Tuesdays 4-6, Evans 939
John Nolan kindly agreed to share his live-TeXed notes publicly.
| date | speaker | title | abstract |
|---|---|---|---|
| 8/27 | David Eisenbud | Resolutions over Golod Rings | Recent work of mine with Dao, Cuong, Polini, Takahashi and Ulrich has uncovered surprising new structure in the resolution of the residue field of a Golod ring. I will explain what Golod rings are, why they have been studied, and what we now know and suspect about the structure of their resolutions. |
| Sterling Saint Rain | Sullivan Minimal Models and Free Torus Actions | This is an expository talk introducing differential graded algebras and Sullivan minimal models, as well as outlining their connection through rational homotopy theory to the existence of free torus actions on reasonable (i.e. finite CW-complex) topological spaces. | |
| 9/3 | Bernd Sturmfels | Maximal Mumford Curves from Planar Graphs | We introduce a recent project (arXiv:2404.11838) with Mario Kummer and Raluca Vlad in real non-archimedean geometry. A curve of genus \(g\) is maximal Mumford (MM) if it has \(g+1\) ovals and \(g\) tropical cycles. We construct full-dimensional families of MM curves in the Hilbert scheme of canonical curves. This rests on first-order deformations of graph curves whose graph is planar. |
| Feiyang Lin | Abelian covers | An abelian cover is a finite morphism \(X \to Y\) which is the quotient map for a generically faithful action of a finite abelian group G. Unlike the situation of general finite covers, where there is only a structure theorem for degree at most \(5\), there is a practical structure theorem of abelian covers due to Pardini in 1991, in the situation where the target is smooth and proper, and the source is normal. I will explain this structure theorem and give many examples along the way. | |
| 9/10 | -- | -- | No Seminar |
| 9/17 | Peter Haine | Reconstructing schemes from their étale topoi | In Grothendieck’s 1983 letter to Faltings that initiated the study of anabelian geometry, he conjectured that a large class of schemes can be reconstructed from their étale topoi. In this talk, I’ll explain a number of formulations of Grothendieck’s conjecture, and how one might come to make this conjecture. I’ll also discuss joint work with Magnus Carlson and Sebastian Wolf, generalizing work of Voevodsky, that proves Grothendieck’s conjecture. Specifically, we show that over a finitely generated field k of characteristic 0, seminormal finite type k-schemes can be reconstructed from their étale topoi. Over a finitely generated field k of positive characteristic and transcendence degree \(\geq 1\), we show that perfections of finite type k-schemes can be reconstructed from their étale topoi. |
| Hannah Larson | Chow rings of moduli spaces | The Chow ring of a variety captures information about its subvarieties and how those subvarieties intersect each other. I'll start by defining the Chow ring and describing some of its basic properties. Then I'll use the example of projective space to introduce the idea of "tautological classes", which are distinguished classes in the Chow ring of a moduli space. Finally, I'll define tautological classes on the moduli space of curves and tell you what is known (and not known) about them. | |
| 9/24 | Christian Gaetz | Combinatorics of singularities of Schubert varieties and torus orbit closures therein | The singularities of Schubert varieties in the flag variety have deep connections to geometric representation theory via Kazhdan-Lusztig theory. I will describe joint work with Yibo Gao in which we resolve a conjecture of Billey-Postnikov by quantifying these singularities by determining the first homological degree in which Poincare duality fails. Equivalently, this determines the smallest power of \(q\) appearing in a Kazhdan-Lusztig polynomial. Our methods rely on the combinatorics of permutation patterns. I'll also explain how this can be combined with my other work on generic torus orbit closures in Schubert varieties in order to determine when these are smooth, thereby proving a conjecture of Lee-Masuda-Park. |
| 10/1 | Daigo Ito | A new proof of the Bondal-Orlov reconstruction theorem | In 1997, Bondal and Orlov showed that a smooth (anti-)Fano variety \(X\) can be reconstructed from the triangulated category structure of its derived category. On the other hand, in 2004, Balmer showed that any variety \(X\) can be reconstructed from its derived category together with the monoidal structure given by the derived tensor product. This indicates that for a smooth (anti-)Fano variety, Balmer's reconstruction should be able to be done without actually using the monoidal structure and from this perspective, we will provide a new proof of the Bondal-Orlov reconstruction theorem. Furthermore, we offer new insights and institutions on derived categories from the techniques we developed as well as generalizations of several other reconstruction theorems. |
| Noah Olander | Fully faithful functors and dimension | Can one embed the derived category of a higher dimensional variety into the derived category of a lower dimensional variety? The expected answer was no. We give a simple proof and prove new cases of a conjecture of Orlov along the way. | |
| 10/8 | Hannah Larson | Chow rings of moduli spaces of pointed hyperelliptic curves | In this talk, I describe the geometry of the moduli space \(\mathcal{H}_{g,n}\) of n-pointed, genus g hyperelliptic curves. As n grows relative to g, work of Casnati, Barros--Mullane, and Schwarz shows that \(\mathcal{H}_{g,n}\) goes from being rational (the simplest kind of variety) to general type (quite complicated). This suggests that we have hope of probing finer invariants of \(\mathcal{H}_{g,n}\) when \(n\) is small relative to \(g\). The Chow ring of \(\mathcal{H}_{g,n}\) is one such invariant. I will describe an inductive procedure for stratifying \(\mathcal{H}_{g,n}\) into nice pieces, which allows us to calculate its rational Chow ring when \(n\) is less than or equal to \(2g + 6\). This is joint work with Samir Canning. |
| 10/15 | Smita Rajan | Kinematic Varieties for Massless Particles | We study algebraic varieties that encode the kinematic data for \(n\) massless particles in \(d\)-dimensional spacetime subject to momentum conservation. Their coordinates are spinor brackets, which we derive from the Clifford algebra associated to the Lorentz group. This was proposed for \(d = 5\) in the recent physics literature. Our kinematic varieties are given by polynomial constraints on tensors with both symmetric and skew symmetric slices. This is joint work with Bernd Sturmfels and Svala Sverrisdóttir. |
| John Nolan | Toric Varieties as Quotients of Affine Spaces | A toric variety is a normal algebraic variety \(X\) equipped with the action of an algebraic torus \(T\) and a \(T\)-equivariant dense open immersion \(T \hookrightarrow X\). In 1992, Cox showed that reasonably nice (e.g. smooth) toric varieties have canonical descriptions as quotients of affine space (minus a union of coordinate subspaces) by tori. I will present the basic theory of toric varieties alongside Cox's construction, using projective space as a running example. If time permits (it won't), I will also discuss Ballard-Favero-Katzarkov's application of related machinery to constructing full exceptional collections on smooth projective toric varieties. | |
| 10/22 | Will Fisher | Introduction to Hochschild Homology | We will define Hochschild homology and introduce basic computational techniques such as the HKR isomorphism. Towards the end we will briefly discuss more modern perspectives including THH and Bokstedt's computation and the relation to loop spaces. This talk should be accessible to all listeners. |
| 10/29 | Catherine Cannizzo | Homological Mirror Symmetry for Theta Divisors | Symplectic geometry is a relatively new branch of geometry. However, a string theory-inspired duality known as “mirror symmetry” reveals more about symplectic geometry from its mirror counterparts in complex geometry. M. Kontsevich conjectured an algebraic version of mirror symmetry called “homological mirror symmetry” (HMS) in his 1994 ICM address. HMS results were then proved for symplectic mirrors to Calabi-Yau and Fano manifolds. Those mirrors to general type manifolds have been studied in more recent years, including my research. In this talk, we will introduce HMS through the example of the 2-torus \(T^2\). We will then outline how it relates to HMS for a hypersurface of a 4-torus \(T^4\) and generalize to hypersurfaces of higher dimensional tori, otherwise known as “theta divisors.” This is joint work with Haniya Azam, Heather Lee, and Chiu-Chu Melissa Liu. |
| 11/5 | Martin Olsson | Ample vector bundles and projective geometry of stacks | I will report on joint work with Bragg and Webb on the following two questions:
(1) What is the analogue of a projective embedding for an algebraic stack? (2) How do we describe embeddings into such stacks in terms of vector bundles and sections? I will explain how exploration of these two questions leads to a notion of ample vector bundle on an algebraic stack. As an application we construct moduli of stacky curves. |
| 11/12 | Nathaniel Gallup | The Grothendieck Ring of Certain Non-Noetherian Multigraded Algebras via Hilbert Series | Inspired by work of Knutson, Miller, and Sturmfels, and with the goal of defining Grothendieck polynomials for infinite matrix Schubert varieties, we introduce a new category of graded “BDF modules” (containing the syzygies of infinite matrix Schubert varieties, which need not be finitely generated) over a new type of \(\Gamma\)-graded “PDCF algebra” (which includes infinite polynomial rings). If \(R\) is a PDCF \(\Gamma\)-graded \(k\)-algebra, we prove that every projective BDF \(\Gamma\)-graded \(R\)-module is free, and that when \(R\) is regular, a version of the Hilbert Syzygy Theorem holds: every BDF \(R\)-module has a resolution of free BDF modules which, though not finite, has the property that each graded piece is eventually zero. We use this to show that the natural map from the Grothendieck group of projective BDF \(R\)-modules to the Grothendieck group of all BDF \(R\)-modules is an isomorphism. We describe this Grothendieck group explicitly by using Hilbert series to give an isomorphism with a certain space of formal Laurent series. Finally we give a version of Serre's formula for the product in the Grothendieck group, making it into a ring. |
| 11/19 | David Eisenbud | Some one-dimensional local rings | One-dimensional local rings, also known as curve singularities, are often mistakenly overlooked as ``trivial'' in algebraic geometry, but there is a rich theory with interesting open problems. I'll explain some of the theory around one interesting class that has recently arisen in my research: Arf rings, named after the Turkish mathematician Cahit Arf, and studied in depth by Joe Lipman.<\th> |
| Cameron Chang | Applications of Molien's Formula | Given a finite group \(G\subset \operatorname{GL}(V)\), the ring of invariants \(\operatorname{Sym}(V)^G\) is the subalgebra of the polynomial ring \(\operatorname{Sym}(V)\) fixed under the \(G\)-action. In 1897, Theodore Molien wrote down a beautiful formula for the generating function of \(\dim \operatorname{Sym}^k(V)^G\). I will explain this formula and apply it to as many beautiful applications as I can before time runs out, such as: every irreducible representation is contained in \(\operatorname{Sym}(V)\), the only groups where \(\operatorname{Sym}(V)^G\) is a polynomial ring are pseudo-reflection groups, the Polya-Redfield counting formula, and more! | |
| 11/26 | Serkan Hoşten | Matroid Stratification of the Maximum Likelihood Degree of Segre Products | abstract |
| Dustin Ross | title | abstract | |
| 12/3 | Kabir Kapoor | title | abstract |
| Dawei Chen | title | abstract | |
| 12/10 | Adam Boocher | From Classical Commutative Algebra to Some Diophantine Equations | In a first course in commutative algebra one might encounter the "Principal Ideal Theorem" or the "Auslander-Buchsbaum Formula". It turns out that these are both implied by a longstanding conjecture about lower bounds for ranks of syzygies - the Buchsbaum-Eisenbud-Horrocks Rank Conjecture. In this talk I'll discuss historical progress on the conjecture as well as a related (and rather mysterious) conjecture about even larger bounds. Later, I'll discuss some recent work about what happens if one looks at the special case of pure modules, using techniques from Boij-Soederberg Theory. This approach leads to some interesting diophantine equations, which may shed light on the original conjectures. This is joint work with my two undergraduate students Noah Huang and Harrison Wolf. |
| Tejas Rao | title | abstract |
Tuesdays 4-6, Evans 748
| date | speaker | title | abstract |
|---|---|---|---|
| 1/16 | Bernd Sturmfels | Taylor Polynomials of Rational Functions | A Taylor variety consists of all fixed order Taylor polynomials of rational functions, where the number of variables and degrees of numerators and denominators are fixed. In one variable, this leads us to Padé approximation and rank constraints on Hankel matrices. We study the dimension and defining ideals of Taylor varieties. In three and more variables, there exist defective Taylor varieties, and we explain this with Fröberg's Conjecture in commutative algebra. This is joint work with Aldo Conca, Simone Naldi and Giorgio Ottaviani. |
| Hai Long Dao | The dual of the canonical module and applications | Let \( R \) be a Noetherian ring with canonical module \( W \). We will use the \( R \)-dual of \( W \) to define new invariants of \( R \). We will discuss some surprising connections of these invariants to additive number theory and extremal components of Hilbert schemes. | |
| 1/23 | (note: SLMath intro workshop) | ||
| 1/30 | Sameera Vemulapalli | The relationship between scrollar invariants of curves and successive minima of orders in number fields, and related counting problems | Let \( n \geq 2 \) be an integer. To a degree \( n \) cover of \( \mathbb P^1 \), we may attach \( n-1 \) integers called the scrollar invariants. Similarly, to an order in a degree \( n \) number field, we may attach \( n-1 \) real numbers called the successive minima. In this talk I will explain the relationship between scrollar invariants of curves and successive minima of orders in number fields. Surprisingly, we don't know the answers to many fundamental questions about these invariants. In this talk, I prove bounds on these invariants. When \( n < 6 \), I'll also discuss a related counting question: how many orders in degree \( n \) number fields are there with bounded discriminant and prescribed successive minima? If time permits, we will also discuss some partial results on which scrollar invariants arise from smooth curves. |
| Frank-Olaf Schreyer | Extensions of varieties, the adjunction process for surfaces and maximal extensions of paracanonical curves of genus 6. | Let \( X \) be a subvariety of \( \mathbb P^n \). An \( e \)-extension \( Y \) is a subvariety of \( \mathbb P^{n+e} \), which is not a cone, such that there exists a regular sequence \( y_1,\ldots,y_e \) of linear forms for the homogeneous coordinate ring \( S_Y \) of \( Y \) such that \( S_Y/(y_1,\ldots,y_e) = S_X \) is the coordinate ring of \( X \). In this talk I will discribe a computationally easy to use method to compute maximal extensions of certain varieties and illustrate this method by describing maximal extensions of paracanonical curves \( C \) of genus 6. A paracanonical curve is a curve embedded by a line bundle \( \mathscr L \), which differs from the canonical bundle by a topologically trivial line bundle \( \eta \), i.e, an \( \eta \) in \( \operatorname{Pic}^0(C) \). In case of genus 6, the general ones are curves of degree 10 in \( \mathbb P^4 \) with a cubic linear resolution \( S \leftarrow S^{10}(-3) \leftarrow S^{15}(-4) \leftarrow S^6(-5) \leftarrow 0 \). To identify the smooth 1-extensions \( Y \) in \( \mathbb P^5 \) of \( C \) we use the adjunction process of Sommese and Van de Ven for surfaces. | |
| 2/6 | (note: SLMath intro workshop) | ||
| 2/13 | Anand Patel | Counting cubic surfaces | The moduli space of cubic surfaces is 4-dimensional, and hence in a 4-dimensional family of such surfaces we expect to find the general cubic finitely many times. Using equivariant geometry I will describe a general formula which, when evaluated over most families, computes this number. This is joint work with Anand Deopurkar and Dennis Tseng. |
| Claudia Miller | Residue field summands of syzygies via canonical resolutions | In joint work with Michael DeBellevue, we give extensions of results of Dao and Eisenbud showing the existence of direct summands isomorphic to the residue field in all high syzygies of any module whenever the ring satisfies a certain condition. This implies that some amount of linearity is always present in the resolutions. Our method of proof is via the relative bar resolution of Iyengar and gives some idea of why these summands should exist in such abundance and why they appear from a certain degree onward. In addition, we go on to find an exponential number of these explicitly in the Golod setting using instead the bar resolution formed from A-infinity resolutions due to Burke. | |
| 2/20 | Lizzie Pratt | The Chow-Lam Form | The classical Chow form encodes any projective variety by a single equation. In this talk, I will introduce the Chow-Lam form, which is a generalization of the Chow form to subvarieties of arbitrary Grassmannians. Like the Chow form, it has useful computational properties: for example, it gives us universal formulas for certain linear projections between Grassmannians. Such formulas were pioneered by Thomas Lam for positroid varieties in the study of amplituhedra. This is joint work with Bernd Sturmfels. |
| Hannah Kerner Larson | Cohomology of moduli spaces of curves | The moduli space \( M_g \) of genus g curves (or Riemann surfaces) is a central object of study in algebraic geometry. Its cohomology is important in many fields. For example, the cohomology of \( M_g \) is the same as the cohomology of the mapping class group, and is also related to spaces of modular forms. Using its properties as a moduli space, Mumford defined a distinguished subring of the cohomology of \( M_g \) called the tautological ring. The definition of the tautological ring was later extended to the compactification \( \overline{M_g} \) and the moduli spaces with marked points \( \overline{M_{g,n}} \). While the full cohomology ring of \( \overline{M_{g,n}} \) is quite mysterious, the tautological subring is relatively well understood, and conjecturally completely understood. In this talk, I'll ask the question: which cohomology groups \( H^k(\overline{M_{g,n}}) \) are tautological? And when they are not, how can we better understand them? This is joint work with Samir Canning and Sam Payne. | |
| 2/27 | Jack Jeffries | Local cohomology of determinantal nullcones | The coordinate ring \( R \) of the variety of \( m\times n \) matrices with rank at most \( t \) can be realized in a natural way as an invariant ring of a polynomial ring \( S \); the same is true if one replaces "\( m\times n \) matrices" by "\( n\times n \) symmetric matrices" or "\( n\times n \) alternating matrices". This realization as a ring of invariants explains many of the nice algebraic properties enjoyed by determinantal rings, at least in characteristic zero, since \( R \) is a retract of \( S \) via the Reynolds operator. Motivated by understanding the relationship between \( R \) and \( S \) in arbitrary characteristic, we consider the ideal generated by positive degree invariants \( R_+ \) inside of the ambient polynomial ring \( S \). For example, certain varieties of complexes as introduced by Buchsbaum and Eisenbud occur like so. In this talk, we will discuss some aspects of the behavior of local cohomology with support in these ideals in different characteristics and some applications. This is based on work in progress with Pandey, Singh, and Walther, and earlier joint work with Hochster, Pandey, and Singh. |
| Uli Walther | Matroidal Polynomials and their Singularities | (joint with Dan Bath) We introduce a class of polynomials attached to matroids (or flags of matroids) and the choice of a base field that includes various other polynomials that have appeared in the literature, including Kirchhoff polynomials, configuration polynomials, matroid basis polynomials, and multivariable Tutte polynomials. The common property of matroidal polynomials is a Deletion-Contraction formula. In the talk I will explain various geometric properties of matroidal polynomials; in particular, for connected matroids they have rational singularities over the complex numbers, and (if homogeneous coming from a single matroid) are \( F \)-regular in positive characteristic. The idea of the proof in characteristic zero can be extended to polynomials called Feynman integrands, and which show up in certain integrals in scattering theory. In particular, these do have rational singularities as well under certain genericity hypotheses on masses and momenta. The plan is to explain all these words, and hint at the proofs, which involve jets and the Frobenius. | |
| 3/5 | Yairon Cid-Ruiz | Numerical criteria for integral dependence and their behavior in families | A classical theorem of Rees tells us that, in an equidimensional and universally catenary Noetherian local ring, two zero-dimensional ideals \( I \subset J \) have the same integral closure if and only if they have the same Hilbert-Samuel multiplicity. This seminal result sparked much interest and has become an important research topic in commutative algebra, singularity theory, and algebraic geometry. For instance, an important consequence is Teissier’s principle of specialization of integral dependence. We will present new criteria for the integrality and birationality of an extension of graded algebras in terms of the general notion of polar multiplicities due to Kleiman and Thorup. We will discuss the behavior in families of ideals of certain invariants studied by Gaffney and Gassler: the polar multiplicities and Segre numbers of ideals. I will report on ongoing joint work with Claudia Polini and Bernd Ulrich. |
| Benjamin Briggs | A-infinity tricks in local algebra | This talk will be mostly expository: the plan is to explain what an A-infinity algebra is and how you might use them to prove things about commutative rings. I'll focus on how they can be used the construct Iyengar and Burke's bar resolution that achieves Serre's bound on the Betti numbers of a module, what they have to do with the Eisenbud-Shamash resolution as well, and, if there's time, I'll talk about some new analogues of these resolutions, also using A-infinity tricks. This last part is joint work with James Cameron, Janina Letz, and Josh Pollitz on something we call Koszul homomorphisms. | |
| 3/12 | Note: DE out of town | ||
| 3/19 | Frank Sottile | Welschinger Signs and the Wronski Map (New conjectured reality) | A general real rational plane curve \( C \) of degree \( d \) has \( 3(d-2) \) flexes and \( (d-1)(d-2)/2 \) complex double points. Those double points lying in \( \mathbb{RP}^2 \) are either nodes or solitary points. The Welschinger sign of \( C \) is \( (-1)^s \), where \( s \) is the number of solitary points. When all flexes of \( C \) are real, its parameterization comes from a point on the Grassmannian under the Wronskii map, and every parameterized curve with those flexes is real (this is the Mukhin-Tarasov-Varchenko Theorem). Thus to \( C \) we may associate the local degree of the Wronskii map, which is also \( 1 \) or \( -1 \). My talk will discuss work with Brazelton and McKean towards a possible conjecture that that these two signs associated to \( C \) agree, and the challenges to gathering evidence for this. |
| Liana Sega | Relations between Poincare series for quasi-complete intersection homomorphisms | Quasi-complete intersection (q.c.i.) homomorphisms are surjective homomorphisms of local rings for which the Koszul homology on a minimal generating set of the kernel is an exterior algebra. We study base change results for Poincare series along a q.c.i. homomorphism in situations that extend results known for complete intersection (c.i.) homomorphisms. The main new result is joint work with Josh Pollitz, and generalizes a well-known result of Shamash for c.i. homomorphisms which makes use of systems of higher homotopies. Our proof develops base change results involving Poincare series over the Koszul complex. | |
| 3/26 | Mohamed Barakat | Doctrine-specific ur-algorithms | Various constructions of categories have a universal property expressing the freeness/initiality of the construction within a specific categorical doctrine. Expressed in an algorithmic framework, it turns out that this universal property is in a certain sense a doctrine-specific “ur-algorithm” from which various known categorical constructions/algorithms (including spectral sequences of bicomplexes) can be derived in a purely computational way. This can be viewed as a categorical version of the Curry-Howard correspondence to extract programs from proofs. |
| Holger Brenner | Module schemes in invariant theory | Let \( G \) be a finite group acting linearly on the polynomial ring with invariant ring \( R \). If the action is small, then a classical result of Auslander gives in dimension two a correspondence between linear representations of \( G \) and maximal Cohen-Macaulay \( R \)-modules. We establish a correspondence for all linear actions, in particular for actions of a group generated by reflections, between representations and objects over the invariant ring by looking at quotient module schemes (up to modification) instead of the modules of covariants. | |
| 4/2 | Mike Stillman | Kuramoto oscillators: dynamical systems meet algebra | Coupled oscillators appear in a large number of applications: e.g. in biological, chemical sciences, neuro science, power grids, and many more fields. They appear in nature: fireflies flashing in sync with each other is one fun situation. In 1974, Yoshiki Kuramoto proposed a simple, yet surprisingly effective model for oscillators. We consider homogeneous Kuramoto systems (we will define these notions!). They are determined from a finite graph. In this talk, we describe some of what is known about long term behavior of such systems (do the oscillators self-synchronize? or are there other, "exotic" solutions?), and then relate these systems to systems of polynomial equations. We use algebra, computations in algebraic geometry, and algebraic geometry to study equilibrium solutions to these systems. We will see how computations using algebraic geometry and my computer algebra system Macaulay2 finds all graphs with at most 8 vertices (i.e. 8 oscillators) which have exotic solutions. Note: we assume essentially NO dynamical systems in this talk! The parts of the talk that are new represent joint work with Heather Harrington and Hal Schenck, and also Steve Strogatz and Alex Townsend. |
| Craig Huneke | Number of Generators of Licci Ideals | This talk will discuss a somewhat surprising conjectured bound on the number of generators of a licci (in the linkage class of a complete intersection) ideal, namely that the number of generators of a homogeneous licci ideal is bounded above by the greatest last twist in a minimal graded free resolution of the ideal. This is continuing joint work with Claudia Polini and Bernd Ulrich. We will give a brief introduction to licci ideals, discuss where this conjectured bound comes from, why it is useful, and then describe various cases in which we have been able to prove the bound, as time permits. These cases include monomial licci ideals of finite colength, ideals with a maximal regular sequence of quadrics, and licci ideals with nearly pure resolutions. In the non-homogeneous case, we state a related conjecture and introduce new classes of licci ideals. The techniques used are varied, including Golod rings, Boij-Soederberg theory, and the Eisenbud-Green-Harris conjecture. | |
| 4/9 | (notes: DE out of town. SLMath NCAG recent developments workshop) | ||
| 4/16 | (note: SLMath COMA recent developments workshop) | ||
| 4/23 | Frank Sottile | Galois groups in Enumerative Geometry | In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem. I will describe this background and discuss some work of many to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. |
| Mats Boij | The Kähler package for finite geometries and modular lattices | In this joint work with Bill Huang, June Huh and Greg Smith we give very explicit proofs of the existence of a Kähler package for the graded Möbius algebra associated to the lattice of subspaces of a vector space over a finite field \( \mathbb F_q \). There are fascinating connections to other areas such as the theory of Gelfand pairs and generalized Radon transforms. | |
| 4/30 (in 939) | Mahrud Sayrafi | Splitting of vector bundles on toric varieties | In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces. |
| Feiyang Lin | Finding special line bundles on special tetragonal curves | There is a canonical way to associate to a degree 4 cover of \( \mathbb P^1 \) two vector bundles \( E \) and \( F \), which give rise to a stratification of the Hurwitz space \( H_{4,g} \). It is natural to ask whether the Brill-Noether theory of tetragonal curves is controlled by this data. I will describe a procedure for producing a particular line bundle on tetragonal covers in special strata, which is expected to be special in the Hurwitz-Brill-Noether sense. The main technique is the realization of an inflation of vector bundles on \( \mathbb P^1 \) as a blow-up and blow-down of the associated projective bundle. | |
| 5/7 (in 939) | Christopher O'Neill | Classifying numerical semigroups using polyhedral geometry | A numerical semigroup is a subset of the natural numbers that is closed under addition. There is a family of polyhedral cones \( C_m \), called Kunz cones, for which each numerical semigroup with smallest positive element \( m \) corresponds to an integer point in \( C_m \). It has been shown that if two numerical semigroups correspond to points in the same face of \( C_m \), they share many important properties, such as the number of minimal generators and the Betti numbers of their defining toric ideals. In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure. In this talk, we survey what is known about the face structure of Kunz cones, and how studying Kunz cones can inform the classification of numerical semigroups. |
| Aldo Conca | Two bounds on Castelnuovo-Mumford regularity | I will report on bounds on the Castelnuovo-Mumford regularity for ideals with polynomial parametrization (joint work with F.Cioffi) and for ideals associated with general subspace arrangements (joint work with M.Tsakiris). |
Tuesdays 4-6, Evans 748
| date | speaker | title | abstract |
|---|---|---|---|
| 8/29 | Hannah Kerner Larson | The embedding theorem in Hurwitz-Brill-Noether theory | Abstract: Brill-Noether theory studies the maps of general curves to projective spaces. The embedding theorem of Eisenbud and Harris states that a general degree \(d\) map \(C\) to \(\mathbb P^r\) is an embedding when \(r\) is at least 3. Hurwitz-Brill-Noether theory starts with a curve \(C\) already equipped with a fixed map \(C\) to \(\mathbb P^1\) (which often forces \(C\) to be special) and then studies the maps of \(C\) to other projective spaces. In this setting, the appropriate analogue of the invariants \(d\) and \(r\) is a finer invariant called the splitting type. Our embedding theorem determines the splitting types \(\vec{e}\) such that a general map of splitting type \(\vec{e}\) is an embedding. This is joint work with Kaelin Cook-Powel, Dave Jensen, Eric Larson, and Isabel Vogt. |
| 9/5 | David Eisenbud | Socle Summands in Syzygies | I'll discuss two open problems about infinite resolutions, explain the "Burch index", and prove that, if a local Artinian ring has Burch index at least 2, then the 7th syzygy of every module has a summand isomorphic to the residue field. This is joint work with Hai Long Dao. |
| 9/12 | Xianglong Ni | Weyman's generic free resolutions of length three | One approach to studying the structure of finite free resolutions is to construct and analyze universal examples. Unfortunately, Bruns proved that these examples typically do not exist---but they do if one weakens the standard notion of universality to allow for non-unique specialization. Weyman constructed such "generic" examples for length three resolutions, with a careful handling of this non-uniqueness. Moreover, this apparent defect of the construction actually endows the generic example with additional symmetry, from which a surprising connection to the ADE classification arises. I'll explain how this extra symmetry can be leveraged to better understand Weyman's generic example, and some of its applications to linkage and the structure theory of perfect ideals. |
| 9/19 | Bernd Ulrich | Linkage I | The first of a series of talks explaining the elements of this geometric theory that began with the classification of curves in \(\mathbb P^3\). The series will continue with an exposition of still-unpublished work giving new invariants of linkage in higher dimensions. |
| 9/26 | (No seminar: DE out of town) | ||
| 10/3 | Bernd Ulrich | Linkage II | |
| Daigo Ito | Structures of the derived category on an elliptic curve | Although the definition of derived categories (of coherent sheaves on a variety) requires some understanding of homological algebra, derived categories themselves can be treated geometrically in many sense. In this talk, I will first talk about how and why people use derived categories in algebraic geometry and then I will briefly explain how we can indeed define derived categories. Then, as an example, we will explore the derived category on an elliptic curve. If time permits, I will also compare it with the derived category on a projective line. | |
| 10/10 | Bernd Ulrich | Linkage III | |
| Robin Hartshorne | On the maximum genus of space curves | The study of space curves goes well back into the 19th century, with the great papers of Max Noether and Georges Halphen in 1882. The determination of all possible pairs \((d,g)\) of degree and genus of space curves was correctly stated by Halphen in his paper, but not proved until the work of Gruson and Peskine in 1982. I will focus on just one part of this problem, namely, what is the maximum genus of a space curve of degree \(d\)? In its unadorned form, this question has a simple answer, but when you add consideration of the degree of a surface containing the curve, it is more complicated, and still not completely known. I will discuss the current state of this problem, also known as Halphen’s problem. | |
| 10/17 | Bernd Ulrich | Linkage IV | |
| Xianglong Ni | Gorenstein ideals of codimension four | The structure of Gorenstein ideals of codimension three is described by the Buchsbaum-Eisenbud structure theorem. In codimension four, there is a mostly settled conjecture that such an ideal of deviation two (the simplest non-trivial case) is a hypersurface section of one in codimension three. But such ideals with more generators remain mysterious. Unlike in codimension three, these ideals need not be in the linkage class of a complete intersection (licci). Kustin conjectured that the licci property could be detected by a sequence of "higher order products" generalizing the multiplication on the free resolution of such an ideal, but he only defined a few of these products. I will explain a conjecture, coming from Weyman's work on generic resolutions, that gives a new perspective on the preceding ideas. Then I will explain how similar ideas have been used to investigate previously unexplored territory in codimension three. | |
| 10/24 | Bernd Ulrich | Linkage V | |
| 10/31 | Bernd Ulrich | Linkage VI | |
| Daigo Ito | Symmetries on derived categories and reconstruction of elliptic curves | We know the structure of the derived category of coherent sheaves on an elliptic curve very well and moreover it has been known that an elliptic curve can be reconstructed from categorical structures of its derived category. In this talk, we will discuss a version of reconstruction of elliptic curves that was announced this year, using symmetry of derived categories coming from their autoequivalences. Moreover, if time permits, we will see possible generalization and obstructions to higher dimensional cases, which were discussed in my recent paper in relation with monoidal structures on derived categories. | |
| 11/8 (Wednesday) 11AM, Evans 748 | Feiyang Lin | Measures of irrationality for hypersurfaces of large degree | Determining whether a variety is rational is a classical and longstanding problem in birational geometry. A paper by Bastianelli et al. proposes a different angle to this question by considering various measures of irrationality that quantify how far a variety is from being rational. Under this framework, rather than asking whether a variety is rational, it makes sense to ask: what are the values of these measures of irrationality of a variety? I will outline the proof of the following result of Bastianelli et al.: Let \( \mathrm{irr}(X) \) denote the least degree of dominant rational maps \( X \) to \( \mathbb P^{\dim X} \). Then for \( X \) a very general smooth hypersurface in \( \mathbb P^{n+1} \) of degree \( d \geq 2n+1 \), the irrationality degree \( \mathrm{irr}(X) \) is \( d-1 \), and any rational map achieving the irrationality degree is given by projection from a point if \( d \geq 2n+2 \). This recovers a theorem of M. Noether about the gonality of smooth plane curves. The key insight is that positivity of the canonical bundle and Mumford's technique of induced differentials allows us to say the fibers of a dominant map to \( \mathbb P^n \) must lie on a line, which converts the problem to a study of first order line congruences in \( \mathbb P^{n+1} \). In the case of surfaces in \( \mathbb P^3 \) and threefolds in \( \mathbb P^4 \), an earlier paper of Bastianelli, Cortini and de Poi characterizes explicitly which hypersurfaces are not very general in this sense. If time permits, I will also discuss some open problems related to this circle of ideas. We work over the complex numbers throughout. |
| 11/14 | Emily Clader | Permutohedral complexes and curves with cyclic action | Although the moduli space of genus-zero curves is not toric, it shares an intriguing amount of the combinatorial structure that a toric variety would enjoy. In fact, by adjusting the moduli problem slightly, one finds a moduli space that is indeed toric, known as Losev—Manin space. The associated polytope is the permutohedron, which has a wealth of other connections: notably, to the structure of the symmetric group and to the combinatorics of matroids. Batyrev and Blume generalized this story by constructing a type-B version of Losev—Manin space whose associated polytope is a signed permutohedron that relates to the group of signed permutations and the combinatorics of delta-matroids. I will discuss joint work with C. Damiolini, C. Eur, D. Huang, S. Li, and R. Ramadas in which we carry out the next stage of generalization, defining a family of moduli spaces of rational curves with \( \mathbb{Z}_r \)-action that can be encoded by a "permutohedral complex" for a more general complex reflection group, and which relates to the combinatorics of multimatroids. |
| 11/21 | (Thanksgiving week) | ||
| 11/28 | Joseph Hlavinka | An Introduction to the Theory of Moduli | Given a functorial association of a collection of geometric objects to every variety, when is this functor actually represented by a (nice) scheme \( S \)? More pointedly, what can such representability properties tell us about \( S \), and what can such a scheme tell us about the so-called "moduli problem" we began with? This talk will be a crash course in answering such questions! |
| Sterling Saint Rain | The Bruns-Eisenbud-Evans Generalized Principal Ideal Theorem | Krull’s Altitude Theorem is a classic result which tells us that the height of an ideal of a Noetherian ring \( R \) is bounded above by the number of its generators. Over the years, this statement has been generalized a number of times; the subject of this talk will be describing and proving the much stronger Bruns-Eisenbud-Evans Principal Ideal Theorem. In addition, we will recall the fundamentals of determinantal ideals, ideals generated by the \( t \times t \) minors of an \( m \times n \) matrix over \( R \), and discuss Bruns’ sharpening of the Eagon-Northcott bound in the special case \( I_t(\phi) \neq R \) with \( I_{t+1}(\phi) = 0\). | |
| 12/5--last seminar of the semester | Swapnil Garg | Why projectivity is not affine-local | I will first show why the definition of a projective morphism is not affine-local. I will then explain the blow-up construction and why it would likely help us construct a counterexample, and then exhibit Hironaka's famous example of a smooth proper non-projective complex threefold. Finally, I will discuss why nevertheless, affine-locality holds in dimension 1 and (assuming smoothness) in dimension 2. |
Tuesdays 3:10-4:30, Evans 939
| date | speaker | title |
|---|---|---|
| 01/24 | Lauren Cranton Heller | Short virtual resolutions |
| 01/31 | no seminar | |
| 02/07 | no seminar | |
| 02/14 | David Eisenbud | Differential graded algebra resolutions |
| 02/21 | Jay Yang | Virtual resolutions of monomial ideals and virtual shellability |
| 02/28 | David Swinarski | Singular curves in Mukai's model of \(\bar M_7\) |
| 03/07 | Anurag Singh | When are the natural embeddings of classical invariant rings pure? |
| 03/14 | Anthony Várilly-Alvarado | Using algebraic geometry to probe the Earth's mantle |
| 03/21 | Xianglong Ni | Schubert varieties and the structure of codimension three perfect ideals |
| 03/28 | spring break | |
| 04/04 | Madeleine Weinstein | Metric algebraic geometry |
| 04/11 | Eric Larson | Interpolation for Brill-Noether curves |
| 04/18 | Eric Larson | The Minimal Resolution Conjecture for points on general Brill--Noether curves |
| 04/25 | Owen Barrett | Semistable reduction for curves over valuation rings |
| 05/02 | Yukari Ito | G-Hilbert scheme and the McKay correspondence |
| 08/31 | David Eisenbud | Summands in high syzygies |
| 09/07 | Lauren Cranton Heller | Multigraded regularity and betti numbers |
| 09/14 | Xianglong Ni | Linkage in codimension three |
| 10/11 | Franny Dean | Log concavity of sequences and a Hodge theory on matroids |
| 10/24 | Robin Hartshorne | History of the complete intersection problem |
| 11/01 | Luis Giraldo | An algebraic geometric approach to study polynomial vector fields in \(\mathbb C^2\) |
| 11/08 | Frank-Olaf Schreyer | Extensions of paracanonical curves of genus 6 |