Letters of recommendation

I follow Ravi Vakil's rules on letters of recommendation

Former UMass postdocs:

Luca Schaffler (Roma Tre University): [30]

Giancarlo Urzua(Universidad Catolica de Chile) : [35], [26], [25]

Ana-Maria Castravet (Versailles): [33], [31], [28], [27], [24], [22], [21], [13]

Former graduate students and their dissertations:

Sebastian Torres, Windows in Algebraic Geometry and Applications to Moduli

Abstract. We apply the theory of windows, as developed by Halpern-Leistner and by Ballard, Favero and Katzarkov, to study certain moduli spaces and their derived categories. Using quantization and other techniques we show that stable GIT quotients of (P1)n by PGL2 over an algebraically closed field of characteristic zero satisfy a rare property called Bott vanishing, which states that Ω jY⊗L has no higher cohomology for every j and every ample line bundle L. Similar techniques are used to reprove the well known fact that toric varieties satisfy Bott vanishing. We also use windows to explore derived categories of moduli spaces of rank-two vector bundles on a curve. By applying these methods to Thaddeus' moduli spaces, we find a four-term sequence of semi-orthogonal blocks in the derived category of the moduli space of rank-two vector bundles on a curve of genus at least 3 and determinant of odd degree, a result in the direction of the Narasimhan conjecture.

Arie Stern Gonzalez, Anticanonical Models of Smoothings of Cyclic Quotient Singularities

Abstract. In this thesis we study anticanonical models of smoothings of cyclic quotient singularities. Given a surface cyclic quotient singularity Y, it is an open problem to determine all smoothings of Y that admit an anticanonical model and to compute it. In [25], Hacking, Tevelev and Urzua studied certain irreducible components of the versal deformation space of Y , and within these components, they found one parameter smoothings of Y that that admit an anticanonical model and proved that the total spaces have canonical singularities. Moreover, they compute explicitly the anticanonical models that have terminal singularities using Mori's division algorithm. We study one parameter smoothings in these components that admit an anticanonical model with canonical but non-terminal singularities with the goal of classifying them completely. We identify certain class of "diagonal" smoothings where the total space is a toric threefold and we construct the anticanonical model explicitly using the toric MMP.

Tassos Vogiannou, Spherical Tropicalization

Abstract. In this thesis, I extend tropicalization of subvarieties of algebraic tori over a trivially valued algebraically closed field to subvarieties of spherical homogeneous spaces. I show the existence of tropical compactifications in a general setting. Given a tropical compactification of a closed subvariety of a spherical homogeneous space, I show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the closed subvariety. I provide examples of tropicalization of subvarieties of GL(n), SL(n), and PGL(n).

Julie Rana, Boundary Divisors in the Moduli Space of Stable Quintic Surfaces

Abstract. I give a bound on which singularities may appear on KSBA stable surfaces for a wide range of topological invariants, and use this result to describe all stable numerical quintic surfaces, i.e. stable surfaces with K2= 5, pg=4, and q=0, whose unique non Du Val singularity is a Wahl singularity. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry. I then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space of KSBA stable numerical quintic surfaces corresponding to these surfaces.

Research and training program in Undergraduate Algebraic Geometry

Since 2007, a rigorous year-long training program has been established and supported by the National Science Foundation and internal grants. The program is open to motivated undergraduate students who are interested in pursuing a career in STEM. Those who are interested in participating are encouraged to apply in December by submitting their CV, unofficial transcript, and a statement of interest.

Applicants who have taken graduate courses or are from groups underrepresented in STEM will receive preference during the selection process. The program kicks off in the Spring semester with a comprehensive reading course on algebraic geometry. Following that, the students will engage in a Research Experience for Undergraduates (REU) in the summer.

In the Fall semester, the program offers another independent study to help students concentrate on their writing skills and complete either an Honors thesis or a paper. The program also provides support and guidance for graduate school and scholarship applications, along with mentoring to help students achieve their career goals. Overall, this program offers a comprehensive and in-depth training experience that is tailored to the unique needs of each individual student.

As a Math Alliance mentor , I support underrepresented undergraduate students in pursuing doctoral studies in math. I offer guidance, personalized mentoring, and connect them to resources and opportunities to develop their skills and succeed in their field.

Advice for UMass undergraduate students pursuing research in pure mathematics

Program participants and alumni (for citations see the research page)

Elias Sink

Aditya Khurmi

Viviene Do studies scattering amplitudes of stable M-curves of genus 1 and 2, see [32] .

Pranav Ramakrishnan studies density of polyhedral primes for blow-ups of P2, see[33] .

Lizzie Pratt studied toric elliptic pairs (see [33] ). An elliptic pair (X, C) is a projective rational surface X with log terminal singularities, and an irreducible curve C contained in the smooth locus of X, with arithmetic genus 1 and self-intersection 0. They are a useful tool for determining whether the pseudo-effective cone of X is polyhedral, and interesting algebraic and geometric objects in their own right. Especially of interest are toric elliptic pairs, where X is the blow-up of a projective toric surface at the identity element of the torus. In her paper

The Geometry Of Elliptic Pairs, J. Pure Appl. Algebra (2023),

Lizzie classified all toric elliptic pairs of Picard number 2. Strikingly, it turns out that there are only three of these. Furthermore, she studied a class of non-toric elliptic pairs coming from the blow-up of P2 at nine points on a nodal cubic, in characteristic p. This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes p of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes p of positive density. Lizzie is currently a graduate student at UC Berkeley with the NSF postdoctoral fellowship.

Shelby Cox explored log canonical models of hyperplane arrangement complements, focusing on the braid arrangement (see [19] and [15]) and produced an Honors Thesis on this topic. Her contributions were recognized with the NSF Graduate Research Fellowship (GRFP) and UMass Rising Researcher award. Shelby is now continuing her studies as a graduate student at the University of Michigan. .

Greg McGrath studied Seshadri constants on irrational surfaces. An interpolation problem is a game where you want to find some geometric object that passes through a collection of sufficiently general points with certain multiplicities. Such questions date all the way back to the beginning, when Euclid postulated that there is a line that passes through any two points. For being such a fundamental question, we know very little about polynomial interpolation. For curves in P2, we have a good prediction of what will happen through Nagata's Conjecture, although we are very far from being able to prove it. In investigating interpolation on more complicated varieties, an interesting problem is computing Seshadri constants, which measure the maximum multiplicity an irreducible curve can have at a given general point. Greg wrote an Honors Thesis about Seshadri constants of the symmetric square of an elliptic curve. More recent results related to this variety can be found in the paper Effective cone of the blow up of the symmetric product of a curve by A. Laface and L. Ugaglia. Greg is a graduate student at UC Santa Barbara.

Stephen Obinna studied effective cones of blow-ups of toric surfaces in the identity element of the torus(see [24]). While examples of toric surfaces of Picard number 3 such that its blow-up at the identity element has a non-polyhedral effective cone were later constructed in [33] , the case of Picard number 2 studied by Stephen is still open. He created an on-line database of these surfaces containing many curves with a point of high multiplicity. Stephen is a graduate student at Brown University.

Morgan Opie studied effective divisors on M0,n and wrote a paper

Extremal divisors on M0,n, Michigan Math. J., V. 65(2) (2016), 251-285,

where she found a counterexample to an over-optimistic conjecture from [22], see also [33]. Morgan also created an on-line database of hypertree divisors, building on the earlier work by Ilya Scheidwasser, see also [32] For her achievements, Morgan received the Churchill scholarship, the NSF GRFP. and the UMass Rising Researcher award. She was also a runner-up for the Alice T. Schafer prize, a national prize for excellence in mathematics by an undergraduate woman. Morgan received a PhD from Harvard and is currently a Hedrick Assistant Adjunct Professor and NSF Postdoc at UCLA.

Nicky Reyes studied extremal P-resolutions of cyclic quotient singularities, confirming a later proved conjecture (see [25]) that they come in (at most) pairs. He received the NSF GRFP. and the PhD degree from UT Austin, where he currently works as Assistant Professor of Instruction.

Nate Harman studied combinatorics of hypertrees, contributing several results to the paper [22]. He received the NSF GRFP and a PhD degree from MIT. He is currently a Postdoctoral Assistant Professor and IBL Fellow at the University of Michigan.

Charles Boyd provided a critical contribution to the paper [26] by finding a Q-Gorenstein degeneration of the Craighero-Gattazzo surface to a surface with a Wahl singularity in characteristic 7. He had an internship at Macaulay 2, a computer algebra system, and is currently a Systems Software Developer at cPanel.

Ilya Scheidwasser studied hypergraph curves and hypertrees (see [22], [32]), and wrote a Capstone thesis about them, as well as a C++ program that found all hypertrees with 7, 8, 9, 10, and 11 vertices. Ilya received a PhD degree from Northeastern University and is currently a Principal Software Developer at athenahealth.

Alex Levin studied the dual complex of the boundary of the Naruki moduli space of cubic surfaces (see [20]) and proved that it is shellable. He received a PhD degree from University of New Hampshire and is currently a lecturer at University of Vermont.