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