In Fall 2021 I'm teaching Math 150 (Calculus I) and Math 395 (Problem Solving Seminar). I've been at the University of San Diego since 2018. Before that I was a postdoc at the University of Utah and the University of Edinburgh. I completed my PhD at UC Berkeley under the direction of David Eisenbud. I like to think about algebra, geometry and combinatorics with a focus on minimal free resolutions and Betti numbers.
I am passionate about incorporating undergraduates into my research and love teaching courses at all levels. In 2008, I directed a summer program in Algebraic Geometry and in 2013 I led an REU on commutative algebra and combinatorics and have led one-on-one research projects with undergraduates in Utah. Please follow the links to teaching and research for information and for publications coming from these experiences.
Email: aboocher@sandiego.edu | Office: Saints Hall 161 |
I usually base my course websites on Blackboard.
I enjoy working on research projects with undergraduates in a variety of settings. If you are interested in a reading course or summer project please let me know! I have led the following REUs:
In Summer 2016, I supervised Jimmy Seiner (U. Michigan) on a project in commutative algebra and homological algebra. We continued working after the summer on variants of the Buchsbaum-Eisenbud-Horrocks Rank Conjecture. Our paper is available here.
In 2013, I co-organized an REU at UC Berkeley with two other graduate students. We directed a total of 17 undergraduates in three research projects. The REU was funded by the Geometry and Topology RTG. My group of 6 students studied a problem in Combinatorial Commutative Algebra concerning toric ideals.
Our paper Robust Graph Ideals appears in the Annals of Combinatorics. The students gave two presentations at a conference we organized with Stanford University. Their presentations:
Presentation on Robust Graph Ideals
Presentation on Regularity
(Students advised: Bryan Brown, Timothy Duff, Laura Lyman, Takumi Murayama, Amy Nesky,
Karl Schaefer)
Jimmy after his talk in the Utah Commutative Algebra Seminar
A graph whose toric ideal has primitive but not indispensible binomials (with 2013 REU students)
The Berkeley REU 2013
(The Moebius function on the lattice of flats of matroid computes multi-graded Betti numbers of an associated ideal)
My research broadly concerns interactions between algebraic geometry, combinatorics, and commutative algebra. I am particularly interested
in studying the way in which geometric information is preserved (or changed) upon deformation. Using these techniques I've been able to better understand the minimal free resolution and minimal generating sets of many classes of ideals arising from determinants, matroids, and graphs. It has also allowed us to better understand the way that the deviations of algebras (which are determined by its Poincaré series) behave.
Recently, I've been very interested in bounding the Betti numbers of certain classes of algebras.
For instance in work with Srikanth Iyengar and Hamid Hassanzadeh we look at whether the upper bounds
on the Betti numbers implied by the Taylor Complex hold for arbitrary Koszul algebras.
On the other hand, with Jimmy Seiner, I've studied lower bounds for Betti numbers of monomial ideals.
Electronic copies of my papers are linked below.
Here is my CV.
Two Calculus Projects that I have used in my Math 150 course. The first is a treasure-map oriented (mostly linear) project concerning our hero Percy Precal aand the second is an open-ended detective story about the mysterious orb incident in the land of Calculia
Notes from the Undergraduate Program at Notre Dame in May 2019. The Lecture Notes and exercises offer an introduction to homological algebra and the Hilbert Syzygy Theorem. from a mini-course I ran at Notre Dame.
Some notes (ca 2007) on Algebraic Geometry from a mini-course I ran at Notre Dame.
In Summer 2017, with Troy Jones and Ray Barton, I organized a workshop for high school teachers in Sandy, Utah. The theme was symmetry, and here is a link to some of the math activities we did each morning.
I have recently become a big fan of Mathologer on Youtube. My favorite video helps make sense of the "equality" $$ 1 + 2 + 3 + 4 + \cdots = -\frac{1}{12}$$ by first looking at different types of convergence and then discussing analytic continuations and the gamma function! For those interested in an introduction to higher level math without brushing anything under the rug - this is the channel for you!