BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Mike Stillman (Cornell University)
DTSTART;VALUE=DATE-TIME:20200423T203000Z
DTEND;VALUE=DATE-TIME:20200423T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/1
DESCRIPTION:Title: Qua
dratic Gorenstein rings and the Koszul property\nby Mike Stillman (Cor
nell 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\, an
d the top betti number is one. R is called Koszul\nif the free resolution
of k = R/(maximal homogeneous ideal) over R is\nlinear. Any Koszul algebra
is defined by quadratic relations\, but the\nconverse is false\, and no o
ne knows a finitely computable criterion.\nBoth types of rings have dualit
y properties\, and occur in many\nsituations in algebraic geometry and com
mutative algebra\, and in many\ncases\, a Gorenstein quadratic algebra com
ing from geometry is often\nKoszul (e.g. homogeneous coordinate rings of m
ost canonical curves).\n\nIn 2001\, Conca\, Rossi\, and Valla asked the qu
estion: must a (graded)\nquadratic Gorenstein algebra of regularity 3 be K
oszul?\n\nIn the first 45 minutes\, we will define these notions\, and giv
e\nexamples of quadratic Gorenstein algebras and Koszul algebras. We\nwill
give methods for their construction\, e.g. via inverse systems.\nAfter a
short break\, we will use these techniques to answer negatively\nthe above
question\, as well as see how to construct many other\nexamples of quadra
tic Gorenstein algebras which are not Koszul.\n
LOCATION:https://researchseminars.org/talk/FOTR/1/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christine Berkesch (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20200430T203000Z
DTEND;VALUE=DATE-TIME:20200430T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/2
DESCRIPTION:Title: The
geometry of toric syzygies\nby Christine Berkesch (University of Minn
esota) as part of Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Jeffries (CIMAT)
DTSTART;VALUE=DATE-TIME:20200507T203000Z
DTEND;VALUE=DATE-TIME:20200507T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/3
DESCRIPTION:Title: Two
applications of $p$-derivations to commutative algebra\nby Jack Jeffr
ies (CIMAT) as part of Fellowship of the Ring\n\n\nAbstract\nThe notions o
f derivations and modules of differentials have been central in commutativ
e algebra for much of its history. A somewhat more exotic notion is that o
f $p$-derivations: these are maps that satisfy functional equations simila
rly to derivations\, but are not even additive. Nonetheless\, $$-derivatio
ns and related constructions have found applications in arithmetic geometr
y. In this talk\, we will give a basic introduction to p-derivations\, and
discuss two applications to commutative algebra\, based on projects with
Melvin Hochster and Anurag K. Singh (each).\n
LOCATION:https://researchseminars.org/talk/FOTR/3/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Erman (University of Wisconsin)
DTSTART;VALUE=DATE-TIME:20200514T203000Z
DTEND;VALUE=DATE-TIME:20200514T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/4
DESCRIPTION:Title: Bou
ndedness questions for polynomials in many variables\nby Dan Erman (Un
iversity of Wisconsin) as part of Fellowship of the Ring\n\n\nAbstract\nI
will discuss questions and results related to polynomials in a large numbe
r of variables\, starting with classical results of Hilbert and moving to
Stillman's conjecture and its proof by Ananyan and Hochster. Then I will d
escribe ways to think about the limit of polynomial rings as the number of
variables goes to infinity\, and how this can be applied to obtain new fi
niteness results. The original work covered in this talk is all joint with
Steven V Sam and Andrew Snowden.\n
LOCATION:https://researchseminars.org/talk/FOTR/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudiu Raicu (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20200521T203000Z
DTEND;VALUE=DATE-TIME:20200521T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/5
DESCRIPTION:Title: Com
mutative algebra with $S_n$-invariant monomial ideals\nby Claudiu Raic
u (University of Notre Dame) as part of Fellowship of the Ring\n\n\nAbstra
ct\nConsider a polynomial ring in $n$ variables\, together with the action
of the symmetric group by coordinate permutations. In my talk I will desc
ribe many familiar notions in Commutative Algebra in the context of monomi
al ideals that are preserved by the action of the symmetric group. These i
nclude Castelnuovo-Mumford regularity\, projective dimension\, saturation\
, symbolic powers\, or the Cohen-Macaulay property. My goal is to explain
how changing focus from minimal resolutions to Ext modules can lead to a s
implified picture of the homological algebra\, and to provide concrete com
binatorial recipes to determine the relevant homological invariants.\n
LOCATION:https://researchseminars.org/talk/FOTR/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eloísa Grifo (University of California\, Riverside)
DTSTART;VALUE=DATE-TIME:20200528T203000Z
DTEND;VALUE=DATE-TIME:20200528T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/6
DESCRIPTION:Title: Sym
bolic powers\, stable containments\, and degree bounds\nby Eloísa Gri
fo (University of California\, Riverside) as part of Fellowship of the Rin
g\n\n\nAbstract\nWhat's the smallest degree of a homogeneous polynomial th
at vanishes to order n on a given finite set of points\, or more generally
on some algebraic variety in projective space? A classical result of Zari
ski and Nagata tells us the set of such polynomials is the nth symbolic po
wer of the ideal I corresponding to our variety. To bound degrees of eleme
nts in the symbolic powers of I\, we can look for containments between sym
bolic powers and other better understood ideals\, such as powers of I. We
will take a tour through the history of the containment problem and some o
f its variations\, with an eye towards lower bounds for degrees of symboli
c powers. Our story will include joint work with Craig Huneke and Vivek Mu
kundan\, and with Sankhaneel Bisui\, Tài Huy Hà\, and Thái Thành Nguy
ễn.\n
LOCATION:https://researchseminars.org/talk/FOTR/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason McCullough (Iowa state)
DTSTART;VALUE=DATE-TIME:20200604T203000Z
DTEND;VALUE=DATE-TIME:20200604T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/7
DESCRIPTION:Title: Sub
additivity of syzygies and related problems\nby Jason McCullough (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 $S$-ideal.
There are many interesting questions about the maximal graded shifts of $S
/I$\, denoted $t_i$. In the first part of my talk\, I will discuss two cl
assical constructions that turn a (graded) S-module into an ideal with sim
ilar properties\, namely idealizations and Bourbaki ideals\, and what they
say about maximal graded shifts of ideals. In the second part of the tal
k\, I will discuss restrictions on maximal graded shifts of ideals. In pa
rticular\, an ideal $I$ is said to satisfy the subadditivity condition if
$t_a + t_b ≥ t_(a+b)$ for all $a\,b$. This condition fails for arbitrar
y\, even Cohen-Macaulay\, ideals but is open for certain nice classes of i
deals\, such as Koszul and monomial ideals. I will present a construction
(joint with A. Seceleanu) showing that subadditivity can fail for Gorenst
ein ideals. \n\nIf time allows\, I will talk about some results that hold
more generally\, including a linear bound on the maximal graded shifts in
terms of the first $p-c$ shifts\, where $p = pd(S/I)$ and $c = codim(I)$.
I hope to include several examples and open questions as well.\n
LOCATION:https://researchseminars.org/talk/FOTR/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Claudia Polini (University of Notre Dame)
DTSTART;VALUE=DATE-TIME:20200618T203000Z
DTEND;VALUE=DATE-TIME:20200618T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/8
DESCRIPTION:Title: Ree
s algebras of ideals generated by 2x2 minors\nby Claudia Polini (Unive
rsity of Notre Dame) as part of Fellowship of the Ring\n\n\nAbstract\nRees
algebras of ideals of maximal minors of generic matrices are very well un
derstood. In this\ntalk we deal with the case of sparse matrices with two
rows and with the case of two by two \nminors of generic matrices. We inve
stigate the defining ideals of these algebras\, and in the first \ncase we
prove that they are Koszul and have rational singularities or are F-ratio
nal\, respectively. \nThis is a report on joint work with Ela Celikbas\, E
milie Dufresne\, Louiza Fouli\, Elisa Gorla\, Kuei-Nuan \nLin\, and Irena
Swanson and with Hang Huang\, Michael Perlman\, Claudiu Raicu\, and Alessi
o Sammartano.\n
LOCATION:https://researchseminars.org/talk/FOTR/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matteo Varbaro (University of Genoa\, Italy)
DTSTART;VALUE=DATE-TIME:20200611T203000Z
DTEND;VALUE=DATE-TIME:20200611T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
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. H
ochster-Huneke graph) G(R) of a Noetherian ring R of dimension d is the fi
nite simple graph whose vertices correspond to the minimal primes of 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 Grot
hendieck\, Hartshorne\, and Hochster-Huneke\, concerning the connectedness
of G(R). We will also see\, given a finite simple graph G\, how to constr
uct a Noetherian ring R such that G(R)=R.\n\nIn the second part of the tal
k\, we will discuss some recent developments related to the following two
questions:\n1) How many paths are there between two minimal primes of R?\n
2) What is the shortest path between two minimal primes of R?\nBy taking t
he 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 connectivity
and its diameter. Most of the things that I will discuss are contained in
works written together with Bruno Benedetti\, Barbara Bolognese and Miche
la Di Marca.\n
LOCATION:https://researchseminars.org/talk/FOTR/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hailong Dao (University of Kansas)
DTSTART;VALUE=DATE-TIME:20200625T203000Z
DTEND;VALUE=DATE-TIME:20200625T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/10
DESCRIPTION:Title: A
truly mutually beneficial friendship: how Stanley-Reisner theory enhanced
both combinatorics and algebra\nby Hailong Dao (University of Kansas)
as part of Fellowship of the Ring\n\n\nAbstract\nGiven a simplicial comple
x on n vertices\, one can associate to it a quotient of the polynomial rin
g in n variables\, called the Stanley-Reisner ring. Starting with the proo
f of the Upper Bound Conjecture for spheres\, this approach has been spect
acularly useful in bringing tools from commutative algebra to the study of
simplicial complexes. In the first part of the talk I will sketch some re
levant parts of this story. In the second\, I will describe how modern too
ls\, including cohomological vanishing results and characteristic p method
s\, have inspired new developments. At the same time\, results obtained on
the combinatorics side now can be brought back to induce interesting 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://researchseminars.org/talk/FOTR/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Polstra (University of Utah)
DTSTART;VALUE=DATE-TIME:20200723T203000Z
DTEND;VALUE=DATE-TIME:20200723T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/11
DESCRIPTION:Title: Th
e weak implies strong conjecture and finite generation of symbolic Rees al
gebras\nby Thomas Polstra (University of Utah) as part of Fellowship o
f the Ring\n\n\nAbstract\nTight closure theory is prominent to prime chara
cteristic commutative algebra. Historically\, tight closure has been used
to simplify proofs of landmark theorems in commutative algebra\, provide t
ests to determine when an element of a ring belongs to a particular ideal\
, and provides proofs of results in prime characteristic which otherwise c
an only be proved in equal characteristic 0. The first half of the talk wi
ll be devoted to basic definitions\, properties\, and consequences of tigh
t closure. The second half of the talk we will go over several important c
lasses of rings defined via tight closure\, recent progress on the weak im
plies strong conjecture appearing in a joint paper with Ian Aberbach\, and
a conjecture that certain symbolic Rees algebras are Noetherian.\n
LOCATION:https://researchseminars.org/talk/FOTR/11/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bhargav Bhatt (University of Michigan)
DTSTART;VALUE=DATE-TIME:20200827T203000Z
DTEND;VALUE=DATE-TIME:20200827T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/12
DESCRIPTION:Title: Co
hen-Macaulayness of absolute integral closures\nby Bhargav Bhatt (Univ
ersity of Michigan) as part of Fellowship of the Ring\n\n\nAbstract\nA dee
p theorem of Hochster-Huneke in F-singularity theory is that the absolute
integral closure of an excellent noetherian local domain R over F_p is Coh
en-Macaulay. In other words\, all relations on a system of parameters 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 som
e consequences.\n
LOCATION:https://researchseminars.org/talk/FOTR/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Seceleanu (University of Nebraska)
DTSTART;VALUE=DATE-TIME:20200716T203000Z
DTEND;VALUE=DATE-TIME:20200716T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/13
DESCRIPTION:Title: Re
flection arrangements\, syzygies\, and the containment problem\nby Ale
xandra Seceleanu (University of Nebraska) as part of Fellowship of the Rin
g\n\n\nAbstract\nInvariant theory\, that is the art of finding polynomials
invariant under the action of a given group\, has played a major role in
the historical development of commutative algebra. In this theory reflecti
on groups are singled out for having rings of invariants that are isomorph
ic to polynomial rings. From a geometric perspective\, reflection groups g
ive rise to beautiful and very symmetric arrangements of hyperplanes terme
d reflection arrangements.\n\nThis talk will take a close look at the idea
ls defining the singular loci of reflection arrangements\, which are in tu
rn symmetric subspace arrangements. We describe their syzygies in terms of
invariant polynomials for the relevant reflection groups. We leverage thi
s information to settle many aspects of the containment problem asking for
containments between the ordinary and the symbolic powers of the ideals i
n this family. This talk is based on joint work with Ben Drabkin.\n
LOCATION:https://researchseminars.org/talk/FOTR/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Briggs (University of Utah)
DTSTART;VALUE=DATE-TIME:20200806T203000Z
DTEND;VALUE=DATE-TIME:20200806T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/14
DESCRIPTION:Title: Th
e homotopy Lie algebra and the conormal module\nby Benjamin Briggs (Un
iversity of Utah) as part of Fellowship of the Ring\n\n\nAbstract\nI 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 Vasconcel
os\, after he and (separately) Ferrand established the case that the conor
mal module is free.\n\nThe key tool is the homotopy Lie algebra. This is a
graded Lie algebra naturally associated with any local homomorphism. It s
its at the centre of a longstanding friendship between commutative algebra
and rational homotopy theory\, through which ideas and results have been
passed back and forth for decades.\n\nI'll go through the construction of
the homotopy Lie algebra and how it's been used in commutative algebra in
the past\, before explaining how its structure detects when the conormal m
odule has finite projective dimension. I'll also talk about ongoing work w
ith Srikanth Iyengar comparing the cotangent complex with the homotopy Lie
algebra.\n
LOCATION:https://researchseminars.org/talk/FOTR/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eleonore Faber (University of Leeds)
DTSTART;VALUE=DATE-TIME:20200820T203000Z
DTEND;VALUE=DATE-TIME:20200820T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/15
DESCRIPTION:Title: Gr
assmannian categories of infinite rank and countable Cohen-Macaulay type\nby Eleonore Faber (University of Leeds) as part of Fellowship of the R
ing\n\n\nAbstract\nThis talk is about a categorification of the coordinate
rings of Grassmannians of infinite rank in terms of graded maximal Cohen-
Macaulay modules over a hypersurface singularity. This gives an infinite r
ank analogue of the Grassmannian cluster categories introduced by Jensen\,
King\, and Su. In a special case\, when the hypersurface singularity is a
curve of countable Cohen-Macaulay type\, our category has a combinatorial
model by an ``infinity-gon'' and we can determine triangulations of 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 Sibyll
e Schroll.\n
LOCATION:https://researchseminars.org/talk/FOTR/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aldo Conca (University of Genoa)
DTSTART;VALUE=DATE-TIME:20200903T190000Z
DTEND;VALUE=DATE-TIME:20200903T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/16
DESCRIPTION:Title: Id
eals associated to subspace arrangements\nby Aldo Conca (University 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 of linea
r forms has a linear resolution. Only recently Tsakiris and I managed to
describe such a resolution. It is supported on a polymatroid naturally a
ttached to the associated subspace arrangements.\n
LOCATION:https://researchseminars.org/talk/FOTR/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Juliette Bruce (Syracuse University)
DTSTART;VALUE=DATE-TIME:20200910T190000Z
DTEND;VALUE=DATE-TIME:20200910T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/17
DESCRIPTION:Title: Sy
zygies of Products of Projective Space\nby Juliette Bruce (Syracuse Un
iversity) as part of Fellowship of the Ring\n\n\nAbstract\nI will discuss
the asymptotic non-vanishing of syzygies for products of projective spaces
\, generalizing the monomial methods of Ein-Erman-Lazarsfeld. This provide
s the first example of how the asymptotic syzygies of a smooth projective
variety whose embedding line bundle grows in a semi-ample fashion behave i
n nuanced and previously unseen ways.\n
LOCATION:https://researchseminars.org/talk/FOTR/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dale Cutkosky (University of Missouri)
DTSTART;VALUE=DATE-TIME:20200702T203000Z
DTEND;VALUE=DATE-TIME:20200702T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/18
DESCRIPTION:Title: Mi
xed multiplicities of filtrations\nby Dale Cutkosky (University of Mis
souri) as part of Fellowship of the Ring\n\n\nAbstract\nWe discuss the the
ory of multiplicities and mixed multiplicities of filtrations of m-primary
ideals. We show that many classical formulas are true in this setting. We
also consider the case of equality in Minkowski's inequality. We give som
e general theorems characterizing when this condition holds\, giving gener
alizations of classical theorems of Rees\, Sharp\, Teissier\, Katz and oth
ers.\n
LOCATION:https://researchseminars.org/talk/FOTR/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Núñez-Betancourt (CIMAT\, Guanajuato)
DTSTART;VALUE=DATE-TIME:20200709T203000Z
DTEND;VALUE=DATE-TIME:20200709T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/19
DESCRIPTION:Title: Di
fferential powers of ideals\nby Luis Núñez-Betancourt (CIMAT\, Guana
juato) as part of Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brooke Ullery (Emory University)
DTSTART;VALUE=DATE-TIME:20200813T203000Z
DTEND;VALUE=DATE-TIME:20200813T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/20
DESCRIPTION:Title: Ca
yley-Bacharach theorems and measures of irrationality\nby Brooke Uller
y (Emory University) as part of Fellowship of the Ring\n\n\nAbstract\nIf Z
is a set of points in projective space\, we can ask which polynomials of
degree d vanish at every point in Z. If P is one point of Z\, the vanishin
g of a polynomial at P imposes one linear condition on the coefficients. T
hus\, the vanishing of a polynomial on all of Z imposes |Z| linear conditi
ons on the coefficients. A classical question in algebraic geometry\, dati
ng back to at least the 4th century\, is how many of those linear conditio
ns are independent? For instance\, if we look at the space of lines throug
h 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-Bacharach T
heorem and Castelnuovo’s Lemma about points on rational curves. I’ll t
hen describe some recent results and conjectures about points satisfying t
he so-called Cayley-Bacharach condition and show how they connect to sever
al seemingly unrelated questions in contemporary algebraic geometry relati
ng to the gonality of curves and measures of irrationality of higher dimen
sional varieties.\n
LOCATION:https://researchseminars.org/talk/FOTR/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Federico Ardila (San Francisco State University)
DTSTART;VALUE=DATE-TIME:20200730T203000Z
DTEND;VALUE=DATE-TIME:20200730T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/21
DESCRIPTION:Title: La
grangian Geometry of Matroids\nby Federico Ardila (San Francisco State
University) as part of Fellowship of the Ring\n\n\nAbstract\nMatroid theo
ry had its origins in linear algebra and graph theory. In recent years\, t
he geometric roots of the field have grown much deeper\, bearing many new
fruits. The interplay between matroid theory and algebraic geometry has op
ened up interesting research directions at the intersection of combinatori
cs\, algebra\, and geometry\, and led to the solution of long-standing que
stions. \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 Ch
ow ring\, we find simple interpretations of the Chern-Schwartz-MacPherson
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 prov
e Brylawski's and Dawson's conjectures that the h-vector of a matroid is l
og-concave.\n\nI will make the talk as self-contained as possible\, and as
sume no previous knowledge of matroid theory.\n
LOCATION:https://researchseminars.org/talk/FOTR/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Andrew Snowden (University of Michigan)
DTSTART;VALUE=DATE-TIME:20200917T190000Z
DTEND;VALUE=DATE-TIME:20200917T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/22
DESCRIPTION:Title: In
finite dimensional equivariant commutative algebra\nby Andrew Snowden
(University of Michigan) as part of Fellowship of the Ring\n\n\nAbstract\n
Certain infinite variable polynomial rings equipped with actions of large
groups (like the infinite symmetric group or the infinite general linear g
roup) behave in many ways like finitely generated algebras\; for instance\
, one sometimes has an "equivariant noetherian" property. I will discuss s
ome deeper parallels with commutative algebra\, and how these somewhat exo
tic objects can be applied to study classical questions.\n
LOCATION:https://researchseminars.org/talk/FOTR/22/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Cris Negron (University of North Carolina)
DTSTART;VALUE=DATE-TIME:20200924T190000Z
DTEND;VALUE=DATE-TIME:20200924T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/23
DESCRIPTION:Title: No
ncommutative hypersurfaces and support theory for Hopf algebras\nby Cr
is Negron (University of North Carolina) as part of Fellowship of the Ring
\n\n\nAbstract\nI will talk about a new approach to support theory for non
commutative (Hopf) algebras which mirrors Avramov and Buchweitz’ support
theory for commutative local complete intersections. I will explain what
this support theory entails for ``noncommutative complete intersections"\
, and relevant examples\ncoming from quantum linear spaces\, functions on
finite group schemes\, and quantum groups. I will also explain how this s
upport theory is used to classify thick ideals in the associated stable re
presentation categories. No familiarity with these topics is assumed\, an
d everything in the talk should be explained in relatively basic terms. T
his is joint work with Julia Pevtsova.\n
LOCATION:https://researchseminars.org/talk/FOTR/23/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Diane Maclagan (University of Warwick)
DTSTART;VALUE=DATE-TIME:20201001T190000Z
DTEND;VALUE=DATE-TIME:20201001T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/24
DESCRIPTION:Title: Tr
opical ideals\nby Diane Maclagan (University of Warwick) as part of Fe
llowship of the Ring\n\n\nAbstract\nOne consequence of the recent push to
develop a scheme theory in tropical geometry has been the development of a
tropical commutative algebra. This starts with the commutative algebra o
f semirings\, but in order to get a theory that interacts with geometry\,
we are lead to impose some combinatorial\, matroid-theoretic\, conditions.
I will introduce these ideas\, and discuss the current\nstate of our und
erstanding.\n
LOCATION:https://researchseminars.org/talk/FOTR/24/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Takumi Murayama (Princeton University)
DTSTART;VALUE=DATE-TIME:20201008T190000Z
DTEND;VALUE=DATE-TIME:20201008T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/25
DESCRIPTION:Title: Gr
othendieck's localization problem\nby Takumi Murayama (Princeton Unive
rsity) 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 resol
ution of singularities Grothendieck and Dieudonné showed that if the clos
ed fiber of the map $A\\to B$ is Cohen-Macaulay and if $A$ is of equal cha
racteristic zero\, then all the fibers of the map are Cohen-Macaulay. Thre
e decades later\, Avramov and Foxby showed that the same statement holds w
ithout the characteristic assumption on A. Grothendieck's localization pro
blem asks whether a similar statement holds with Cohen-Macaulayness replac
ed by other local properties of noetherian local rings. We solve Grothendi
eck's localization problem for all sufficiently well-behaved properties of
noetherian local rings. Our proof provides a uniform treatment of previou
sly known special cases of Grothendieck's problem\, in particular giving a
new proof of Avramov and Foxby's result. As an application\, we show that
if the closed fibers of a flat morphism of algebraic varieties are smooth
\, then all fibers are smooth.\n
LOCATION:https://researchseminars.org/talk/FOTR/25/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jenna Rajchgot (McMaster University)
DTSTART;VALUE=DATE-TIME:20201015T190000Z
DTEND;VALUE=DATE-TIME:20201015T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/26
DESCRIPTION:Title: Ge
ometric vertex decomposition and liaison\nby Jenna Rajchgot (McMaster
University) as part of Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/26/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gennady Lyubeznik (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20201022T190000Z
DTEND;VALUE=DATE-TIME:20201022T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/27
DESCRIPTION:Title: A
characteristic-free definition of holonomic D-modules\nby Gennady Lyub
eznik (University of Minnesota) as part of Fellowship of the Ring\n\n\nAbs
tract\nMost of the theory of D-modules has been developed only in characte
ristic zero. This includes holonomic modules. Some candidates for holonomi
c modules in characteristic p>0 have been proposed using definitions speci
fic to characteristic p>0. The first characteristic-free definition of hol
onomicity was given in 2010 by the speaker\, but only for modules over pol
ynomial rings. In the talk I am going to describe an extension of this def
inition to arbitrary non-singular varieties.\n
LOCATION:https://researchseminars.org/talk/FOTR/27/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alessandro de Stefani (University of Genoa)
DTSTART;VALUE=DATE-TIME:20201029T190000Z
DTEND;VALUE=DATE-TIME:20201029T203000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/28
DESCRIPTION:Title: De
formation and stability of F-singularities\nby Alessandro de Stefani (
University of Genoa) as part of Fellowship of the Ring\n\n\nAbstract\nGive
n 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-singular
ities in prime characteristic\, and present some recent results on deforma
tion of F-injectivity\, obtained in joint work with Linquan Ma. In the sec
ond part of the talk\, we will discuss another problem\, called stability.
We will present some results obtained in collaboration with Ilya Smirnov\
, and outline a general relation between the two problems of deformation a
nd stability.\n
LOCATION:https://researchseminars.org/talk/FOTR/28/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Henry Schenck (Auburn University)
DTSTART;VALUE=DATE-TIME:20201105T200000Z
DTEND;VALUE=DATE-TIME:20201105T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/29
DESCRIPTION:Title: Ca
labi-Yau threefolds in P^n and Gorenstein rings\nby Henry Schenck (Aub
urn University) as part of Fellowship of the Ring\n\n\nAbstract\nA project
ively normal Calabi-Yau threefold $X \\subseteq \\mathbb{P}^n$ has an idea
l $I_X$ which is arithmetically Gorenstein\, of Castelnuovo-Mumford regula
rity four. Such ideals have been intensively studied when $I_X$ is a compl
ete 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 gi
ven by the Pfaffians of a skew-symmetric matrix. A number of recent papers
study the situation when $I_X$ has codimension four. We prove there are 1
6 possible betti tables for an arithmetically Gorenstein ideal I with codi
m(I) = 4 = regularity(I)\, and that 9 of these arise for prime nondegenera
te threefolds. We investigate the situation in codimension five or more\,
obtaining examples of X with $h^{p\,q}(X)$ not among those appearing for $
I_X$ of lower codimension or as complete intersections in toric Fano varie
ties--in other words\, Calabi-Yau's with Hodge numbers not previously know
n to occur. A main feature of our approach is the use of inverse systems t
o identify possible betti tables for X. This is joint work with M. Stillma
n and B. Yuan\n
LOCATION:https://researchseminars.org/talk/FOTR/29/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Michael Brown (Auburn University)
DTSTART;VALUE=DATE-TIME:20201119T200000Z
DTEND;VALUE=DATE-TIME:20201119T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
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 work with
David Eisenbud\, Daniel Erman\, and Frank-Olaf Schreyer. The Bernstein-Gel
'fand-Gel'fand (BGG) correspondence is a derived equivalence between a sta
ndard graded polynomial ring and its Koszul dual exterior algebra. One of
the many important applications of the BGG correspondence is an algorithm\
, due to Eisenbud-Fløystad-Schreyer\, for computing sheaf cohomology on p
rojective space that is\, in some cases\, the fastest available. The goal
of this talk is to discuss a generalization of the BGG correspondence from
standard graded to multigraded polynomial rings and how it leads to an Ei
senbud-Fløystad-Schreyer-type algorithm for computing sheaf cohomology ov
er certain projective toric varieties.\n
LOCATION:https://researchseminars.org/talk/FOTR/31/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mircea Mustata (University of Michigan)
DTSTART;VALUE=DATE-TIME:20201112T200000Z
DTEND;VALUE=DATE-TIME:20201112T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/32
DESCRIPTION:Title: Mi
nimal exponents of hypersurfaces and a conjecture of Teissier\nby Mirc
ea Mustata (University of Michigan) as part of Fellowship of the Ring\n\n\
nAbstract\nThe minimal exponent of a hypersurface is an invariant of singu
larities defined via the Bernstein-Sato polynomial. It is a refinement of
the log canonical threshold (a fundamental invariant in birational geomet
ry)\, that can be used to measure rational singularities. In the first pa
rt of the talk I will give an introduction to these and related invariants
. The second part of the talk will describe joint work with Eva Elduque an
d Bradley Dirks on a conjecture of Teissier\, relating the minimal exponen
t of a hypersurface with that of a hyperplane section.\n
LOCATION:https://researchseminars.org/talk/FOTR/32/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Grigoriy Blekherman (Georgia Institute of Technology)
DTSTART;VALUE=DATE-TIME:20210114T213000Z
DTEND;VALUE=DATE-TIME:20210114T230000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/33
DESCRIPTION:Title: Su
ms of Squares: From Real to Commutative Algebra\nby Grigoriy Blekherma
n (Georgia Institute of Technology) as part of Fellowship of the Ring\n\n\
nAbstract\nA real polynomial is called nonnegative if it takes only nonneg
ative values. A sum of squares or real polynomials is clearly nonnegative.
The relationship between nonnegative polynomials and sums of squares is o
ne of the central questions in real algebraic geometry. A modern approach
is to look at nonnegative polynomials and sums of squares on a real variet
y X\, where unexpected links to complex algebraic geometry and commutative
algebra appear.\n\nIn the first half of the talk I will review the histor
y of the problem\, do some examples\, and provide a brief overview of the
results. Our two guiding questions will be: the relationship between nonne
gative polynomials and sums of squares\, and the number of squares needed
to write any sum of squares on X. I will explain the connection between th
ese questions and properties of the free resolution of the ideal of X: the
number of of steps that the resolution only has linear syzygies (property
$N_{2\,p}$) and the number of steps that linear syzygies persist (the len
gth of the linear strand).\n\nIn the second half\, I will concentrate on t
he number of squares\, and introduce an invariant of X we call quadratic p
ersistence. Quadratic persistence of X is equal to the least number of poi
nts in X such that after projecting from (the span of) these points the id
eal of the resulting variety has no quadrics. I will explain how quadratic
persistence connects real algebraic geometry and commutative algebra. Joi
nt work with Rainer Sinn\, Greg Smith and Mauricio Velasco.\n
LOCATION:https://researchseminars.org/talk/FOTR/33/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shunsuki Takagi (University of Tokyo)
DTSTART;VALUE=DATE-TIME:20210122T010000Z
DTEND;VALUE=DATE-TIME:20210122T023000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/34
DESCRIPTION:Title: Ar
ithmetic deformations of F-singularities\nby Shunsuki Takagi (Universi
ty of Tokyo) as part of Fellowship of the Ring\n\n\nAbstract\nF-singularit
ies are singularities in positive characteristic defined via the Frobenius
map. In the first half of the talk\, I will survey a connection between F
-singularities and singularities in complex birational geometry. In the se
cond half of the talk\, I will present a new application of Ma-Schwede’s
theory of singularities in mixed characteristic. They proved that a Q-Gor
enstein affine domain over a field of characteristic zero has log terminal
singularities if its mod p reduction is F-regular for one single prime p.
I will discuss the analog of their result for log canonical singularities
. This talk is based on joint work with Kenta Sato.\n
LOCATION:https://researchseminars.org/talk/FOTR/34/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ilya Smirnov (Stockholm University)
DTSTART;VALUE=DATE-TIME:20210128T210000Z
DTEND;VALUE=DATE-TIME:20210128T223000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/35
DESCRIPTION:Title: Th
e quest for F-rational signature\nby Ilya Smirnov (Stockholm Universit
y) as part of Fellowship of the Ring\n\n\nAbstract\nStrongly F-regular sin
gularities are one of the fundamental classes of singularities defined by
the properties of Frobenius endomorphism. This class of mild singularities
can be detected using F-signature\, an invariant of a local ring with man
y good properties. Through this connection we obtain a powerful tool for s
tudying strongly F-regular singularities\, for example\, several results o
n "mildness" of F-regular can be quantified using F-signature. \n\nAnothe
r fundamental class of singularities in positive characteristic are F-rati
onal singularities. While generally more severe\, this class of singularit
ies is in many aspects analogous to strongly F-regular singularities and c
an be even understood by "moving" the definition of F-regularity to take p
lace in the dualizing module. Naturally\, there has been interest in adapt
ing the definition of F-signature to work with F-rational singularities. \
n\nWhile there is no complete solution yet\, I am convinced that such a th
eory should exist. As my evidence\, I will present results of a joint work
with Kevin Tucker\, and prior works of Hochster and Yao\, and Sannai.\n\n
My talk will be self-contained. I will discuss all necessary background\,
such as definitions\, properties\, and relations between these notions\, i
n the first half and then proceed to more technical results in the second
half.\n
LOCATION:https://researchseminars.org/talk/FOTR/35/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Elisa Gorla (University of Neuchatel)
DTSTART;VALUE=DATE-TIME:20210204T213000Z
DTEND;VALUE=DATE-TIME:20210204T230000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/36
DESCRIPTION:Title: Id
eals with a radical generic initial ideal\nby Elisa Gorla (University
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 ideal\, c
alled the initial ideal of I. The initial ideal of I depends not only on t
he choice of a term order\, but also on the system of coordinates. Neverth
eless\, many properties of I can be inferred from those of its initial ide
al(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 gen
eric 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 the term
order and of the system of coordinates. These are exactly the multigraded
homogeneous ideals which have a radical generic initial ideal. Multigraded
ideals which have a radical generic initial ideal show interesting rigidi
ty properties\, which e.g. allow us to deduce information on their univers
al Groebner bases. In the talk\, I will present examples of ideals which b
elong to this class and of what we can deduce about them using this machin
ery. The original work in the talk is joint with Aldo Conca and Emanuela D
e Negri.\n
LOCATION:https://researchseminars.org/talk/FOTR/36/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Liana Sega (University of Missouri-Kansas City)
DTSTART;VALUE=DATE-TIME:20210211T220000Z
DTEND;VALUE=DATE-TIME:20210211T233000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/37
DESCRIPTION:Title: Si
mplicial resolutions of powers of square-free monomial ideals\nby Lian
a Sega (University of Missouri-Kansas City) as part of Fellowship of the R
ing\n\n\nAbstract\nA free resolution of an ideal I generated by p monomial
s can be described using the simplicial chain maps of a simplex on p verti
ces. This resolution is called the Taylor resolution of the ideal and was
constructed by Diana Taylor in her thesis (1966). Work of Bayer\, Peeva an
d Sturmfels further established a criterion for deciding whether the simpl
icial chain maps of a smaller simplicial complex on p vertices describes
a free resolution of I\, and since then there has been a significant amoun
t of work on trimming down the Taylor resolution to simplicial resolutions
that are minimal (for certain classes of ideals) or closer to being minim
al. In this talk we will discuss a class of simplicial complexes\, indexe
d by the positive integers\, where the r-th complex in this class supports
a resolution of the r-th power of I^r\, where I is a square-free monomial
ideal.\n \nThis work is joint with Susan Cooper\, Sabine El Khoury\, Sara
Faridi\, Sara Mayes-Tang\, Susan Morey\, and Sandra Spiroff\, and was sta
rted at the "Women in Commutative Algebra" workshop in Banff\, 2019.\n
LOCATION:https://researchseminars.org/talk/FOTR/37/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gregory G. Smith (Queen's University)
DTSTART;VALUE=DATE-TIME:20210218T213000Z
DTEND;VALUE=DATE-TIME:20210218T230000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/38
DESCRIPTION:Title: Sm
ooth Hilbert schemes\nby Gregory G. Smith (Queen's University) as part
of Fellowship of the Ring\n\n\nAbstract\nHow can we understand all satura
ted homogeneous ideals in a polynomial ring?\nHilbert schemes provide a ge
ometric answer to this question. After surveying\nthe key features of the
se natural parameter spaces\, we will present a complete\ncombinatorial cl
assification of the smooth Hilbert schemes. We will also\nreinterpret the
smooth Hilbert schemes as suitable generalizations of partial\nflag varie
ties. This talk is based on joint work with Roy Skjelnes (KTH).\n
LOCATION:https://researchseminars.org/talk/FOTR/38/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Satoshi Murai (Waseda University)
DTSTART;VALUE=DATE-TIME:20210325T220000Z
DTEND;VALUE=DATE-TIME:20210325T233000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/39
DESCRIPTION:Title: Be
tti numbers of monomial ideals fixed by permutations of the variables\
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 to R_n.
In this talk\, I will mainly talk about the following problem: Fix monomia
ls 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 when n inc
reases?\nI will explain that there is a simple way to determine non-zero p
ositions of the Betti table of I_n when n is sufficiently large. I also ex
plain 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 motivated by
recent studies of algebraic properties of S_n-invariant ideals and is insp
ired by studies of Noetherianity up to symmetry. I will explain this motiv
ation and basic combinatorial properties of S_n-invariant ideals in the fi
rst part of the talk.\nThis talk includes a joint work with Claudiu Raicu.
\n
LOCATION:https://researchseminars.org/talk/FOTR/39/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Mark Walker (University of Nebraska)
DTSTART;VALUE=DATE-TIME:20210225T213000Z
DTEND;VALUE=DATE-TIME:20210225T230000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/40
DESCRIPTION:Title: Ho
w short can a module of finite projective dimension be?\nby Mark Walke
r (University of Nebraska) as part of Fellowship of the Ring\n\n\nAbstract
\nThis is joint work with Srikanth Iyengar and Linquan Ma. I will discuss
the question:\n\nFor a given Cohen-Macaulay local ring R\, what is the min
imum non-zero value of length(M)\, where M ranges over those R-modules hav
ing finite projective dimension?\n\nIn investigating this question\, one i
s quickly led to conjecture that the answer is e(R)\, the Hilbert-Samuel m
ultiplicity of R. It turns out that this can be established for rings havi
ng Ulrich modules\, or\, more generally\, lim Ulrich sequences of modules\
, with certain properties. Moreover\, there is a related conjecture concer
ning length(M) and the Betti numbers of M\, and a conjecture concerning th
e Dutta multiplicity of M\, which can also be established when certain Ulr
ich modules (or lim Ulrich sequences) exist.\n
LOCATION:https://researchseminars.org/talk/FOTR/40/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adela Vraicu (University of South Carolina)
DTSTART;VALUE=DATE-TIME:20210311T213000Z
DTEND;VALUE=DATE-TIME:20210311T230000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/41
DESCRIPTION:Title: Cl
assification of extremal hypersurfaces in positive characteristic\nby
Adela Vraicu (University of South Carolina) as part of Fellowship of the R
ing\n\n\nAbstract\nThe log canonical threshold is an invariant that measur
es how singular a hypersurface over an algebraically closed field of chara
cteristic zero is. The F-pure threshold is the positive characteristic ana
log. Hypersurfaces with smaller threshold are more singular.\n\nI will dis
cuss a lower bound for a homogeneous polynomial in characteristic p\, rela
tive to its degree\, and describe the classification of the hypersurfaces
that achieve this bound up to change of coordinates. These results were ob
tained as part of a project started at the A.W.M. Workshop ``Women in Comm
utative Algebra” at B.I.R.S.\; joint work with Zhibek Kadyrsizova\, Jenn
ifer Kenkel\, Janet Page\, Jyoti Singh\, Karen E. Smith and Emily Witt.\n
LOCATION:https://researchseminars.org/talk/FOTR/41/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robert Lazarsfeld (Stony Brook University)
DTSTART;VALUE=DATE-TIME:20210304T213000Z
DTEND;VALUE=DATE-TIME:20210304T230000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/42
DESCRIPTION:Title: Sa
turation bounds for smooth varieties\nby Robert Lazarsfeld (Stony Broo
k University) as part of Fellowship of the Ring\n\n\nAbstract\nLet X be a
smooth complex projective variety with homogeneous ideal I. We consider th
e question of bounding the saturation degree of the powers I^a of I\, ie t
he 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 results i
n two situations:\n\n— When I is the ideal of a smooth curve C\, we g
ive a bound in terms of the regularity of C\, extending results of Geramit
a 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 terms
of the degrees of defining equations that has the same general shape as (b
ut a rather different proof than) regularity bounds established many years
ago with Bertram and Ein.\n
LOCATION:https://researchseminars.org/talk/FOTR/42/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kei-ichi Watanabe (Nihon University and Meiji University)
DTSTART;VALUE=DATE-TIME:20210319T000000Z
DTEND;VALUE=DATE-TIME:20210319T013000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/43
DESCRIPTION:Title: No
rmal reduction numbers\, normal Hilbert coefficients and elliptic ideals i
n normal 2-dimensional local domains\nby Kei-ichi Watanabe (Nihon Univ
ersity and Meiji University) as part of Fellowship of the Ring\n\n\nAbstra
ct\nThis is a joint work with T. Okuma (Yamagata Univ.)\, M.E. Rossi (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 integrall
y closed $\\mathfrak{m}$-primary ideal and $Q$ be a minimal reduction of $
I$ (a parameter ideal with $I^{r+1} = Q I^{r}$ for some $r ≥ 1$). Then t
he 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\\]\na
re important invariants of the ideal and the singularity. Also the normal
Hilbert coefficients $\\overline{e}_i(I)$\, for $i = 0\, 1\, 2$\, are defi
ned by\n\\[\n\\ell_A(A/\\overline{I^{n+1}}) = \\overline{e}_0(I)\\binom{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 by these
invariants. Namely\, $A$ is a\nrational singularity if and only if $\\ove
rline{r}(A) = 1$\, or equivalently\, $\\overline{e}_2(I) = 0$ for every $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 integra
lly closed $\\mathfrak{m}$ primary ideal is a$p_g$ ideal.\n\nOur aim is to
know the behavior of these invariants for every integrally closed $\\math
frak{m}$ primary ideal $I$ of a given ring $A$.\n\nIf $A$ is an elliptic s
ingularity\, 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 i
f $\\overline{r}(I) = 2$ and strongly elliptic ideal if $\\overline{e}_2 =
1$.\n\nWe will show several nice equivalent properties for $I$ to be an e
lliptic or a strongly elliptic ideal.\n\nOur tool is resolution of singula
rities of $\\mathrm{Spec}(A)$. Let $I$ be an $\\mathfrak{m}$-primary integ
rally closed ideal in $A$. We can take $f\\colon X \\to \\mathrm{Spec}(A)$
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 190
9.13190.\n
LOCATION:https://researchseminars.org/talk/FOTR/43/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kevin Tucker (University of Illinois at Chicago)
DTSTART;VALUE=DATE-TIME:20210401T203000Z
DTEND;VALUE=DATE-TIME:20210401T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/44
DESCRIPTION:Title: Gl
obal +- regularity\nby Kevin Tucker (University of Illinois at Chicago
) as part of Fellowship of the Ring\n\n\nAbstract\nOver a field of charact
eristic $p > 0$\, a globally F-regular algebraic variety is a special type
of Frobenius split variety. They are necessarily locally (strongly) F-reg
ular\, hence normal and Cohen-Macaulay\, but also satisfy a number of part
icularly nice global properties as well. A smooth projective variety is gl
obally F-regular if its (normalized) coordinate rings are F-regular\, a co
ndition which imposes strong positivity properties and implies Kodaira-typ
e vanishing results. Globally F-regular varieties are closely related to c
omplex log Fano varieties via reduction to characteristic $p > 0$.\n\nIn t
his talk\, I will describe an analog of global F-regularity in the mixed c
haracteristic setting called global +-regularity and introduce certain sta
ble sections of adjoint line bundles. This is inspired by recent work of B
hatt on the Cohen-Macaulayness of the absolute integral closure\, and has
applications to birational geometry in mixed characteristic. This is based
on arXiv:2012.15801 and is joint work with Bhargav Bhatt\, Linquan Ma\, Z
solt Patakfalvi\, Karl Schwede\, Joe Waldron\, and Jakub Witaszek.\n
LOCATION:https://researchseminars.org/talk/FOTR/44/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Martina Juhnke-Kubitzke (University of Osanbrueck)
DTSTART;VALUE=DATE-TIME:20210408T200000Z
DTEND;VALUE=DATE-TIME:20210408T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/45
DESCRIPTION:Title: Th
e antiprism triangulation\nby Martina Juhnke-Kubitzke (University of O
sanbrueck) as part of Fellowship of the Ring\n\n\nAbstract\nThe antiprism
triangulation provides a natural way to subdivide a simplicial complex $\\
Delta$\, similar to barycentric subdivision\, which appeared independently
in combinatorial algebraic topology and computer science. It can be defin
ed as the simplicial complex of chains of multi-pointed faces of $\\Delta$
from a combinatorial point of view\, and by successively applying the ant
iprism construction\, or balanced stellar subdivisions\, on the faces of $
\\Delta$ from a geometric point of view.\nIn this talk\, we will study enu
merative invariants associated to this triangulation\, such as the transfo
rmation of the $h$-vector of $\\Delta$ under antiprism triangulation\, the
local $h$-vector\, and algebraic properties of its Stanley--Reisner ring.
Among other results\, it is shown that the $h$-polynomial of the antipris
m triangulation of a simplex is real-rooted and that the antiprism triangu
lation of $\\Delta$ has the almost strong Lefschetz property over $\\mathb
b{R}$ for every shellable complex $\\Delta$.\n\nI will make the talk as se
lf-contained as possible\, and assume no previous knowledge of combinatori
cs of subdivisions. This is joint work with Christos Athanasiadis and Jan-
Marten Brunink.\n
LOCATION:https://researchseminars.org/talk/FOTR/45/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Adam Boocher (University of San Diego)
DTSTART;VALUE=DATE-TIME:20210415T203000Z
DTEND;VALUE=DATE-TIME:20210415T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/46
DESCRIPTION:Title: On
the size and shape of betti numbers\nby Adam Boocher (University of S
an Diego) as part of Fellowship of the Ring\n\n\nAbstract\nGiven a finitel
y-generated graded module over a polynomial ring\, there are many results
and conjectures concerning lower bounds for its betti numbers. Major playe
rs in this story include the Syzygy Theorem\, the Buchsbaum-Eisenbud-Horro
cks Rank Conjecture\, and the Total Rank Conjecture. In this talk I'll su
rvey these results and conjectures including a collection of intricately c
onnected recent results that point to even stronger bounds.\n
LOCATION:https://researchseminars.org/talk/FOTR/46/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Paolo Mantero (University of Arkansas)
DTSTART;VALUE=DATE-TIME:20210422T203000Z
DTEND;VALUE=DATE-TIME:20210422T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/47
DESCRIPTION:by Paolo Mantero (University of Arkansas) as part of Fellowshi
p of the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/47/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jonathan Montaño (New Mexico State University)
DTSTART;VALUE=DATE-TIME:20210429T203000Z
DTEND;VALUE=DATE-TIME:20210429T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/48
DESCRIPTION:Title: Wh
en are multidegrees positive?\nby Jonathan Montaño (New Mexico State
University) as part of Fellowship of the Ring\n\n\nAbstract\nMultidegrees
of multiprojective varieties extend the notion of degree of projective var
ieties. These invariants can be defined via intersection theory\, or algeb
raically as the leading coefficients of multivariate Hilbert polynomials.
It follows that multidegrees are nonnegative integers\, so a fundamental q
uestion is: When are multidegrees positive?\nIn the first part of the talk
\, I will survey definitions and key properties of degrees and multidegree
s\, including some examples. \n\nIn the second part\, I will present a
complete characterization of the positivity of multidegrees\, and establis
h a combinatorial description using convex geometry. I will also show appl
ications of our result to mixed multiplicities of ideals and to the suppor
t of Schubert polynomials. The talk is based on joint work with F. Castill
o\, Y. Cid-Ruiz\, B. Li\, and N. Zhang.\n
LOCATION:https://researchseminars.org/talk/FOTR/48/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Irena Swanson (Purdue University)
DTSTART;VALUE=DATE-TIME:20210506T203000Z
DTEND;VALUE=DATE-TIME:20210506T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/49
DESCRIPTION:Title: Nu
mbers of associated primes of powers of ideals\nby Irena Swanson (Purd
ue University) as part of Fellowship of the Ring\n\n\nAbstract\nThis talk
is about associated primes of powers of ideals in Noetherian commutative r
ings. 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 general\, the
\nnumber of associated primes can go up or down as $n$ increases. This ta
lk 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 a
nd 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 projects with
Sarah Weinstein\, Jesse Kim\, Robert Walker\, and ongoing work with Roswit
ha Rissner.\n
LOCATION:https://researchseminars.org/talk/FOTR/49/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Karen Smith (University of Michigan)
DTSTART;VALUE=DATE-TIME:20210513T203000Z
DTEND;VALUE=DATE-TIME:20210513T220000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/50
DESCRIPTION:Title: Ex
tremal singularities in prime characteristic\nby Karen Smith (Universi
ty of Michigan) as part of Fellowship of the Ring\n\n\nAbstract\nWhat is t
he most singular possible singularity? What can we say about it's geometri
c and algebraic properties? This seemingly naive question has a sensible a
nswer in characteristic p. The "F-pure threshold\," which is an analog of
the log canonical threshold\, can be used to "measure" how bad a singula
rity is. The F-pure threshold is a numerical invariant of a point on (say
) a hypersurface---a positive rational number that is 1 at any smooth poi
nt (or more generally\, any F-pure point) but less than one in general\, w
ith "more singular" points having smaller F-pure thresholds. We explain a
recently proved lower bound on the F-pure threshold in terms of the multi
plicity of the singularity. We also show that there is a nice class of hyp
ersurfaces--which we call "Extremal hypersurfaces"---for which this bound
is achieved. These have very nice (extreme!) geometric properties. For exa
mple\, the affine cone over a non Frobenius split cubic surface of charact
eristic two is one example of an "extremal singularity". Geometrically\, t
hese are the only cubic surfaces with the property that *every* triple of
coplanar lines on the surface meets in a single point (rather than a "tria
ngle" as expected)--a very extreme property indeed.\n
LOCATION:https://researchseminars.org/talk/FOTR/50/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Wenliang Zhang (University of Illinois-Chicago)
DTSTART;VALUE=DATE-TIME:20210914T200000Z
DTEND;VALUE=DATE-TIME:20210914T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/51
DESCRIPTION:Title: Va
nishing of local cohomology modules\nby Wenliang Zhang (University of
Illinois-Chicago) as part of Fellowship of the Ring\n\n\nAbstract\nStudyin
g the vanishing of local cohomology modules has a long and rich history\,
and is still an active research area. In this talk\, we will discuss class
ic theorems (due to Grothendieck\, Hartshorne\, Peskine-Szpiro\, and Ogus)
\, recent developments\, and some open problems.\n
LOCATION:https://researchseminars.org/talk/FOTR/51/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Steven Sam (University of California\, San Diego)
DTSTART;VALUE=DATE-TIME:20210928T200000Z
DTEND;VALUE=DATE-TIME:20210928T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
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://researchseminars.org/talk/FOTR/52/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Yairon Cid-Ruiz (Ghent University)
DTSTART;VALUE=DATE-TIME:20211012T200000Z
DTEND;VALUE=DATE-TIME:20211012T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/53
DESCRIPTION:by Yairon Cid-Ruiz (Ghent University) as part of Fellowship of
the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/53/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Julia Pevtsova (University of Washington\, Seattle)
DTSTART;VALUE=DATE-TIME:20211026T200000Z
DTEND;VALUE=DATE-TIME:20211026T213000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/54
DESCRIPTION:by Julia Pevtsova (University of Washington\, Seattle) as part
of Fellowship of the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/54/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Patricia Klein (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20211109T210000Z
DTEND;VALUE=DATE-TIME:20211109T223000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/55
DESCRIPTION:by Patricia Klein (University of Minnesota) as part of Fellows
hip of the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Daniel Hernández (University of Kansas)
DTSTART;VALUE=DATE-TIME:20211207T210000Z
DTEND;VALUE=DATE-TIME:20211207T223000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/56
DESCRIPTION:by Daniel Hernández (University of Kansas) as part of Fellows
hip of the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Robin Baidya (University of Tennessee)
DTSTART;VALUE=DATE-TIME:20211130T210000Z
DTEND;VALUE=DATE-TIME:20211130T223000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/58
DESCRIPTION:by Robin Baidya (University of Tennessee) as part of Fellowshi
p of the Ring\n\nAbstract: TBA\n
LOCATION:https://researchseminars.org/talk/FOTR/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Kriti Goel (University of Utah)
DTSTART;VALUE=DATE-TIME:20221107T210000Z
DTEND;VALUE=DATE-TIME:20221107T223000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/60
DESCRIPTION:Title: Hi
lbert-Kunz function and Hilbert-Kunz multiplicity of ideals and Rees algeb
ras\nby Kriti Goel (University of Utah) as part of Fellowship of the R
ing\n\n\nAbstract\nHilbert-Kunz functions were introduced by E. Kunz in 19
69 in his work characterizing regular local rings in the prime characteris
tic setting. The existence of Hilbert-Kunz multiplicity was proved later b
y P. Monsky in 1983. Since then\, Hilbert-Kunz functions and Hilbert-Kunz
multiplicities have been extensively studied\, partly because of their con
nections with the theory of tight closure and their unpredictable behaviou
r. Unlike the Hilbert-Samuel function\, the Hilbert-Kunz function need not
be a polynomial function.\n\nIn this talk\, we consider the Hilbert-Kunz
function of Rees algebra of ideals and show that\, in certain cases\, it b
ehaves as a quasi-polynomial\, a piece-wise polynomial\, or even a polynom
ial. We also consider Hilbert-Kunz multiplicity of powers of an ideal\, in
an attempt to write it as a function of the power of the ideal. This invo
lves a surprising connection with the Hilbert-Samuel coefficients of Frobe
nius powers of an ideal.\n
LOCATION:https://researchseminars.org/talk/FOTR/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Janet Page (North Dakota State Unviersity)
DTSTART;VALUE=DATE-TIME:20221212T210000Z
DTEND;VALUE=DATE-TIME:20221212T223000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/61
DESCRIPTION:Title: Ex
tremal Surfaces in Positive Characteristic\nby Janet Page (North Dakot
a State Unviersity) as part of Fellowship of the Ring\n\n\nAbstract\nWhat
is the most singular possible point on any variety in positive characteris
tic? In recent joint work with Zhibek Kadyrsizova\, Jennifer Kenkel\, Jyo
ti Singh\, Karen Smith\, Adela Vraciu\, and Emily Witt\, we gave a precise
answer to this question for cone points on hypersurfaces using a measure
of singularity called the F-pure threshold\, and we called these “most s
ingular” hypersurfaces extremal hypersurfaces. In this talk\, I’ll fo
cus on the special case of extremal surfaces and discuss some of their oth
er surprising properties. This talk is based on joint work with Anna Bros
owsky\, Tim Ryan\, and Karen Smith.\n
LOCATION:https://researchseminars.org/talk/FOTR/61/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Matt Mastroeni (Iowa State University)
DTSTART;VALUE=DATE-TIME:20230127T213000Z
DTEND;VALUE=DATE-TIME:20230127T230000Z
DTSTAMP;VALUE=DATE-TIME:20230208T080618Z
UID:FOTR/62
DESCRIPTION:Title: Ch
ow rings of matroids are Koszul\nby Matt Mastroeni (Iowa State Univers
ity) as part of Fellowship of the Ring\n\n\nAbstract\nKoszul algebras have
long been studied in connection with topology and representation theory f
or their exceptional homological and duality properties\, and they appear
with incredible frequency among rings at the intersection of commutative a
lgebra\, algebraic geometry\, and combinatorics. The Chow ring of a matro
id is just such a ring - a commutative\, graded\, Artinian\, Gorenstein al
gebra with linear and quadratic relations defined by the matroid\, which r
ecently played an important role in establishing a number of long-standing
conjectures on the combinatorics of matroids.\n\nIn this talk\, I will di
scuss joint work with Jason McCullough affirmatively answering a conjectur
e of Dotsenko that the Chow ring of any matroid is Koszul. Time permittin
g\, I will also mention some potential implications of this fact. No prio
r experience with matroids is necessary for the talk.\n
LOCATION:https://researchseminars.org/talk/FOTR/62/
END:VEVENT
END:VCALENDAR