Algebra and Algebraic Geometry Seminar

@ McMaster University

This is the homepage of the Algebra and Algebraic Geometry Seminar at McMaster University. Talks take place Mondays at 10:30-11:20am (Eastern). During Fall 2020, all talks will be over Zoom.

Schedule for Winter 2021

Feb 22, 2021: Claudia Miller (Syracuse) **This talk will be at 10:00am**

Title: Torsion in exterior powers of differentials over complete intersection rings

Abstract: In this talk, after a review of the definition of and facts about Kaehler differentials, I will give some history behind the classic Lipman-Zariski Conjecture and the generalized Lipman-Zariski questions of Graf. Then I’ll give some results on the torsion and cotorsion of exterior powers of the module of Kaehler differentials over complete intersection rings, and say how these are used to prove a generalized Lipman-Zariski result under certain conditions. This is joint work with Sophia Vassiliadou.

March 1: Emmanuel Neye

Title: Grobner bases for Kazhdan-Lusztig ideals

Abstract: Schubert determinantal ideals are a class of generalized determinantal ideals which include the classical determinantal

ideals. In this talk, ​we use the approach of "Grobner basis via linkage" ​to ​give a new proof of a well known result ​of Knutson and Miller: the essential minors of every Schubert determinantal ideal form a Grobner basis with respect to a certain term order. We also adapt the Grobner basis via linkage technique ​to the multigraded setting and use this to show that the essential minors of every Kazhdan-Lusztig ideal form a Grobner basis with respect to a certain term order, thereby giving a new proof of a result of Woo and Yong.

March 15: Bradd Hart

March 29: Student talks

Past Talks:

Sept. 14, 2020: Jeremy Lane (McMaster)

Title: Toric degenerations and symplectic geometry

Abstract: In a recent pre-print (arXiv:2008.13656), Benjamin Hoffman and I extended some earlier results of Harada and Kaveh about toric degenerations and gradient-Hamiltonian vector fields. Our main application is to symplectic manifolds equipped with Hamiltonian group actions.

Sept. 28, 2020: Graham Keiper (McMaster)

Title: Regularity and h-Polynomials of Toric Ideals of Graphs

Abstract: I will discuss recent work of Favacchio, Van Tuyl and myself which demonstrates that we may construct toric ideals associated with graphs which have regularity r and h-polynomials of degree d for any 4 \leq r\leq d. The ideas involved rely on Gröbner bases and combinatorics. I will also discuss how we can use this to recover a result of Hibi, Higashitani, Kimura, and O’Keefe.

Oct. 26, 2020: Sergio Da Silva (McMaster)

Title: Understanding Schubert varieties using pattern avoidance

Abstract: Many geometric properties of Schubert varieties in the full flag manifold GL_n/B can be described using pattern avoidance or interval pattern avoidance conditions. These convenient characterizations reduce difficult computations to simple combinatorics involving permutations. For example, a Schubert variety associated to a permutation w is smooth if and only if w pattern avoids 3412 and 4231. Similar descriptions exist for checking factoriality or Gorensteinness, among other properties.

I will start by providing any necessary background on Schubert varieties followed by an overview of known results regarding pattern avoidance. I will also discuss newer research on a combinatorial subword description for the Gorenstein property.

Nov. 9, 2020: Cam Franc (McMaster)

Title: Eisenstein metrics

Abstract: Harmonic metrics on vector bundles mediate the nonabelian Hodge correspondence between categories of irreducible representations of fundamental groups and semi-stable Higgs bundles. By uniformizing the underlying base one obtains a description of harmonic metrics as smooth matrix-valued automorphic forms satisfying a nonlinear PDE. The main step in establishing the nonabelian Hodge correspondence lies in proving the existence of solutions to these PDEs.

In this talk we will discuss a construction of (families of) metrics for bundles on hyperbolic curves that is inspired by the theory of Eisenstein series. We will discuss cases where harmonic metrics can be obtained explicitly as residues of these series. This talk will be partly expository and will assume no prior knowledge of the subject. In particular, we will begin with a summary of basic facts on the Riemann zeta function and classical Eisenstein series.

Nov. 23, 2020: Jenna Rajchgot (McMaster)

Title: Geometric vertex decomposition and liaison

Abstract: Geometric vertex decomposition (a degeneration technique) and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this talk, I will review these two techniques and explain why they are useful for studying certain types of questions in algebraic geometry and commutative algebra. I will then discuss recent joint work with Patricia Klein on an explicit connection between geometric vertex decomposition and liaison, as well as applications of this connection.

Dec. 7, 2020: Craig Kohne (McMaster)

Title: Symbolic powers and the Waldschmidt constant of monomial ideals

Abstract: For an ideal I in a commutative Noetherian ring we can define its symbolic power denoted by I^(n). We will discuss how the symbolic power compares to the ordinary power (i.e. I^n) in the context of polynomial ideals and their associated varieties. The Waldschmidt constant is an invariant of I which measures the growth of the I^(n) relative to I^n as n increases. We will demonstrate how the Waldschmidt constant can be computed as the value of a linear program when I is a monomial ideal.

Zoom Information

This seminar is primarily intended for McMaster University graduate students, postdocs, and faculty with an interest in algebra, algebraic geometry, number theory, or related areas. If you are interested in attending a talk, please contact one of the faculty members affiliated with the seminar (Cam Franc, Megumi Harada, Jenna Rajchgot, Adam Van Tuyl).

Zoom House Rules:

  1. Please enter the meeting 5-10 minutes before the talks you want to attend since we need to approve each participant and we will not allow new participants to join once the talk starts.

  2. Please keep your microphones muted to avoid background noise. (Having your video on is encouraged but not required.)

  3. The preferred way to ask a question is using the Chat.

  4. Alternatively, to ask a question you can raise your hand (click on Participants and then Raise Hand at the bottom of the Participants window). Either the speaker or the hosts will see that you have raised your hand and will get to you at the earliest opportunity.

To quickly mute/unmute yourself, you can use any of the following keyboard shortcuts:

Push to Talk. If you are using the Zoom desktop client on a computer, you can temporarily unmute yourself by pressing and holding the SPACE key on your keyboard. You will be unmuted for as long as you hold the space key.

Mac: Command(⌘)+Shift(⇧)+A

Windows: Alt+M

Linux: Alt+A

iOS/iPadOS with keyboard: Command(⌘)+Shift(⇧)+A