action of the symmetric group by coordinate permutations. In my talk I wi
ll describe many familiar notions in Commutative Algebra in the context of
monomial ideals that are preserved by the action of the symmetric group.
These include Castelnuovo-Mumford regularity\, projective dimension\, satu
ration\, symbolic powers\, or the Cohen-Macaulay property. My goal is to e
xplain how changing focus from minimal resolutions to Ext modules can lead
to a simplified picture of the homological algebra\, and to provide concr
ete combinatorial recipes to determine the relevant homological invariants
.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eloísa Grifo (University of California\, Riverside)
DTSTART:20200528T203000Z
DTEND:20200528T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/6
DESCRIPTION:Title: Symbolic powers\, stable containments\, and degree bounds\nby Elo
ísa Grifo (University of California\, Riverside) as part of Fellowship of
the Ring\n\n\nAbstract\nWhat's the smallest degree of a homogeneous polyn
omial that vanishes to order n on a given finite set of points\, or more g
enerally on some algebraic variety in projective space? A classical result
of Zariski and Nagata tells us the set of such polynomials is the nth sym
bolic power of the ideal I corresponding to our variety. To bound degrees
of elements in the symbolic powers of I\, we can look for containments bet
ween symbolic powers and other better understood ideals\, such as powers o
f I. We will take a tour through the history of the containment problem an
d some of its variations\, with an eye towards lower bounds for degrees of
symbolic powers. Our story will include joint work with Craig Huneke and
Vivek Mukundan\, and with Sankhaneel Bisui\, Tài Huy Hà\, and Thái Thà
nh Nguyễn.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason McCullough (Iowa state)
DTSTART:20200604T203000Z
DTEND:20200604T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/7
DESCRIPTION:Title: Subadditivity of syzygies and related problems\nby Jason McCulloug
h (Iowa state) as part of Fellowship of the Ring\n\n\nAbstract\nLet $S = K
[x_1\,...\,x_n]$ be a polynomial ring over a field and $I$ a graded $S$-i
deal. There are many interesting questions about the maximal graded shift
s of $S/I$\, denoted $t_i$. In the first part of my talk\, I will discuss
two classical constructions that turn a (graded) S-module into an ideal w
ith similar properties\, namely idealizations and Bourbaki ideals\, and wh
at they say about maximal graded shifts of ideals. In the second part of
the talk\, I will discuss restrictions on maximal graded shifts of ideals.
In particular\, an ideal $I$ is said to satisfy the subadditivity condit
ion if $t_a + t_b ≥ t_(a+b)$ for all $a\,b$. This condition fails for a
rbitrary\, even Cohen-Macaulay\, ideals but is open for certain nice class
es of ideals\, such as Koszul and monomial ideals. I will present a const
ruction (joint with A. Seceleanu) showing that subadditivity can fail for
Gorenstein ideals. \n\nIf time allows\, I will talk about some results th
at hold more generally\, including a linear bound on the maximal graded sh
ifts in terms of the first $p-c$ shifts\, where $p = pd(S/I)$ and $c = cod
im(I)$. I hope to include several examples and open questions as well.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudia Polini (University of Notre Dame)
DTSTART:20200618T203000Z
DTEND:20200618T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/8
DESCRIPTION:Title: Rees algebras of ideals generated by 2x2 minors\nby Claudia Polini
(University of Notre Dame) as part of Fellowship of the Ring\n\n\nAbstrac
t\nRees algebras of ideals of maximal minors of generic matrices are very
well understood. In this\ntalk we deal with the case of sparse matrices wi
th two rows and with the case of two by two \nminors of generic matrices.
We investigate the defining ideals of these algebras\, and in the first \n
case we prove that they are Koszul and have rational singularities or are
F-rational\, respectively. \nThis is a report on joint work with Ela Celik
bas\, Emilie Dufresne\, Louiza Fouli\, Elisa Gorla\, Kuei-Nuan \nLin\, and
Irena Swanson and with Hang Huang\, Michael Perlman\, Claudiu Raicu\, and
Alessio Sammartano.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Varbaro (University of Genoa\, Italy)
DTSTART:20200611T203000Z
DTEND:20200611T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/9
DESCRIPTION:Title: The dual graph of a ring\nby Matteo Varbaro (University of Genoa\,
Italy) as part of Fellowship of the Ring\n\n\nAbstract\nThe dual graph (a
.k.a. Hochster-Huneke graph) G(R) of a Noetherian ring R of dimension d is
the finite simple graph whose vertices correspond to the minimal primes o
f R and such that {P\,Q} is an edge iff R/(P+Q) has dimension d-1.\nAfter
showing some basic properties\, we will discuss three fundamental results
of Grothendieck\, Hartshorne\, and Hochster-Huneke\, concerning the connec
tedness of G(R). We will also see\, given a finite simple graph G\, how to
construct a Noetherian ring R such that G(R)=R.\n\nIn the second part of
the talk\, we will discuss some recent developments related to the followi
ng two questions:\n1) How many paths are there between two minimal primes
of R?\n2) What is the shortest path between two minimal primes of R?\nBy t
aking the minimum in 1) and the maximum in 2) varying the pair of minimal
primes we get two important invariants of the graph G(R): its vertex conne
ctivity and its diameter. Most of the things that I will discuss are conta
ined in works written together with Bruno Benedetti\, Barbara Bolognese an
d Michela Di Marca.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hailong Dao (University of Kansas)
DTSTART:20200625T203000Z
DTEND:20200625T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/10
DESCRIPTION:Title: A truly mutually beneficial friendship: how Stanley-Reisner theory en
hanced both combinatorics and algebra\nby Hailong Dao (University of K
ansas) as part of Fellowship of the Ring\n\n\nAbstract\nGiven a simplicial
complex on n vertices\, one can associate to it a quotient of the polynom
ial ring in n variables\, called the Stanley-Reisner ring. Starting with t
he proof of the Upper Bound Conjecture for spheres\, this approach has bee
n spectacularly useful in bringing tools from commutative algebra to the s
tudy of simplicial complexes. In the first part of the talk I will sketch
some relevant parts of this story. In the second\, I will describe how mod
ern tools\, including cohomological vanishing results and characteristic p
methods\, have inspired new developments. At the same time\, results obta
ined on the combinatorics side now can be brought back to induce interesti
ng new questions and theorems on the algebra side. One thing I really like
about this topic is that it can be used to generate good problems at all
levels\, including for high school students.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Polstra (University of Utah)
DTSTART:20200723T203000Z
DTEND:20200723T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/11
DESCRIPTION:Title: The weak implies strong conjecture and finite generation of symbolic
Rees algebras\nby Thomas Polstra (University of Utah) as part of Fello
wship of the Ring\n\n\nAbstract\nTight closure theory is prominent to prim
e characteristic commutative algebra. Historically\, tight closure has bee
n used to simplify proofs of landmark theorems in commutative algebra\, pr
ovide tests to determine when an element of a ring belongs to a particular
ideal\, and provides proofs of results in prime characteristic which othe
rwise can only be proved in equal characteristic 0. The first half of the
talk will be devoted to basic definitions\, properties\, and consequences
of tight closure. The second half of the talk we will go over several impo
rtant classes of rings defined via tight closure\, recent progress on the
weak implies strong conjecture appearing in a joint paper with Ian Aberbac
h\, and a conjecture that certain symbolic Rees algebras are Noetherian.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bhargav Bhatt (University of Michigan)
DTSTART:20200827T203000Z
DTEND:20200827T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/12
DESCRIPTION:Title: Cohen-Macaulayness of absolute integral closures\nby Bhargav Bhat
t (University of Michigan) as part of Fellowship of the Ring\n\n\nAbstract
\nA deep theorem of Hochster-Huneke in F-singularity theory is that the ab
solute integral closure of an excellent noetherian local domain R over F_p
is Cohen-Macaulay. In other words\, all relations on a system of paramete
rs in R become trivial after passing to a finite cover of R. In this talk\
, I'll discuss the analog of this result in mixed characteristic\, as well
as some consequences.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Seceleanu (University of Nebraska)
DTSTART:20200716T203000Z
DTEND:20200716T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/13
DESCRIPTION:Title: Reflection arrangements\, syzygies\, and the containment problem\
nby Alexandra Seceleanu (University of Nebraska) as part of Fellowship of
the Ring\n\n\nAbstract\nInvariant theory\, that is the art of finding poly
nomials invariant under the action of a given group\, has played a major r
ole in the historical development of commutative algebra. In this theory r
eflection groups are singled out for having rings of invariants that are i
somorphic to polynomial rings. From a geometric perspective\, reflection g
roups give rise to beautiful and very symmetric arrangements of hyperplane
s termed reflection arrangements.\n\nThis talk will take a close look at t
he ideals defining the singular loci of reflection arrangements\, which ar
e in turn symmetric subspace arrangements. We describe their syzygies in t
erms of invariant polynomials for the relevant reflection groups. We lever
age this information to settle many aspects of the containment problem ask
ing for containments between the ordinary and the symbolic powers of the i
deals in this family. This talk is based on joint work with Ben Drabkin.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Briggs (University of Utah)
DTSTART:20200806T203000Z
DTEND:20200806T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/14
DESCRIPTION:Title: The homotopy Lie algebra and the conormal module\nby Benjamin Bri
ggs (University of Utah) as part of Fellowship of the Ring\n\n\nAbstract\n
I will do my best to explain what goes in to proving the following theorem
: if $I$ is an ideal of finite projective dimension in a local ring $R$\,
and the conormal module $I/I^2$ has finite projective dimension over $R/I$
\, then $I$ is generated by a regular sequence. This was conjectured by Va
sconcelos\, after he and (separately) Ferrand established the case that th
e conormal module is free.\n\nThe key tool is the homotopy Lie algebra. Th
is is a graded Lie algebra naturally associated with any local homomorphis
m. It sits at the centre of a longstanding friendship between commutative
algebra and rational homotopy theory\, through which ideas and results hav
e been passed back and forth for decades.\n\nI'll go through the construct
ion of the homotopy Lie algebra and how it's been used in commutative alge
bra in the past\, before explaining how its structure detects when the con
ormal module has finite projective dimension. I'll also talk about ongoing
work with Srikanth Iyengar comparing the cotangent complex with the homot
opy Lie algebra.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eleonore Faber (University of Leeds)
DTSTART:20200820T203000Z
DTEND:20200820T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/15
DESCRIPTION:Title: Grassmannian categories of infinite rank and countable Cohen-Macaulay
type\nby Eleonore Faber (University of Leeds) as part of Fellowship o
f the Ring\n\n\nAbstract\nThis talk is about a categorification of the coo
rdinate rings of Grassmannians of infinite rank in terms of graded maximal
Cohen-Macaulay modules over a hypersurface singularity. This gives an inf
inite rank analogue of the Grassmannian cluster categories introduced by J
ensen\, King\, and Su. In a special case\, when the hypersurface singulari
ty is a curve of countable Cohen-Macaulay type\, our category has a combin
atorial model by an ``infinity-gon'' and we can determine triangulations o
f this infinity-gon.\n\nI will first give an introduction to Grassmannian
cluster algebras and categories\, and then explain our limit constructions
. This is joint work with Jenny August\, Man-Wai Cheung\, Sira Gratz\, and
Sibylle Schroll.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aldo Conca (University of Genoa)
DTSTART:20200903T190000Z
DTEND:20200903T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/16
DESCRIPTION:Title: Ideals associated to subspace arrangements\nby Aldo Conca (Univer
sity of Genoa) as part of Fellowship of the Ring\n\n\nAbstract\nMotivated
by the study of the Castelnuovo-Mumford regularity of products of ideals
Herzog and I proved\, about twenty years ago\, that a product of ideals o
f linear forms has a linear resolution. Only recently Tsakiris and I mana
ged to describe such a resolution. It is supported on a polymatroid natu
rally attached to the associated subspace arrangements.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juliette Bruce (Syracuse University)
DTSTART:20200910T190000Z
DTEND:20200910T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/17
DESCRIPTION:Title: Syzygies of Products of Projective Space\nby Juliette Bruce (Syra
cuse University) as part of Fellowship of the Ring\n\n\nAbstract\nI will d
iscuss the asymptotic non-vanishing of syzygies for products of projective
spaces\, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This
provides the first example of how the asymptotic syzygies of a smooth proj
ective variety whose embedding line bundle grows in a semi-ample fashion b
ehave in nuanced and previously unseen ways.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dale Cutkosky (University of Missouri)
DTSTART:20200702T203000Z
DTEND:20200702T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/18
DESCRIPTION:Title: Mixed multiplicities of filtrations\nby Dale Cutkosky (University
of Missouri) as part of Fellowship of the Ring\n\n\nAbstract\nWe discuss
the theory of multiplicities and mixed multiplicities of filtrations of m-
primary ideals. We show that many classical formulas are true in this sett
ing. We also consider the case of equality in Minkowski's inequality. We g
ive some general theorems characterizing when this condition holds\, givin
g generalizations of classical theorems of Rees\, Sharp\, Teissier\, Katz
and others.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Núñez-Betancourt (CIMAT\, Guanajuato)
DTSTART:20200709T203000Z
DTEND:20200709T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/19
DESCRIPTION:Title: Differential powers of ideals\nby Luis Núñez-Betancourt (CIMAT\
, Guanajuato) as part of Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brooke Ullery (Emory University)
DTSTART:20200813T203000Z
DTEND:20200813T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/20
DESCRIPTION:Title: Cayley-Bacharach theorems and measures of irrationality\nby Brook
e Ullery (Emory University) as part of Fellowship of the Ring\n\n\nAbstrac
t\nIf Z is a set of points in projective space\, we can ask which polynomi
als of degree d vanish at every point in Z. If P is one point of Z\, the v
anishing of a polynomial at P imposes one linear condition on the coeffici
ents. Thus\, the vanishing of a polynomial on all of Z imposes |Z| linear
conditions on the coefficients. A classical question in algebraic geometry
\, dating back to at least the 4th century\, is how many of those linear c
onditions are independent? For instance\, if we look at the space of lines
through three collinear points in the plane\, the unique line through two
of the points is exactly the one through all three\; i.e. the conditions
imposed by any two of the points imply those of the third. In this talk\,
I will survey several classical results including the original Cayley-Bach
arach Theorem and Castelnuovo’s Lemma about points on rational curves. I
’ll then describe some recent results and conjectures about points satis
fying the so-called Cayley-Bacharach condition and show how they connect t
o several seemingly unrelated questions in contemporary algebraic geometry
relating to the gonality of curves and measures of irrationality of highe
r dimensional varieties.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Ardila (San Francisco State University)
DTSTART:20200730T203000Z
DTEND:20200730T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/21
DESCRIPTION:Title: Lagrangian Geometry of Matroids\nby Federico Ardila (San Francisc
o State University) as part of Fellowship of the Ring\n\n\nAbstract\nMatro
id theory had its origins in linear algebra and graph theory. In recent ye
ars\, the geometric roots of the field have grown much deeper\, bearing ma
ny new fruits. The interplay between matroid theory and algebraic geometry
has opened up interesting research directions at the intersection of comb
inatorics\, algebra\, and geometry\, and led to the solution of long-stand
ing questions. \n\nThis talk will discuss my recent joint work with Graham
Denham and June Huh. We introduce the conormal fan of a matroid M. Inside
its Chow ring\, we find simple interpretations of the Chern-Schwartz-MacP
herson cycle of M (a tropical geometric construction) and of the h-vector
of M (a combinatorial invariant). We then use the Hodge-Riemann relations
to prove Brylawski's and Dawson's conjectures that the h-vector of a matro
id is log-concave.\n\nI will make the talk as self-contained as possible\,
and assume no previous knowledge of matroid theory.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Snowden (University of Michigan)
DTSTART:20200917T190000Z
DTEND:20200917T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/22
DESCRIPTION:Title: Infinite dimensional equivariant commutative algebra\nby Andrew S
nowden (University of Michigan) as part of Fellowship of the Ring\n\n\nAbs
tract\nCertain infinite variable polynomial rings equipped with actions of
large groups (like the infinite symmetric group or the infinite general l
inear group) behave in many ways like finitely generated algebras\; for in
stance\, one sometimes has an "equivariant noetherian" property. I will di
scuss some deeper parallels with commutative algebra\, and how these somew
hat exotic objects can be applied to study classical questions.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cris Negron (University of North Carolina)
DTSTART:20200924T190000Z
DTEND:20200924T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/23
DESCRIPTION:Title: Noncommutative hypersurfaces and support theory for Hopf algebras
\nby Cris Negron (University of North Carolina) as part of Fellowship of t
he Ring\n\n\nAbstract\nI will talk about a new approach to support theory
for noncommutative (Hopf) algebras which mirrors Avramov and Buchweitz’
support theory for commutative local complete intersections. I will expla
in what this support theory entails for ``noncommutative complete intersec
tions"\, and relevant examples\ncoming from quantum linear spaces\, functi
ons on finite group schemes\, and quantum groups. I will also explain how
this support theory is used to classify thick ideals in the associated st
able representation categories. No familiarity with these topics is assum
ed\, and everything in the talk should be explained in relatively basic te
rms. This is joint work with Julia Pevtsova.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Diane Maclagan (University of Warwick)
DTSTART:20201001T190000Z
DTEND:20201001T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/24
DESCRIPTION:Title: Tropical ideals\nby Diane Maclagan (University of Warwick) as par
t of Fellowship of the Ring\n\n\nAbstract\nOne consequence of the recent p
ush to develop a scheme theory in tropical geometry has been the developme
nt of a tropical commutative algebra. This starts with the commutative al
gebra of semirings\, but in order to get a theory that interacts with geom
etry\, we are lead to impose some combinatorial\, matroid-theoretic\, cond
itions. I will introduce these ideas\, and discuss the current\nstate of
our understanding.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takumi Murayama (Princeton University)
DTSTART:20201008T190000Z
DTEND:20201008T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/25
DESCRIPTION:Title: Grothendieck's localization problem\nby Takumi Murayama (Princeto
n University) as part of Fellowship of the Ring\n\n\nAbstract\nLet $A\\to
B$ be a flat local map of noetherian complete local rings. Using Hironaka'
s resolution of singularities Grothendieck and Dieudonné showed that if t
he closed fiber of the map $A\\to B$ is Cohen-Macaulay and if $A$ is of eq
ual characteristic zero\, then all the fibers of the map are Cohen-Macaula
y. Three decades later\, Avramov and Foxby showed that the same statement
holds without the characteristic assumption on A. Grothendieck's localizat
ion problem asks whether a similar statement holds with Cohen-Macaulayness
replaced by other local properties of noetherian local rings. We solve Gr
othendieck's localization problem for all sufficiently well-behaved proper
ties of noetherian local rings. Our proof provides a uniform treatment of
previously known special cases of Grothendieck's problem\, in particular g
iving a new proof of Avramov and Foxby's result. As an application\, we sh
ow that if the closed fibers of a flat morphism of algebraic varieties are
smooth\, then all fibers are smooth.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jenna Rajchgot (McMaster University)
DTSTART:20201015T190000Z
DTEND:20201015T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/26
DESCRIPTION:Title: Geometric vertex decomposition and liaison\nby Jenna Rajchgot (Mc
Master University) as part of Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gennady Lyubeznik (University of Minnesota)
DTSTART:20201022T190000Z
DTEND:20201022T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/27
DESCRIPTION:Title: A characteristic-free definition of holonomic D-modules\nby Genna
dy Lyubeznik (University of Minnesota) as part of Fellowship of the Ring\n
\n\nAbstract\nMost of the theory of D-modules has been developed only in c
haracteristic zero. This includes holonomic modules. Some candidates for h
olonomic modules in characteristic p>0 have been proposed using definition
s specific to characteristic p>0. The first characteristic-free definition
of holonomicity was given in 2010 by the speaker\, but only for modules o
ver polynomial rings. In the talk I am going to describe an extension of t
his definition to arbitrary non-singular varieties.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro de Stefani (University of Genoa)
DTSTART:20201029T190000Z
DTEND:20201029T203000Z
DTSTAMP:20260404T131152Z
UID:FOTR/28
DESCRIPTION:Title: Deformation and stability of F-singularities\nby Alessandro de St
efani (University of Genoa) as part of Fellowship of the Ring\n\n\nAbstrac
t\nGiven a property (P)\, the deformation problem asks whether\, whenever
x is a regular element of a ring R such that R/xR satisfies (P)\, then so
does R. We will survey some known facts on the deformation problem for F-s
ingularities in prime characteristic\, and present some recent results on
deformation of F-injectivity\, obtained in joint work with Linquan Ma. In
the second part of the talk\, we will discuss another problem\, called sta
bility. We will present some results obtained in collaboration with Ilya S
mirnov\, and outline a general relation between the two problems of deform
ation and stability.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henry Schenck (Auburn University)
DTSTART:20201105T200000Z
DTEND:20201105T213000Z
DTSTAMP:20260404T131152Z
UID:FOTR/29
DESCRIPTION:Title: Calabi-Yau threefolds in P^n and Gorenstein rings\nby Henry Schen
ck (Auburn University) as part of Fellowship of the Ring\n\n\nAbstract\nA
projectively normal Calabi-Yau threefold $X \\subseteq \\mathbb{P}^n$ has
an ideal $I_X$ which is arithmetically Gorenstein\, of Castelnuovo-Mumford
regularity four. Such ideals have been intensively studied when $I_X$ is
a complete intersection\, as well as in the case were $X$ has codimension
three. In the latter case\, the Buchsbaum-Eisenbud theorem shows that $I_X
$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent
papers study the situation when $I_X$ has codimension four. We prove ther
e are 16 possible betti tables for an arithmetically Gorenstein ideal I wi
th codim(I) = 4 = regularity(I)\, and that 9 of these arise for prime nond
egenerate threefolds. We investigate the situation in codimension five or
more\, obtaining examples of X with $h^{p\,q}(X)$ not among those appearin
g for $I_X$ of lower codimension or as complete intersections in toric Fan
o varieties--in other words\, Calabi-Yau's with Hodge numbers not previous
ly known to occur. A main feature of our approach is the use of inverse sy
stems to identify possible betti tables for X. This is joint work with M.
Stillman and B. Yuan\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Brown (Auburn University)
DTSTART:20201119T200000Z
DTEND:20201119T213000Z
DTSTAMP:20260404T131152Z
UID:FOTR/31
DESCRIPTION:Title: A toric BGG correspondence\nby Michael Brown (Auburn University)
as part of Fellowship of the Ring\n\n\nAbstract\nThis is ongoing joint wor
k with David Eisenbud\, Daniel Erman\, and Frank-Olaf Schreyer. The Bernst
ein-Gel'fand-Gel'fand (BGG) correspondence is a derived equivalence betwee
n a standard graded polynomial ring and its Koszul dual exterior algebra.
One of the many important applications of the BGG correspondence is an alg
orithm\, due to Eisenbud-Fløystad-Schreyer\, for computing sheaf cohomolo
gy on projective space that is\, in some cases\, the fastest available. Th
e goal of this talk is to discuss a generalization of the BGG corresponden
ce from standard graded to multigraded polynomial rings and how it leads t
o an Eisenbud-Fløystad-Schreyer-type algorithm for computing sheaf cohomo
logy over certain projective toric varieties.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mircea Mustata (University of Michigan)
DTSTART:20201112T200000Z
DTEND:20201112T213000Z
DTSTAMP:20260404T131152Z
UID:FOTR/32
DESCRIPTION:Title: Minimal exponents of hypersurfaces and a conjecture of Teissier\n
by Mircea Mustata (University of Michigan) as part of Fellowship of the Ri
ng\n\n\nAbstract\nThe minimal exponent of a hypersurface is an invariant o
f singularities defined via the Bernstein-Sato polynomial. It is a refine
ment of the log canonical threshold (a fundamental invariant in birational
geometry)\, that can be used to measure rational singularities. In the fi
rst part of the talk I will give an introduction to these and related inv
ariants. The second part of the talk will describe joint work with Eva Eld
uque and Bradley Dirks on a conjecture of Teissier\, relating the minimal
exponent of a hypersurface with that of a hyperplane section.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grigoriy Blekherman (Georgia Institute of Technology)
DTSTART:20210114T213000Z
DTEND:20210114T230000Z
DTSTAMP:20260404T131152Z
UID:FOTR/33
DESCRIPTION:Title: Sums of Squares: From Real to Commutative Algebra\nby Grigoriy Bl
ekherman (Georgia Institute of Technology) as part of Fellowship of the Ri
ng\n\n\nAbstract\nA real polynomial is called nonnegative if it takes only
nonnegative values. A sum of squares or real polynomials is clearly nonne
gative. The relationship between nonnegative polynomials and sums of squar
es is one of the central questions in real algebraic geometry. A modern ap
proach is to look at nonnegative polynomials and sums of squares on a real
variety X\, where unexpected links to complex algebraic geometry and comm
utative algebra appear.\n\nIn the first half of the talk I will review the
history of the problem\, do some examples\, and provide a brief overview
of the results. Our two guiding questions will be: the relationship betwee
n nonnegative polynomials and sums of squares\, and the number of squares
needed to write any sum of squares on X. I will explain the connection bet
ween these questions and properties of the free resolution of the ideal of
X: the number of of steps that the resolution only has linear syzygies (p
roperty $N_{2\,p}$) and the number of steps that linear syzygies persist (
the length of the linear strand).\n\nIn the second half\, I will concentra
te on the number of squares\, and introduce an invariant of X we call quad
ratic persistence. Quadratic persistence of X is equal to the least number
of points in X such that after projecting from (the span of) these points
the ideal of the resulting variety has no quadrics. I will explain how qu
adratic persistence connects real algebraic geometry and commutative algeb
ra. Joint work with Rainer Sinn\, Greg Smith and Mauricio Velasco.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuki Takagi (University of Tokyo)
DTSTART:20210122T010000Z
DTEND:20210122T023000Z
DTSTAMP:20260404T131152Z
UID:FOTR/34
DESCRIPTION:Title: Arithmetic deformations of F-singularities\nby Shunsuki Takagi (U
niversity of Tokyo) as part of Fellowship of the Ring\n\n\nAbstract\nF-sin
gularities are singularities in positive characteristic defined via the Fr
obenius map. In the first half of the talk\, I will survey a connection be
tween F-singularities and singularities in complex birational geometry. In
the second half of the talk\, I will present a new application of Ma-Schw
ede’s theory of singularities in mixed characteristic. They proved that
a Q-Gorenstein affine domain over a field of characteristic zero has log t
erminal singularities if its mod p reduction is F-regular for one single p
rime p. I will discuss the analog of their result for log canonical singul
arities. This talk is based on joint work with Kenta Sato.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Smirnov (Stockholm University)
DTSTART:20210128T210000Z
DTEND:20210128T223000Z
DTSTAMP:20260404T131152Z
UID:FOTR/35
DESCRIPTION:Title: The quest for F-rational signature\nby Ilya Smirnov (Stockholm Un
iversity) as part of Fellowship of the Ring\n\n\nAbstract\nStrongly F-regu
lar singularities are one of the fundamental classes of singularities defi
ned by the properties of Frobenius endomorphism. This class of mild singul
arities can be detected using F-signature\, an invariant of a local ring w
ith many good properties. Through this connection we obtain a powerful too
l for studying strongly F-regular singularities\, for example\, several re
sults on "mildness" of F-regular can be quantified using F-signature. \n\
nAnother fundamental class of singularities in positive characteristic are
F-rational singularities. While generally more severe\, this class of sin
gularities is in many aspects analogous to strongly F-regular singularitie
s and can be even understood by "moving" the definition of F-regularity to
take place in the dualizing module. Naturally\, there has been interest i
n adapting the definition of F-signature to work with F-rational singulari
ties. \n\nWhile there is no complete solution yet\, I am convinced that su
ch a theory should exist. As my evidence\, I will present results of a joi
nt work with Kevin Tucker\, and prior works of Hochster and Yao\, and Sann
ai.\n\nMy talk will be self-contained. I will discuss all necessary backgr
ound\, such as definitions\, properties\, and relations between these noti
ons\, in the first half and then proceed to more technical results in the
second half.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elisa Gorla (University of Neuchatel)
DTSTART:20210204T213000Z
DTEND:20210204T230000Z
DTSTAMP:20260404T131152Z
UID:FOTR/36
DESCRIPTION:Title: Ideals with a radical generic initial ideal\nby Elisa Gorla (Univ
ersity of Neuchatel) as part of Fellowship of the Ring\n\n\nAbstract\nThe
choice of a term order allows us to associate to any ideal I a monomial id
eal\, called the initial ideal of I. The initial ideal of I depends not on
ly on the choice of a term order\, but also on the system of coordinates.
Nevertheless\, many properties of I can be inferred from those of its init
ial ideal(s). For a given term order and in a generic coordinate system\,
however\, the initial ideal of I is always the same and it is then called
the generic initial ideal of I. In my talk\, I will introduce a family of
ideals whose generic initial ideal is independent of the choice of both th
e term order and of the system of coordinates. These are exactly the multi
graded homogeneous ideals which have a radical generic initial ideal. Mult
igraded ideals which have a radical generic initial ideal show interesting
rigidity properties\, which e.g. allow us to deduce information on their
universal Groebner bases. In the talk\, I will present examples of ideals
which belong to this class and of what we can deduce about them using this
machinery. The original work in the talk is joint with Aldo Conca and Ema
nuela De Negri.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liana Sega (University of Missouri-Kansas City)
DTSTART:20210211T220000Z
DTEND:20210211T233000Z
DTSTAMP:20260404T131152Z
UID:FOTR/37
DESCRIPTION:Title: Simplicial resolutions of powers of square-free monomial ideals\n
by Liana Sega (University of Missouri-Kansas City) as part of Fellowship o
f the Ring\n\n\nAbstract\nA free resolution of an ideal I generated by p m
onomials can be described using the simplicial chain maps of a simplex on
p vertices. This resolution is called the Taylor resolution of the ideal a
nd was constructed by Diana Taylor in her thesis (1966). Work of Bayer\, P
eeva and Sturmfels further established a criterion for deciding whether th
e simplicial chain maps of a smaller simplicial complex on p vertices des
cribes a free resolution of I\, and since then there has been a significan
t amount of work on trimming down the Taylor resolution to simplicial reso
lutions that are minimal (for certain classes of ideals) or closer to bein
g minimal. In this talk we will discuss a class of simplicial complexes\,
indexed by the positive integers\, where the r-th complex in this class s
upports a resolution of the r-th power of I^r\, where I is a square-free m
onomial ideal.\n \nThis work is joint with Susan Cooper\, Sabine El Khoury
\, Sara Faridi\, Sara Mayes-Tang\, Susan Morey\, and Sandra Spiroff\, and
was started at the "Women in Commutative Algebra" workshop in Banff\, 2019
.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gregory G. Smith (Queen's University)
DTSTART:20210218T213000Z
DTEND:20210218T230000Z
DTSTAMP:20260404T131152Z
UID:FOTR/38
DESCRIPTION:Title: Smooth Hilbert schemes\nby Gregory G. Smith (Queen's University)
as part of Fellowship of the Ring\n\n\nAbstract\nHow can we understand all
saturated homogeneous ideals in a polynomial ring?\nHilbert schemes provi
de a geometric answer to this question. After surveying\nthe key features
of these natural parameter spaces\, we will present a complete\ncombinato
rial classification of the smooth Hilbert schemes. We will also\nreinterp
ret the smooth Hilbert schemes as suitable generalizations of partial\nfla
g varieties. This talk is based on joint work with Roy Skjelnes (KTH).\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Satoshi Murai (Waseda University)
DTSTART:20210325T220000Z
DTEND:20210325T233000Z
DTSTAMP:20260404T131152Z
UID:FOTR/39
DESCRIPTION:Title: Betti numbers of monomial ideals fixed by permutations of the variabl
es\nby Satoshi Murai (Waseda University) as part of Fellowship of the
Ring\n\n\nAbstract\nLet R_n be the polynomial ring with n variables over a
field K. We consider the natural action of the n-th symmetric group S_n t
o R_n. In this talk\, I will mainly talk about the following problem: Fix
monomials u_1\,\\dots\,u_m and consider the ideal I_n of R_n generated by
the S_n-orbits of these monomials. How the Betti numbers of I_n change whe
n n increases?\nI will explain that there is a simple way to determine non
-zero positions of the Betti table of I_n when n is sufficiently large. I
also explain that we can determine the Betti numbers of I_n by considering
the S_n-module structure of Tor_i(I_n\,K).\n\nThe above problem is motiva
ted by recent studies of algebraic properties of S_n-invariant ideals and
is inspired by studies of Noetherianity up to symmetry. I will explain thi
s motivation and basic combinatorial properties of S_n-invariant ideals in
the first part of the talk.\nThis talk includes a joint work with Claudiu
Raicu.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Walker (University of Nebraska)
DTSTART:20210225T213000Z
DTEND:20210225T230000Z
DTSTAMP:20260404T131152Z
UID:FOTR/40
DESCRIPTION:Title: How short can a module of finite projective dimension be?\nby Mar
k Walker (University of Nebraska) as part of Fellowship of the Ring\n\n\nA
bstract\nThis is joint work with Srikanth Iyengar and Linquan Ma. I will d
iscuss the question:\n\nFor a given Cohen-Macaulay local ring R\, what is
the minimum non-zero value of length(M)\, where M ranges over those R-modu
les having finite projective dimension?\n\nIn investigating this question\
, one is quickly led to conjecture that the answer is e(R)\, the Hilbert-S
amuel multiplicity of R. It turns out that this can be established for rin
gs having Ulrich modules\, or\, more generally\, lim Ulrich sequences of m
odules\, with certain properties. Moreover\, there is a related conjecture
concerning length(M) and the Betti numbers of M\, and a conjecture concer
ning the Dutta multiplicity of M\, which can also be established when cert
ain Ulrich modules (or lim Ulrich sequences) exist.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adela Vraicu (University of South Carolina)
DTSTART:20210311T213000Z
DTEND:20210311T230000Z
DTSTAMP:20260404T131152Z
UID:FOTR/41
DESCRIPTION:Title: Classification of extremal hypersurfaces in positive characteristic\nby Adela Vraicu (University of South Carolina) as part of Fellowship o
f the Ring\n\n\nAbstract\nThe log canonical threshold is an invariant that
measures how singular a hypersurface over an algebraically closed field o
f characteristic zero is. The F-pure threshold is the positive characteris
tic analog. Hypersurfaces with smaller threshold are more singular.\n\nI w
ill discuss a lower bound for a homogeneous polynomial in characteristic p
\, relative to its degree\, and describe the classification of the hypersu
rfaces that achieve this bound up to change of coordinates. These results
were obtained as part of a project started at the A.W.M. Workshop ``Women
in Commutative Algebra” at B.I.R.S.\; joint work with Zhibek Kadyrsizova
\, Jennifer Kenkel\, Janet Page\, Jyoti Singh\, Karen E. Smith and Emily W
itt.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Lazarsfeld (Stony Brook University)
DTSTART:20210304T213000Z
DTEND:20210304T230000Z
DTSTAMP:20260404T131152Z
UID:FOTR/42
DESCRIPTION:Title: Saturation bounds for smooth varieties\nby Robert Lazarsfeld (Sto
ny Brook University) as part of Fellowship of the Ring\n\n\nAbstract\nLet
X be a smooth complex projective variety with homogeneous ideal I. We cons
ider the question of bounding the saturation degree of the powers I^a of I
\, ie the degrees after which this power agrees with the symbolic power I^
(a). I’ll discuss joint work with Lawrence Ein and Tai Huy Ha giving res
ults in two situations:\n\n— When I is the ideal of a smooth curve C\
, we give a bound in terms of the regularity of C\, extending results of G
eramita et al\, Sidman\, Chandler and others in the case of finite sets\;
\n\n— When X is smooth of arbitrary dimension\, we give a bound in te
rms of the degrees of defining equations that has the same general shape a
s (but a rather different proof than) regularity bounds established many y
ears ago with Bertram and Ein.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kei-ichi Watanabe (Nihon University and Meiji University)
DTSTART:20210319T000000Z
DTEND:20210319T013000Z
DTSTAMP:20260404T131152Z
UID:FOTR/43
DESCRIPTION:Title: Normal reduction numbers\, normal Hilbert coefficients and elliptic i
deals in normal 2-dimensional local domains\nby Kei-ichi Watanabe (Nih
on University and Meiji University) as part of Fellowship of the Ring\n\n\
nAbstract\nThis is a joint work with T. Okuma (Yamagata Univ.)\, M.E. Ross
i (Univ. Genova) and K. Yoshida (Nihon Univ.).\n\nLet $(A\, \\mathfrak{m})
$ be an excellent two-dimensional normal local domain and let $I$ be an in
tegrally closed $\\mathfrak{m}$-primary ideal and $Q$ be a minimal reducti
on of $I$ (a parameter ideal with $I^{r+1} = Q I^{r}$ for some $r ≥ 1$).
Then the reduction numbers \n\\[\nnr(I) = \\min\\{ n \\mid \\overline{I^{
n+1}} = Q \\overline{I^n} \\}\, \n\\]\nand\n\\[\n\\overline{r}(I) = \\min
\\{n \\mid \\overline{I^{N+1}} = Q\\overline{I^N}\, \\forall N \\ge n \\}\
n\\]\nare important invariants of the ideal and the singularity. Also the
normal Hilbert coefficients $\\overline{e}_i(I)$\, for $i = 0\, 1\, 2$\, a
re defined by\n\\[\n\\ell_A(A/\\overline{I^{n+1}}) = \\overline{e}_0(I)\\b
inom{n+2}2 - \\overline{e}_1(I)\\binom{n+1}1 + \\overline{e}_2(I)\\\,.\n\\
]\nfor $n\\gg 0$. \n\nWe can characterize certain class of singularities b
y these invariants. Namely\, $A$ is a\nrational singularity if and only if
$\\overline{r}(A) = 1$\, or equivalently\, $\\overline{e}_2(I) = 0$ for e
very $I$. We defined a $p_g$ ideal by the property $\\overline{r}(I) = 1$
and in this language\, $A$ is a rational singularity if and only if every
integrally closed $\\mathfrak{m}$ primary ideal is a$p_g$ ideal.\n\nOur ai
m is to know the behavior of these invariants for every integrally closed
$\\mathfrak{m}$ primary ideal $I$ of a given ring $A$.\n\nIf $A$ is an ell
iptic singularity\, then it is shown by Okuma that $\\overline{r}(I) \\le
2$ for every $I$. Inspired by these facts we define $I$ to be an elliptic
ideal if $\\overline{r}(I) = 2$ and strongly elliptic ideal if $\\overline
{e}_2 = 1$.\n\nWe will show several nice equivalent properties for $I$ to
be an elliptic or a strongly elliptic ideal.\n\nOur tool is resolution of
singularities of $\\mathrm{Spec}(A)$. Let $I$ be an $\\mathfrak{m}$-primar
y integrally closed ideal in $A$. We can take $f\\colon X \\to \\mathrm{Sp
ec}(A)$ a resolution of $A$ such that $I\\mathcal{O}_X = \\mathcal{O}_X(
−Z)$ is invertible. In particular $p_g(A) := h^1(X\,\\mathcal{O}_X)$ and
$q(I) := h^1(X\,\\mathcal{O}_X(−Z))$ play important role in our theory.
\n\nThis talk is based on our joint work appeared in arXiv 2012.05530 and
arXiv 1909.13190.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Tucker (University of Illinois at Chicago)
DTSTART:20210401T203000Z
DTEND:20210401T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/44
DESCRIPTION:Title: Global +- regularity\nby Kevin Tucker (University of Illinois at
Chicago) as part of Fellowship of the Ring\n\n\nAbstract\nOver a field of
characteristic $p > 0$\, a globally F-regular algebraic variety is a speci
al type of Frobenius split variety. They are necessarily locally (strongly
) F-regular\, hence normal and Cohen-Macaulay\, but also satisfy a number
of particularly nice global properties as well. A smooth projective variet
y is globally F-regular if its (normalized) coordinate rings are F-regular
\, a condition which imposes strong positivity properties and implies Koda
ira-type vanishing results. Globally F-regular varieties are closely relat
ed to complex log Fano varieties via reduction to characteristic $p > 0$.\
n\nIn this talk\, I will describe an analog of global F-regularity in the
mixed characteristic setting called global +-regularity and introduce cert
ain stable sections of adjoint line bundles. This is inspired by recent wo
rk of Bhatt on the Cohen-Macaulayness of the absolute integral closure\, a
nd has applications to birational geometry in mixed characteristic. This i
s based on arXiv:2012.15801 and is joint work with Bhargav Bhatt\, Linquan
Ma\, Zsolt Patakfalvi\, Karl Schwede\, Joe Waldron\, and Jakub Witaszek.\
n
LOCATION:https://stable.researchseminars.org/talk/FOTR/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martina Juhnke-Kubitzke (University of Osanbrueck)
DTSTART:20210408T200000Z
DTEND:20210408T213000Z
DTSTAMP:20260404T131152Z
UID:FOTR/45
DESCRIPTION:Title: The antiprism triangulation\nby Martina Juhnke-Kubitzke (Universi
ty of Osanbrueck) as part of Fellowship of the Ring\n\n\nAbstract\nThe ant
iprism triangulation provides a natural way to subdivide a simplicial comp
lex $\\Delta$\, similar to barycentric subdivision\, which appeared indepe
ndently in combinatorial algebraic topology and computer science. It can b
e defined as the simplicial complex of chains of multi-pointed faces of $\
\Delta$ from a combinatorial point of view\, and by successively applying
the antiprism construction\, or balanced stellar subdivisions\, on the fac
es of $\\Delta$ from a geometric point of view.\nIn this talk\, we will st
udy enumerative invariants associated to this triangulation\, such as the
transformation of the $h$-vector of $\\Delta$ under antiprism triangulatio
n\, the local $h$-vector\, and algebraic properties of its Stanley--Reisne
r ring. Among other results\, it is shown that the $h$-polynomial of the a
ntiprism triangulation of a simplex is real-rooted and that the antiprism
triangulation of $\\Delta$ has the almost strong Lefschetz property over $
\\mathbb{R}$ for every shellable complex $\\Delta$.\n\nI will make the tal
k as self-contained as possible\, and assume no previous knowledge of comb
inatorics of subdivisions. This is joint work with Christos Athanasiadis a
nd Jan-Marten Brunink.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Boocher (University of San Diego)
DTSTART:20210415T203000Z
DTEND:20210415T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/46
DESCRIPTION:Title: On the size and shape of betti numbers\nby Adam Boocher (Universi
ty of San Diego) as part of Fellowship of the Ring\n\n\nAbstract\nGiven a
finitely-generated graded module over a polynomial ring\, there are many r
esults and conjectures concerning lower bounds for its betti numbers. Majo
r players in this story include the Syzygy Theorem\, the Buchsbaum-Eisenbu
d-Horrocks Rank Conjecture\, and the Total Rank Conjecture. In this talk
I'll survey these results and conjectures including a collection of intric
ately connected recent results that point to even stronger bounds.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Mantero (University of Arkansas)
DTSTART:20210422T203000Z
DTEND:20210422T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/47
DESCRIPTION:by Paolo Mantero (University of Arkansas) as part of Fellowshi
p of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Montaño (New Mexico State University)
DTSTART:20210429T203000Z
DTEND:20210429T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/48
DESCRIPTION:Title: When are multidegrees positive?\nby Jonathan Montaño (New Mexico
State University) as part of Fellowship of the Ring\n\n\nAbstract\nMultid
egrees of multiprojective varieties extend the notion of degree of project
ive varieties. These invariants can be defined via intersection theory\, o
r algebraically as the leading coefficients of multivariate Hilbert polyno
mials. It follows that multidegrees are nonnegative integers\, so a fundam
ental question is: When are multidegrees positive?\nIn the first part of t
he talk\, I will survey definitions and key properties of degrees and mult
idegrees\, including some examples. \n\nIn the second part\, I will pre
sent a complete characterization of the positivity of multidegrees\, and e
stablish a combinatorial description using convex geometry. I will also sh
ow applications of our result to mixed multiplicities of ideals and to the
support of Schubert polynomials. The talk is based on joint work with F.
Castillo\, Y. Cid-Ruiz\, B. Li\, and N. Zhang.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irena Swanson (Purdue University)
DTSTART:20210506T203000Z
DTEND:20210506T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/49
DESCRIPTION:Title: Numbers of associated primes of powers of ideals\nby Irena Swanso
n (Purdue University) as part of Fellowship of the Ring\n\n\nAbstract\nThi
s talk is about associated primes of powers of ideals in Noetherian commut
ative rings. By a result of Brodmann\, for any ideal $I$ in a ring $R$\,
the set of associated primes of $I^n$ stabilizes for large $n$. In genera
l\, the\nnumber of associated primes can go up or down as $n$ increases.
This talk is about sequences $\\{a_n\\}$ for which there exists an ideal $
I$ in a Noetherian commutative ring $R$ such that the number of associated
primes of $R/I^n$\nis $a_n$. A family of examples shows that $I$ may be
prime and the number of associated primes of $I^2$ need not be polynomial
in the dimension of the ring.\n\nThis is a report on four separate project
s with Sarah Weinstein\, Jesse Kim\, Robert Walker\, and ongoing work with
Roswitha Rissner.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karen Smith (University of Michigan)
DTSTART:20210513T203000Z
DTEND:20210513T220000Z
DTSTAMP:20260404T131152Z
UID:FOTR/50
DESCRIPTION:Title: Extremal singularities in prime characteristic\nby Karen Smith (U
niversity of Michigan) as part of Fellowship of the Ring\n\n\nAbstract\nWh
at is the most singular possible singularity? What can we say about it's g
eometric and algebraic properties? This seemingly naive question has a sen
sible answer in characteristic p. The "F-pure threshold\," which is an an
alog 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 smo
oth point (or more generally\, any F-pure point) but less than one in gene
ral\, with "more singular" points having smaller F-pure thresholds. We exp
lain a recently proved lower bound on the F-pure threshold in terms of th
e 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". Geometrica
lly\, these are the only cubic surfaces with the property that *every* tri
ple of coplanar lines on the surface meets in a single point (rather than
a "triangle" as expected)--a very extreme property indeed.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenliang Zhang (University of Illinois-Chicago)
DTSTART:20210914T200000Z
DTEND:20210914T213000Z
DTSTAMP:20260404T131152Z
UID:FOTR/51
DESCRIPTION:Title: Vanishing of local cohomology modules\nby Wenliang Zhang (Univers
ity of Illinois-Chicago) as part of Fellowship of the Ring\n\n\nAbstract\n
Studying the vanishing of local cohomology modules has a long and rich his
tory\, and is still an active research area. In this talk\, we will discus
s classic theorems (due to Grothendieck\, Hartshorne\, Peskine-Szpiro\, an
d Ogus)\, recent developments\, and some open problems.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Sam (University of California\, San Diego)
DTSTART:20210928T200000Z
DTEND:20210928T213000Z
DTSTAMP:20260404T131152Z
UID:FOTR/52
DESCRIPTION:by Steven Sam (University of California\, San Diego) as part o
f Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yairon Cid-Ruiz (Ghent University)
DTSTART:20211012T200000Z
DTEND:20211012T213000Z
DTSTAMP:20260404T131152Z
UID:FOTR/53
DESCRIPTION:by Yairon Cid-Ruiz (Ghent University) as part of Fellowship of
the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Pevtsova (University of Washington\, Seattle)
DTSTART:20211026T200000Z
DTEND:20211026T213000Z
DTSTAMP:20260404T131152Z
UID:FOTR/54
DESCRIPTION:by Julia Pevtsova (University of Washington\, Seattle) as part
of Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patricia Klein (University of Minnesota)
DTSTART:20211109T210000Z
DTEND:20211109T223000Z
DTSTAMP:20260404T131152Z
UID:FOTR/55
DESCRIPTION:by Patricia Klein (University of Minnesota) as part of Fellows
hip of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Hernández (University of Kansas)
DTSTART:20211207T210000Z
DTEND:20211207T223000Z
DTSTAMP:20260404T131152Z
UID:FOTR/56
DESCRIPTION:by Daniel Hernández (University of Kansas) as part of Fellows
hip of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Baidya (University of Tennessee)
DTSTART:20211130T210000Z
DTEND:20211130T223000Z
DTSTAMP:20260404T131152Z
UID:FOTR/58
DESCRIPTION:by Robin Baidya (University of Tennessee) as part of Fellowshi
p of the Ring\n\nAbstract: TBA\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kriti Goel (University of Utah)
DTSTART:20221107T210000Z
DTEND:20221107T223000Z
DTSTAMP:20260404T131152Z
UID:FOTR/60
DESCRIPTION:Title: Hilbert-Kunz function and Hilbert-Kunz multiplicity of ideals and Ree
s algebras\nby Kriti Goel (University of Utah) as part of Fellowship o
f the Ring\n\n\nAbstract\nHilbert-Kunz functions were introduced by E. Kun
z in 1969 in his work characterizing regular local rings in the prime char
acteristic setting. The existence of Hilbert-Kunz multiplicity was proved
later by P. Monsky in 1983. Since then\, Hilbert-Kunz functions and Hilber
t-Kunz multiplicities have been extensively studied\, partly because of th
eir connections with the theory of tight closure and their unpredictable b
ehaviour. Unlike the Hilbert-Samuel function\, the Hilbert-Kunz function n
eed not be a polynomial function.\n\nIn this talk\, we consider the Hilber
t-Kunz function of Rees algebra of ideals and show that\, in certain cases
\, it behaves as a quasi-polynomial\, a piece-wise polynomial\, or even a
polynomial. We also consider Hilbert-Kunz multiplicity of powers of an ide
al\, in an attempt to write it as a function of the power of the ideal. Th
is involves a surprising connection with the Hilbert-Samuel coefficients o
f Frobenius powers of an ideal.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Janet Page (North Dakota State Unviersity)
DTSTART:20221212T210000Z
DTEND:20221212T223000Z
DTSTAMP:20260404T131152Z
UID:FOTR/61
DESCRIPTION:Title: Extremal Surfaces in Positive Characteristic\nby Janet Page (Nort
h Dakota State Unviersity) as part of Fellowship of the Ring\n\n\nAbstract
\nWhat is the most singular possible point on any variety in positive char
acteristic? In recent joint work with Zhibek Kadyrsizova\, Jennifer Kenke
l\, Jyoti Singh\, Karen Smith\, Adela Vraciu\, and Emily Witt\, we gave a
precise answer to this question for cone points on hypersurfaces using a m
easure of singularity called the F-pure threshold\, and we called these
“most singular” hypersurfaces extremal hypersurfaces. In this talk\,
I’ll focus on the special case of extremal surfaces and discuss some of
their other surprising properties. This talk is based on joint work with
Anna Brosowsky\, Tim Ryan\, and Karen Smith.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Mastroeni (Iowa State University)
DTSTART:20230127T213000Z
DTEND:20230127T230000Z
DTSTAMP:20260404T131152Z
UID:FOTR/62
DESCRIPTION:Title: Chow rings of matroids are Koszul\nby Matt Mastroeni (Iowa State
University) as part of Fellowship of the Ring\n\n\nAbstract\nKoszul algebr
as have long been studied in connection with topology and representation t
heory for their exceptional homological and duality properties\, and they
appear with incredible frequency among rings at the intersection of commut
ative algebra\, algebraic geometry\, and combinatorics. The Chow ring of
a matroid is just such a ring - a commutative\, graded\, Artinian\, Gorens
tein algebra with linear and quadratic relations defined by the matroid\,
which recently played an important role in establishing a number of long-s
tanding conjectures on the combinatorics of matroids.\n\nIn this talk\, I
will discuss joint work with Jason McCullough affirmatively answering a co
njecture of Dotsenko that the Chow ring of any matroid is Koszul. Time pe
rmitting\, I will also mention some potential implications of this fact.
No prior experience with matroids is necessary for the talk.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jan Draisma (Universität Bern)
DTSTART:20230303T213000Z
DTEND:20230303T230000Z
DTSTAMP:20260404T131152Z
UID:FOTR/63
DESCRIPTION:Title: New instances of equivariant Noetherianity\nby Jan Draisma (Unive
rsität Bern) as part of Fellowship of the Ring\n\n\nAbstract\nWhen a grou
p or monoid G acts on a ring R by means of endomorphisms\, we say that R i
s G-Noetherian if every ascending chain of G-stable ideals in R is eventua
lly constant\; and we call R *topologically* G-Noetherian if this conditio
n holds at least for chains of G-stable radical ideals.\n\nOver the last 1
5 years\, many examples of (topologically) G-Noetherian rings have been di
scovered. I will first discuss some of the older results and their motivat
ion. Here G is usually the infinite symmetric group Sym or the infinite ge
neral linear group GL over an infinite field.\n\nAfter that\, I will turn
to recent joint work with Chiu-Danelon-Eggermont-Farooq on examples where
G=Sym x GL\; and with Blatter-Rupniewski on examples where G=GL over a fin
ite field.\n
LOCATION:https://stable.researchseminars.org/talk/FOTR/63/
END:VEVENT
END:VCALENDAR