The annual Route 81 Conference on Commutative Algebra and Algebraic Geometry will be held at Cornell University, November 2, 2024.
Conference speakers
Diane Maclagan, University of Warwick
Title: Tropical Vector Bundles
Tropicalization replaces a variety by a combinatorial shadow that preserves some of its invariants. In this talk I will discuss a definition for the tropicalization of a toric vector bundle. This takes the fibers of the vector bundle to tropical linear spaces, which are determined by (valuated) matroids. This is joint work with Bivas Khan.
Matthew Mastroeni, Suny Polytechnic Institute
Title: Some Koszul Algebras Arising from Combinatorics
Koszul algebras have long been studied for their extraordinary homological properties and their prevalence among many naturally occurring rings throughout commutative algebra and algebraic geometry. In this talk, we discuss characterizations of when two families of rings defined by combinatorial data, graded Möbius algebras and the quotients of binomial edge ideals, are Koszul algebras. The graded Möbius algebra of a matroid is an Artinian algebra defined by quadratic relations whose Hilbert function encodes the number of flats of the matroid of a given rank, while binomial edge ideals of graphs generalize the ideals of 2 x 2 minors appearing in algebraic statistics. Despite their very different origins, we show in joint work with Adam LaClair, Jason McCullough, and Irena Peeva how a certain class of graphs characterizes Koszulness for both families of rings.
Jeff Mermin, Oklahoma State University
Title: Which Borel Ideals are Gotzmann?
Let $J\subset k[x_{1},\dots,x_{n}] be generated entirely in degree $d$. We say that $J$ is Gotzmann if its Hilbert function grows as slowly as possible; equivalently, if the lexicographic ideal with the same Hilbert function is also generated entirely in degree $d$.
We say that $J$ is Borel if it is a monomial ideal with the property that, for all $i<j$, if $fx_{j}\in J$, then $fx_{i}\in j$ as well.
Most proofs of Macaulay’s theorem bounding Hilbert function growth involve Borel ideals in an essential way, making it natural to expect that many Borel ideals are also Gotzmann. We show how to determine whether a Borel ideal is Gotzmann or not.
Josh Pollitz, Syracuse University
Title: Loewy Lengths of Modules of Finite Projective Dimension
By now it is well understood that the existence of nonzero modules of finite length and finite projective dimension force certain restrictions on the singularity of the specified local ring. For example, a consequence of the New Intersection theorem is that such a module exists if and only if the ring is Cohen-Macaulay. There has been a great deal of work on understanding the length/Loewy length of such modules, as any lower bounds act as numerical obstructions to the ring having “nice" singularities. In this talk, I will discuss what is already known regarding such lower bounds, and I will present some new results from joint work with Nawaj KC for lower bounds on the Loewy lengths of nonzero modules of finite projective dimension.
Rachel Webb, Cornell University
Title: Abelian Orbicurves with Smooth Coarse Space
If p is a smooth point of a complex curve C, one can form a root stack at p of order r to obtain a smooth stack D. The orbicurve D moreover has the properties that its coarse space is C and its only nontrivial stabilizer is the group Z/rZ at p. This talk is about a much more general construction of abelian orbicurves from a collection of points on a smooth curve C. The integer r is replaced by a certain kind of monoid and the resulting stabilizer group at p can be arranged to equal any finite abelian group. Unlike the classical case, the resulting orbicurve will in general be singular.
The key features of this talk will be an explanation of the construction (how certain monoids encode a curve singularity with group action), including how it works in families, and a characterization of the orbicurves constructed in this way that arise as degenerations of smooth (i.e. classical) orbicurves. The construction has applications to the theory of orbifold stable maps. This talk is about joint work with Martin Olsson.
Registration
To register go to Route 81 Registration.
Registration closes October 27th. https://cornell.ca1.qualtrics.com/jfe/form/SV_7ae1aeEIMiRZx30 https://cornell.ca1.qualtrics.com/jfe/form/SV_7ae1aeEIMiRZx30
There is no registration fee. Registration is for internal use only and to help planning, food counts, accommodations, etc.
Continental breakfast and lunch will be provided.
Conference dinner
A conference dinner is planned for Saturday evening, and more information will be provided once a venue is confirmed.
Following dinner there will be a wine/dessert party at the faculty organizers residence. Everyone is welcome and encouraged to come. More information will be provided at the conference.
Lodging
Due to high demand during the time of this conference, hotels in the area are higher than usual. However, a room block is reserved at The Hotel Ithaca located downtown Ithaca with a rate that is a little more affordable in comparison to other hotels in the area. If you are interested, please reserve your room by October 23, 2024. After this date the rooms will be released. To reserve a room, go to Route 81 Conference Attendees or call the hotel at 607-272-1000, and indicate you would like to make a reservation with the Route 81 Conference room block. The room block group ID is 7183.
Try room sharing to save cost: If you are interested in the sharing a room with another participant use this Room Sharing Form to place your name and contact information on this list. Those who are interested in room sharing can contact each other to make arrangements.
Other Ithaca and Cornell information
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Accessibility
For help with any disability or mobility issues, please contact Heather Peterson, hko1@cornell.edu.
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Inclusivity
Cornell University is committed to providing a safe, inclusive, and respectful learning, living, and working environment. To this end, Cornell will not tolerate sexual and related misconduct. Through Cornell University Policy 6.4, and the applicable procedures, the university provides means to address bias, discrimination, harassment, and sexual and related misconduct, including gender-based harassment, sexual harassment, sexual assault, domestic and dating violence, stalking, and sexual exploitation. Reports of bias, discrimination, and harassment can be made online at www.biasconcerns.cornell.edu or by contacting the Office of the University Title IX Coordinator at titleix@cornell.edu.
Conference Organizers:
- Irena Peeva
- Ritvik Ramkumar
- Michael Stillman