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:20200812T043317Z
UID:FOTR/1
DESCRIPTION:Title: Quadratic Gorenstein rings and the Koszul property\nby
Mike Stillman (Cornell University) as part of Fellowship of the Ring\n\n\n
Abstract\nA graded ring R = S/I is Gorenstein (S = polynomial ring\, I =\n
homogeneous ideal) if the length of its free resolution over S is its\ncod
imension in S\, and the top betti number is one. R is called Koszul\nif th
e free resolution of k = R/(maximal homogeneous ideal) over R is\nlinear.
Any Koszul algebra is defined by quadratic relations\, but the\nconverse i
s false\, and no one knows a finitely computable criterion.\nBoth types of
rings have duality properties\, and occur in many\nsituations in algebrai
c geometry and commutative algebra\, and in many\ncases\, a Gorenstein qua
dratic algebra coming from geometry is often\nKoszul (e.g. homogeneous coo
rdinate rings 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. We\nwill give methods for their construction\, e.g. via inverse
systems.\nAfter a short break\, we will use these techniques to answer neg
atively\nthe above question\, as well as see how to construct many other\n
examples of quadratic Gorenstein algebras which are not Koszul.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Christine Berkesch (University of Minnesota)
DTSTART;VALUE=DATE-TIME:20200430T203000Z
DTEND;VALUE=DATE-TIME:20200430T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
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: TB
A\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jack Jeffries (CIMAT)
DTSTART;VALUE=DATE-TIME:20200507T203000Z
DTEND;VALUE=DATE-TIME:20200507T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/3
DESCRIPTION:Title: Two applications of $p$-derivations to commutative alge
bra\nby Jack Jeffries (CIMAT) as part of Fellowship of the Ring\n\n\nAbstr
act\nThe notions of derivations and modules of differentials have been cen
tral in commutative algebra for much of its history. A somewhat more exoti
c notion is that of $p$-derivations: these are maps that satisfy functiona
l equations similarly to derivations\, but are not even additive. Nonethel
ess\, $$-derivations and related constructions have found applications in
arithmetic geometry. 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
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dan Erman (University of Wisconsin)
DTSTART;VALUE=DATE-TIME:20200514T203000Z
DTEND;VALUE=DATE-TIME:20200514T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/4
DESCRIPTION:Title: Boundedness questions for polynomials in many variables
\nby Dan Erman (University of Wisconsin) as part of Fellowship of the Ring
\n\n\nAbstract\nI will discuss questions and results related to polynomial
s in a large number of variables\, starting with classical results of Hilb
ert and moving to Stillman's conjecture and its proof by Ananyan and Hochs
ter. Then I will describe ways to think about the limit of polynomial ring
s as the number of variables goes to infinity\, and how this can be applie
d to obtain new finiteness results. The original work covered in this talk
is all joint with Steven V Sam and Andrew Snowden.\n
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:20200812T043317Z
UID:FOTR/5
DESCRIPTION:Title: Commutative algebra with $S_n$-invariant monomial ideal
s\nby Claudiu Raicu (University of Notre Dame) as part of Fellowship of th
e Ring\n\n\nAbstract\nConsider a polynomial ring in $n$ variables\, togeth
er with the action of the symmetric group by coordinate permutations. In m
y talk I will describe many familiar notions in Commutative Algebra in the
context of monomial ideals that are preserved by the action of the symmet
ric group. These include Castelnuovo-Mumford regularity\, projective dimen
sion\, saturation\, symbolic powers\, or the Cohen-Macaulay property. My g
oal is to explain how changing focus from minimal resolutions to Ext modul
es can lead to a simplified picture of the homological algebra\, and to pr
ovide concrete combinatorial recipes to determine the relevant homological
invariants.\n
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:20200812T043317Z
UID:FOTR/6
DESCRIPTION:Title: Symbolic powers\, stable containments\, and degree boun
ds\nby Eloísa Grifo (University of California\, Riverside) as part of Fel
lowship of the Ring\n\n\nAbstract\nWhat's the smallest degree of a homogen
eous polynomial that vanishes to order n on a given finite set of points\,
or more generally on some algebraic variety in projective space? A classi
cal result of Zariski and Nagata tells us the set of such polynomials is t
he nth symbolic power of the ideal I corresponding to our variety. To boun
d degrees of elements in the symbolic powers of I\, we can look for contai
nments between symbolic powers and other better understood ideals\, such a
s powers of I. We will take a tour through the history of the containment
problem and some of its variations\, with an eye towards lower bounds for
degrees of symbolic powers. Our story will include joint work with Craig H
uneke and Vivek Mukundan\, and with Sankhaneel Bisui\, Tài Huy Hà\, and
Thái Thành Nguyễn.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Jason McCullough (Iowa state)
DTSTART;VALUE=DATE-TIME:20200604T203000Z
DTEND;VALUE=DATE-TIME:20200604T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/7
DESCRIPTION:Title: Subadditivity of syzygies and related problems\nby Jaso
n 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 g
raded $S$-ideal. There are many interesting questions about the maximal g
raded shifts of $S/I$\, denoted $t_i$. In the first part of my talk\, I w
ill discuss two classical constructions that turn a (graded) S-module into
an ideal with similar properties\, namely idealizations and Bourbaki idea
ls\, and what they say about maximal graded shifts of ideals. In the seco
nd part of the talk\, I will discuss restrictions on maximal graded shifts
of ideals. In particular\, an ideal $I$ is said to satisfy the subadditi
vity condition if $t_a + t_b ≥ t_(a+b)$ for all $a\,b$. This condition
fails for arbitrary\, even Cohen-Macaulay\, ideals but is open for certain
nice classes of ideals\, such as Koszul and monomial ideals. I will pres
ent a construction (joint with A. Seceleanu) showing that subadditivity ca
n fail for Gorenstein ideals. \n\nIf time allows\, I will talk about some
results that hold more generally\, including a linear bound on the maxima
l graded shifts in terms of the first $p-c$ shifts\, where $p = pd(S/I)$ a
nd $c = codim(I)$. I hope to include several examples and open questions
as well.\n
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:20200812T043317Z
UID:FOTR/8
DESCRIPTION:Title: Rees algebras of ideals generated by 2x2 minors\nby Cla
udia Polini (University of Notre Dame) as part of Fellowship of the Ring\n
\n\nAbstract\nRees algebras of ideals of maximal minors of generic matrice
s are very well understood. 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 investigate the defining ideals of these algebras\, and in t
he first \ncase we prove that they are Koszul and have rational singularit
ies or are F-rational\, respectively. \nThis is a report on joint work wit
h Ela Celikbas\, Emilie Dufresne\, Louiza Fouli\, Elisa Gorla\, Kuei-Nuan
\nLin\, and Irena Swanson and with Hang Huang\, Michael Perlman\, Claudiu
Raicu\, and Alessio Sammartano.\n
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:20200812T043317Z
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 du
al graph (a.k.a. Hochster-Huneke graph) G(R) of a Noetherian ring R of dim
ension d is the finite simple graph whose vertices correspond to the minim
al 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 fundament
al results of Grothendieck\, Hartshorne\, and Hochster-Huneke\, concerning
the connectedness 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 seco
nd part of the talk\, we will discuss some recent developments related to
the following two questions:\n1) How many paths are there between two mini
mal primes of R?\n2) What is the shortest path between two minimal primes
of R?\nBy taking 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 v
ertex connectivity and its diameter. Most of the things that I will discus
s are contained in works written together with Bruno Benedetti\, Barbara B
olognese and Michela Di Marca.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hailong Dao (University of Kansas)
DTSTART;VALUE=DATE-TIME:20200625T203000Z
DTEND;VALUE=DATE-TIME:20200625T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/10
DESCRIPTION:Title: A truly mutually beneficial friendship: how Stanley-Rei
sner theory enhanced both combinatorics and algebra\nby Hailong Dao (Unive
rsity of Kansas) 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 t
he polynomial ring in n variables\, called the Stanley-Reisner ring. Start
ing with the proof of the Upper Bound Conjecture for spheres\, this approa
ch has been spectacularly useful in bringing tools from commutative algebr
a to the study of simplicial complexes. In the first part of the talk I wi
ll sketch some relevant parts of this story. In the second\, I will descri
be how modern tools\, including cohomological vanishing results and charac
teristic p methods\, have inspired new developments. At the same time\, re
sults obtained on the combinatorics side now can be brought back to induce
interesting new questions and theorems on the algebra side. One thing I r
eally like about this topic is that it can be used to generate good proble
ms at all levels\, including for high school students.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Thomas Polstra (University of Utah)
DTSTART;VALUE=DATE-TIME:20200723T203000Z
DTEND;VALUE=DATE-TIME:20200723T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/11
DESCRIPTION:Title: The weak implies strong conjecture and finite generatio
n of symbolic Rees algebras\nby Thomas Polstra (University of Utah) as par
t of Fellowship of the Ring\n\n\nAbstract\nTight closure theory is promine
nt to prime characteristic commutative algebra. Historically\, tight closu
re has been used to simplify proofs of landmark theorems in commutative al
gebra\, provide tests to determine when an element of a ring belongs to a
particular ideal\, and provides proofs of results in prime characteristic
which otherwise can only be proved in equal characteristic 0. The first ha
lf of the talk will be devoted to basic definitions\, properties\, and con
sequences of tight closure. The second half of the talk we will go over se
veral important classes of rings defined via tight closure\, recent progre
ss on the weak implies strong conjecture appearing in a joint paper with I
an Aberbach\, and a conjecture that certain symbolic Rees algebras are Noe
therian.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bhargav Bhatt (University of Michigan)
DTSTART;VALUE=DATE-TIME:20200827T203000Z
DTEND;VALUE=DATE-TIME:20200827T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/12
DESCRIPTION:by Bhargav Bhatt (University of Michigan) as part of Fellowshi
p of the Ring\n\nAbstract: TBA\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexandra Seceleanu (University of Nebraska)
DTSTART;VALUE=DATE-TIME:20200716T203000Z
DTEND;VALUE=DATE-TIME:20200716T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/13
DESCRIPTION:Title: Reflection arrangements\, syzygies\, and the containmen
t problem\nby Alexandra Seceleanu (University of Nebraska) as part of Fell
owship of the Ring\n\n\nAbstract\nInvariant theory\, that is the art of fi
nding polynomials invariant under the action of a given group\, has played
a major role in the historical development of commutative algebra. In thi
s theory reflection groups are singled out for having rings of invariants
that are isomorphic to polynomial rings. From a geometric perspective\, re
flection groups give rise to beautiful and very symmetric arrangements of
hyperplanes termed reflection arrangements.\n\nThis talk will take a close
look at the ideals defining the singular loci of reflection arrangements\
, which are in turn symmetric subspace arrangements. We describe their syz
ygies in terms of invariant polynomials for the relevant reflection groups
. We leverage this information to settle many aspects of the containment p
roblem asking for containments between the ordinary and the symbolic power
s of the ideals in this family. This talk is based on joint work with Ben
Drabkin.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Benjamin Briggs (University of Utah)
DTSTART;VALUE=DATE-TIME:20200806T203000Z
DTEND;VALUE=DATE-TIME:20200806T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/14
DESCRIPTION:Title: The homotopy Lie algebra and the conormal module\nby Be
njamin Briggs (University of Utah) as part of Fellowship of the Ring\n\n\n
Abstract\nI will do my best to explain what goes in to proving the followi
ng theorem: if $I$ is an ideal of finite projective dimension in a local r
ing $R$\, and the conormal module $I/I^2$ has finite projective dimension
over $R/I$\, then $I$ is generated by a regular sequence. This was conject
ured by Vasconcelos\, after he and (separately) Ferrand established the ca
se that the conormal module is free.\n\nThe key tool is the homotopy Lie a
lgebra. This is a graded Lie algebra naturally associated with any local h
omomorphism. It sits at the centre of a longstanding friendship between co
mmutative algebra and rational homotopy theory\, through which ideas and r
esults 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 commut
ative algebra in the past\, before explaining how its structure detects wh
en the conormal module has finite projective dimension. I'll also talk abo
ut ongoing work with Srikanth Iyengar comparing the cotangent complex with
the homotopy Lie algebra.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Eleonore Faber (University of Leeds)
DTSTART;VALUE=DATE-TIME:20200820T203000Z
DTEND;VALUE=DATE-TIME:20200820T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/15
DESCRIPTION:by Eleonore Faber (University of Leeds) as part of Fellowship
of the Ring\n\nAbstract: TBA\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Aldo Conca (University of Genoa)
DTSTART;VALUE=DATE-TIME:20200903T203000Z
DTEND;VALUE=DATE-TIME:20200903T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/16
DESCRIPTION:by Aldo Conca (University of Genoa) as part of Fellowship of t
he Ring\n\nAbstract: TBA\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Graham Leuschke (Syracuse University)
DTSTART;VALUE=DATE-TIME:20200910T203000Z
DTEND;VALUE=DATE-TIME:20200910T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/17
DESCRIPTION:by Graham Leuschke (Syracuse University) as part of Fellowship
of the Ring\n\nAbstract: TBA\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Dale Cutkosky (University of Missouri)
DTSTART;VALUE=DATE-TIME:20200702T203000Z
DTEND;VALUE=DATE-TIME:20200702T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
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\nW
e discuss the theory of multiplicities and mixed multiplicities of filtrat
ions 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 inequa
lity. We give some general theorems characterizing when this condition hol
ds\, giving generalizations of classical theorems of Rees\, Sharp\, Teissi
er\, Katz and others.\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Luis Núñez-Betancourt (CIMAT\, Guanajuato)
DTSTART;VALUE=DATE-TIME:20200709T203000Z
DTEND;VALUE=DATE-TIME:20200709T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/19
DESCRIPTION:Title: Differential powers of ideals\nby Luis Núñez-Betancou
rt (CIMAT\, Guanajuato) as part of Fellowship of the Ring\n\nAbstract: TBA
\n
END:VEVENT
BEGIN:VEVENT
SUMMARY:Brooke Ullery (Emory University)
DTSTART;VALUE=DATE-TIME:20200813T203000Z
DTEND;VALUE=DATE-TIME:20200813T220000Z
DTSTAMP;VALUE=DATE-TIME:20200812T043317Z
UID:FOTR/20
DESCRIPTION:Title: Cayley-Bacharach theorems and measures of irrationality
\nby Brooke Ullery (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 whic
h polynomials of degree d vanish at every point in Z. If P is one point of
Z\, the vanishing of a polynomial at P imposes one linear condition on th
e coefficients. Thus\, the vanishing of a polynomial on all of Z imposes |
Z| linear conditions on the coefficients. A classical question in algebrai
c geometry\, dating back to at least the 4th century\, is how many of thos
e linear conditions are independent? For instance\, if we look at the spac
e of lines through three collinear points in the plane\, the unique line t
hrough two of the points is exactly the one through all three\; i.e. the c
onditions imposed by any two of the points imply those of the third. In th
is talk\, I will survey several classical results including the original C
ayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational
curves. I’ll then describe some recent results and conjectures about po
ints satisfying the so-called Cayley-Bacharach condition and show how they
connect to several seemingly unrelated questions in contemporary algebrai
c geometry relating to the gonality of curves and measures of irrationalit
y of higher dimensional varieties.\n
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:20200812T043317Z
UID:FOTR/21
DESCRIPTION:Title: Lagrangian Geometry of Matroids\nby Federico Ardila (Sa
n Francisco State University) as part of Fellowship of the Ring\n\n\nAbstr
act\nMatroid theory had its origins in linear algebra and graph theory. In
recent years\, the geometric roots of the field have grown much deeper\,
bearing many new fruits. The interplay between matroid theory and algebrai
c geometry has opened up interesting research directions at the intersecti
on of combinatorics\, algebra\, and geometry\, and led to the solution of
long-standing questions. \n\nThis talk will discuss my recent joint work w
ith 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-Sch
wartz-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 prove Brylawski's and Dawson's conjectures that the h-vector
of a matroid is log-concave.\n\nI will make the talk as self-contained as
possible\, and assume no previous knowledge of matroid theory.\n
END:VEVENT
END:VCALENDAR