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Office Hours:
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Best Witchcraft is Geometry
To the magician's mind -
His ordinary acts are feats
To thinking of mankind.
 - Emily Dickinson

Sculptor Barbara Hepworth sanding a curved form sculpture.

Algebraic geometry is one of the most advanced and multifaceted areas of modern mathematics. The objects of study are algebraic varieties: shapes defined by systems of polynomial equations. Polynomial constraints on variables are commonplace both in mathematics (for example, the famous Fermat equation xn+yn=zn in number theory) and in applications ranging from physics, where varieties are used to describe configuration spaces (and even the shape of the universe!), to computer science and statistics. The symmetry and beauty of algebraic varieties, especially curves and surfaces, makes them instantly recognizable in works of art such as the sculptures of Barbara Hepworth.

The mathematicians Jean-Pierre Serre and Alexander Grothendieck embraced the utility of inherently rigid polynomial functions in addition to more flexible continuous and differentiable functions of topology and analysis. Their approach dramatically expanded the geometer's toolkit to include the sophisticated algebraic techniques developed by Emmy Noether and her school. Paraphrasing the modern geometer Claire Voisin, research in algebraic geometry involves studying the variety of perspectives from which one can see the same object, and using the constant moving back and forth between several geometries and several types of tools to prove results in one field or another.

My research focusses on describing algebraic varieties in a concise coordinate-free way that reveals hidden essential features of their geometry. Henri Poincaré's crucial insight was that one could study the geometry of a complicated space by probing it with simpler objects such as circles and spheres. This leads to homology and homotopy theories of algebraic topology with its cornucopia of discrete invariants. For example, Riemann surfaces (such as the surface of a donut or a pretzel) are classified topologically by the number g of holes in them. But Bernhard Riemann also knew that these surfaces depend on n=3g-3 complex parameters, which can be interpreted as coordinates on the n-dimensional "moduli space" that encodes all possible geometries of, say, a pretzel. The geometry of moduli spaces is an exciting and active area of research that has brought many Fields medals in recent years, including to Maryam Mirzakhani, who was the first woman to receive it. I have contributed various new techniques to the study of moduli spaces and related algebraic varieties, which have allowed us to solve important open problems and conjectures, as well as to open up new venues of research.

A curved triangle with dots on the pointed edges.

One approach to solving geometric problems was pioneered by Isaac Newton, who first studied the asymptotic behavior of a system of functions using the convex geometry of shapes such as a triangle or an icosahedron. One can view these forms as a polyhedral approximation of a curved geometry. In a new twist, I used these approximations to compactify a large class of algebraic varieties including many important moduli spaces, including the moduli space of cubic surfaces with its beautiful configuration of 2727 lines discovered in the 19th century by George Salmon and Arthur Cayley (the study of this moduli space was a joint work with Paul Hacking and Sean Keel). Perhaps the most well-known example of a compactification is the projective plane, which was discovered by the Renaissance artists in the process of developing the mathematical theory of drawing in perspective. The projective plane enriches the Euclidean plane by adding vanishing points at infinity along the horizon line. My tropical compactifications add points at infinity in the systematic fashion encoded in the Newton polytope for algebraic varieties determined by a single equation and in the tropicalization for varieties defined by a system of equations. The notion of a tropicalization or a min-max algebra was originally introduced in the computer science community. This unexpected and fruitful connection has found multiple applications ranging from algebraic geometry to combinatorics, algebraic statistics and the development of new algorithms for computer algebra systems.

A diagram of the moduli space.

Another approach is linearization, which on the most basic level is the approximation of the graph of a function by the tangent line with the slope given by the derivative. Any space can be obtained by gluing its local pieces, which induces coordinate transformations between tensor fields of classical mechanics. In algebraic geometry, we can linearize an algebraic variety by its derived category, which encodes the gluing information in the interplay of ancillary geometric structures of the space such as vector bundles or D-branes in the language of string theory. Topological information such as homology is enhanced in the derived category to the point when equivalence of derived categories of two algebraic varieties either forces them to be the same (for example for algebraic curves) or to be related in a non-trivial way (for example for Calabi-Yau geometries). After many years of work and hundreds of pages of proof, I and Ana-Maria Castravet proved the conjecture of Yuri Manin and Dmitri Orlov on the structure of the derived category of the moduli space M0,n of stable rational curves with punctures. This amazing space is extensively studied because it appears in many physical theories, from the scattering amplitudes of elementary particles to mirror symmetry, where Calabi-Yau and related spaces can be probed by punctured Riemann spheres. As a byproduct of our analysis, we have discovered the geometric foundation for the combinatorial notion of derangements immortalized in the dream sequence from the movie "A Beautiful Mind".

All differentiable manifolds are locally alike and only global phenomena are studied, but every algebraic variety is local in its own way and local phenomena affect global ones. This is known as birational geometry, and I have made several contributions to it, from the proof of the Tessier conjecture in the singularity theory to the detailed study for the moduli space M0,n. Much information can be visualized in the effective cone, which is a convex subset of the Euclidean space that, roughly speaking, describes all possible ways to present a model of an algebraic variety by a system of polynomial equations. The study of effective cones of moduli spaces has a long history starting with the work of Joe Harris and David Mumford, who used computations of effective divisors to show that the moduli space of curves of large genus is not covered by rational curves. A finer invariant is the total coordinate ring of an algebraic variety introduced by David Cox. With Ana-Maria Castravet, we proved that this ring is infinitely generated for M0,n. In other words, this space has infinite complexity and can't be understood by a computer. We also described many vertices of its effective cone, which we called hypertrees. For example, every bi-colored triangulation of the sphere gives one. Their equations appear as numerators of scattering amplitude forms for n particles in N=4 Yang--Mills theory in the work of physicists Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka. Rather than a coincidence, this is just the tip of the iceberg of an exciting relation between algebraic geometry and high energy physics, which I interpreted as the study of statistics of images of marked points under a random meromorphic function uniformly distributed with respect to the translation-invariant volume form of the Jacobian torus.

A graph of primarily brown and red converging on a point, gradually becoming yellow to green to blue.

In fact, we proved that the effective cone of M0,n has infinitely many vertices (joint work with Ana-Maria Castravet, Antonio Laface and Luca Ugaglia). Moreover, as most interesting algebraic varieties, M0,n is cut out by equations with integer coefficients, which can be reduced modulo prime numbers. The solution set modulo p has an extremely interesting finite geometry, which echoes the continuous geometry of the real world and provides deep insights about it. This is why the first geometer Euclid included basic number theory in his "Elements" and proved that there are infinitely many prime numbers. In a far-reaching generalization, we prove that the effective cone of the reduction of M0,n modulo p has infinitely many vertices for infinitely many primes p. Our analysis is based on applying tools of modern Galois theory to a fascinating new class of arithmetic objects related to elliptic curves and of interest to cryptography.

A curved, winding structure.

With Julie Rana and Giancarlo Urzua, we used reduction modulo prime number 7 to construct an arithmetic wormhole connecting two faraway points in the moduli space of algebraic surfaces, which is one of my several contributions to the study of higher-dimensional analogues of the moduli space of algebraic curves. We proved that the Craighero-Gattazzo surface is simply-connected, providing the only known explicit example of a negatively curved algebraic variety which can be deformed to a positively curved one continuously but not smoothly.

A textile animal on a woven pillow.

Apart from algebraic geometry, I am studying the history of mathematics in South America. Pre-Columbian Andean cultures did not develop a system of writing and relied instead on the artistic media of ceramics and textiles, as well as large-scale engineering, architectural and agricultural projects, to preserve and propagate their cultural heritage. Paracas funerary mantles, Wari tunics, Chancay gauzes are brimming with patterns, fractals and complicated geometric transformations. I have developed a course on the geometry of Andean textiles, where I reflect on my experience as a geometer eager to find inspiration in the work of people, communities and cultures all over the world, in the present and in the past. The traditional approach to history of mathematics is based on the western tradition but my course emphasizes multiculturalism and a diversity of approaches to mathematics.


Algebraic Geometry

Selected Publications

  • Categorical aspects of the Kollár-Shepherd-Barron correspondence, with Giancarlo Urzua, 43p. (2022), arXiv:2204.13225
  • The BGMN conjecture via stable pairs, with Sebastian Torres, 41p. (2021), arXiv:2108.11951
  • Blown-up toric surfaces with non-polyhedral effective cone, with Ana-Maria Castravet, Antonio Laface and Luca Ugaglia, 55 p. (2020), arXiv:2009.14298
  • Scattering amplitudes of stable curves, 51 p. (2020), arXiv:2007.03831
  • Derived category of moduli of pointed curves - II, with Ana-Maria Castravet 88 p. (2020), arXiv:2002.02889
  • Compactifications of moduli of points and lines in the projective plane, with Luca Schaffler, IMRN (2021), 79p.
  • Spherical Tropicalization with Tassos Vogiannou, Transformation Groups 26 (2021), 691-718 
  • Exceptional collections on certain Hassett spaces, with A-M Castravet, Épijournal de Géométrie Algébrique 4 (2020), 1-34
  • Derived category of moduli of pointed curves - I, with A-M Castravet, Algebraic Geometry 7 (6) (2020), 722-757
  • The Craighero-Gattazzo surface is simply-connected, with J.Rana and G.Urzua, Compositio Math. 153 (2017), 557-585 
  • Flipping Surfaces, with P. Hacking and G.Urzua, Journal of Alg. Geom. 26 (2017), 279-345
  • M0,n is not a Mori Dream Space, with A-M Castravet, Duke Math. Journal, 164 (2015), 1641-1667
  • On a Question of Teissier, Collectanea Math., 65 (2014), 61-66
  • Hypertrees, Projections, and Moduli of Stable Rational Curves, with A-M Castravet, Crelle's Journal, 675 (2013), 121-180
  • Stable Pair, Tropical, and LC Moduli of Del Pezzo Surfaceswith P.Hacking and S.Keel, Inventiones 178 (2009), 173-228
  • Equations for M0,nwith S.Keel, International Journal of Math. 20 (2009), 1--26
  • Elimination Theory for Tropical Varietieswith B.Sturmfels, Math. Res. Lett., 15 (2008)
  • Compactifications of Subvarieties of Tori, American Journal of Math, 129 (2007), 1087-1104
  • Geometry of Chow Quotients of Grassmannianswith S.Keel, Duke Math. J. 134 (2006), 259-311
  • Compactification of Moduli of Hyperplane Arrangementswith P.Hacking and S.Keel, J. of Alg. Geom. 15 (2006), 657-680
  • Hilbert's 14-th Problem and Cox Rings, with A.-M. Castravet, Compositio Math. 142 (2006), 1479-1498
  • Projective Duality and Homogeneous SpacesSpringer, 2005