• Contacts
• Research
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Letters of recommendation

I follow Ravi Vakil's rules on letters of recommendation Please read them before requesting a letter.

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]

Graduate students

• Anibal Aravena
  
  
• Sebastian Torres, Windows in Algebraic Geometry and Applications to Moduli
  
  Sebastian applied the theory of windows, as developed by Teleman, Halpern-Leistner, and by Ballard, Favero and Katzarkov, to study certain moduli spaces and their derived categories. Using quantization and other techniques, he proved that GIT quotients of (P¹)ⁿ by PGL₂ without strictly semistable points satisfy a rare property called Bott vanishing, which states that Ω ^j_Y⊗L has no higher cohomology for every j and every ample line bundle L. Previously this was known only for n=5 due to a result of Totaro. Similar techniques are used in the thesis to reprove the well known fact that toric varieties satisfy Bott vanishing. Sebastian also used the theory of windows to explore derived categories of moduli spaces of rank-two vector bundles on a curve. By applying these methods to Thaddeus' moduli spaces, he found 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
  
  Arie studied 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. Arie studied one parameter smoothings in these components that admit an anticanonical model with canonical but non-terminal singularities with the goal of classifying them completely. He identified certain class of "diagonal" smoothings (smoothings with equal axial multiplicities), where the total space is a toric threefold and constructed the anticanonical model explicitly using the toric MMP.
  
  
• Tassos Vogiannou, Spherical Tropicalization
  
  In his thesis, Tassos extended the theory of tropicalization and tropical compactifications of subvarieties of algebraic tori from [15] to subvarieties of spherical homogeneous spaces. He showed the existence of tropical compactifications in a general setting. Given a tropical compactification of a closed subvariety of a spherical homogeneous space, he showed that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the closed subvariety. He provided many 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
  
  Julie found a bound on which singularities may appear on KSBA stable surfaces for a wide range of topological invariants, and used this result to describe all stable numerical quintic surfaces, i.e. stable surfaces with K²= 5, p_g=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. Julie extended 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

Current program participants and alumni (for citations see the research page)

• Aditya Khurmi studies birational geometry and toric geometry with the focus on toric elliptic pairs [33].
• Elias Sink works on two projects. Configurations of lines in P² over various fields have been studied since the 19th century. Particularly fascinating are n₃ configurations, which consist of n points and n lines, with each line containing exactly 3 points and each point lying on 3 lines. An example is the Fano configuration, a 7₃ configuration. A classical question is: Which configurations are realizable over Q? This query is directly related to the modern study of rational points on Calabi-Yau varieties. It has been shown that the realization space of an n₃ configuration, constructed as a GIT quotient with democratic polarization, is a Calabi-Yau variety (potentially singular) if it has the expected dimension of n-8. Elias Sink has completed the K3 surface case (n=10) in the paper Line configurations and K3 surfaces. There are exactly 10 such configurations, including the Desargues configuration, where the nine collinearities imply the tenth, according to Desargues' theorem, thus making the realization space three-dimensional. There is also a unique non-realizable configuration. Among the rest, four configurations' GIT quotient is a reducible Kulikov degeneration of a K3 surface. Finally, four configurations yield beautiful smooth K3 surfaces with a Picard number of 20. Answering a question posed by Sturmfels in 1991, Elias proves that their realizations over Q are dense in those over R. These K3 surfaces can be precisely identified, including the universal elliptic curve over Γ₁(7).
  
  
  
  
  In his second project, Elias focuses on the moduli space of stable pairs (vector bundles with a non-zero section on an algebraic curve) with a fixed even determinant. Extending the methods used in the odd case (as in the proof of the Narasimhan conjecture in [36]), Elias constructs the semi-orthogonal decomposition of the derived category.
  
  Elias will continue his studies at the PhD program in Harvard with the NSF graduate research fellowship (GRFP).
  
  
• Pranav Ramakrishnan studied the density of polyhedral primes for blow-ups of P². The theory of elliptic pairs, as investigated in the paper [33], provides useful conditions to determine the polyhedrality of the pseudo-effective cone. These conditions lead to intriguing arithmetic questions when reducing the variety modulo p. In his paper The Arithmetic of Elliptic Pairs and An (n+1)-variable Artin Conjecture, Pranav examines a specific case: the blow-up X of 9 points in P² lying on a nodal cubic. He studies the density of primes p for which the pseudo-effective cone of the reduction of X modulo p is polyhedral. This problem is can be reduced to an analogue of Artin's Conjecture on primitive roots, as previously explored by Stephens and later by Moree and Stevenhagen. Pranav's findings show that the density of such "polyhedral primes" approximates a higher analogue of Stephens' Constant, contingent on the assumption of the Generalized Riemann Hypothesis. To determine a precise value for the density of polyhedral primes, he examines the containment of rank 8 root sublattices in E₈.
  
  Pranav has been admitted to the PhD program at Boston College.
  
  
  
  
• Viviene Do studied scattering amplitudes of stable M-curves of genus 1 and 2, see [32]
  
  
• Elizabeth Pratt studied toric elliptic pairs (as detailed in [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. These pairs are valuable for determining whether the pseudo-effective cone of X is polyhedral, and they are interesting algebraic and geometric objects in their own right. Particularly notable 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, published in J. Pure Appl. Algebra (2023), Elizabeth classified all toric elliptic pairs of Picard number 2. Remarkably, only three such pairs exist. Additionally, she explored a class of non-toric elliptic pairs arising from the blow-up of P² at nine points on a nodal cubic, in characteristic p. This study reveals examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes p of positive density, and, under the assumption of the generalized Riemann hypothesis, polyhedral for a set of primes p of positive density.
  
  Elizabeth is currently a graduate student at UC Berkeley, holding an NSF graduate research fellowship (GRFP).
  
  
  
  
• Shelby Cox explored log canonical models of hyperplane arrangement complements, with a particular focus on the braid arrangement (as discussed in [19] and [15]). She developed an Honors Thesis on this subject. Shelby's contributions in this area have been acknowledged with the UMass Rising Researcher award.
  
  Shelby is continuing her academic journey as a graduate student at the University of Michigan with the NSF Graduate Research Fellowship (GRFP) .
  
  
• Greg McGrath studied Seshadri constants on irrational surfaces. An interpolation problem is a challenge in which the goal is to find a geometric object that passes through a set of sufficiently general points with specified multiplicities. These types of questions trace back to the fundamentals of geometry, as exemplified by Euclid's postulation that a line can be drawn through any two points. Despite the fundamental nature of such questions, our understanding of polynomial interpolation remains limited. For curves in P², Nagata's Conjecture offers a good prediction, although a proof is still elusive. In exploring interpolation on more complex varieties, an interesting avenue is computing Seshadri constants, which quantify the maximum multiplicity an irreducible curve can have at a particular general point. Greg authored an Honors Thesis on the Seshadri constants of the symmetric square of an elliptic curve. More recent developments related to this topic 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 currently a graduate student at UC Santa Barbara.
  
  
• Stephen Obinna studied effective cones of blow-ups of toric surfaces at the identity element of the torus (as discussed in [24]). While examples of toric surfaces of Picard number 3, whose blow-ups at the identity element have a non-polyhedral effective cone, were later constructed (see [33]), the case of Picard number 2, which Stephen investigated, remains unresolved. He created an online database of these surfaces, featuring numerous curves with points of high multiplicity.
  
  Stephen is currently a graduate student at Brown University.
  
  
  
  
• Morgan Opie focused her studies on effective divisors on M_0,n and authored a paper titled Extremal divisors on M_0,n, published in the Michigan Math. J., V. 65(2) (2016), pages 251-285. In this work, she identified a counterexample to an overly optimistic conjecture from [22], also discussed in [33]. Morgan developed an online database of hypertree divisors, extending the earlier work by Ilya Scheidwasser (see [32]). Her achievements have been recognized with several awards, including the Churchill Scholarship and the UMass Rising Researcher award. Morgan was also a runner-up for the Alice T. Schafer Prize, a national award for excellence in mathematics by an undergraduate woman.
  
  Morgan earned her PhD from Harvard with the NSF graduate research fellowship (GRFP) and is currently a Hedrick Assistant Adjunct Professor and NSF Postdoc at UCLA.
  
  
• Nicky Reyes focused his research on extremal P-resolutions of cyclic quotient singularities, verifying a conjecture (later proven and discussed in [25]) that these resolutions occur in pairs, at most.
  
  Nicky earned his PhD degree from the University of Texas at Austin with the NSF graduate research fellowship (GRFP). He is currently employed at UT Austin as an Assistant Professor of Instruction.
  
  
  
  
• Nate Harman focused his research on the combinatorics of hypertrees, contributing several significant findings to the paper [22].
  
  Nate earned his PhD degree from MIT with the NSF graduate research fellowship (GRFP). Nate is an Assistant Professor at the University of Georgia.
  
  
  
  
• Charles Boyd made a pivotal contribution to the paper [26] by discovering a Q-Gorenstein degeneration of the Craighero-Gattazzo surface into a surface with a Wahl singularity in characteristic 7.
  
  Charles completed an internship at Macaulay 2, a computer algebra system, and is currently employed as a Systems Software Developer at cPanel.
  
  
  
  
• Ilya Scheidwasser conducted studies on hypergraph curves and hypertrees, as referenced in [22] and [32]. He authored a Capstone thesis on these topics and also developed a C++ program that successfully identified all hypertrees with 7, 8, 9, 10, and 11 vertices.
  
  Ilya received his PhD degree from Northeastern University and is currently a Principal Software Developer at athenahealth.
  
  
• Alex Levin focused his research on the dual complex of the boundary of the Naruki moduli space of cubic surfaces, as discussed in [20]. He successfully proved that this complex is shellable.
  
  Alex received his PhD degree from the University of New Hampshire and is currently serving as a lecturer at the University of Vermont.