Brendan Hassett - US grants

The investigator and his colleagues
will address a number of
fundamental problems about
the geometry and arithmetic of moduli spaces:
What is the canonical model of the moduli space of curves of genus g?
Can one construct natural spaces that interpolate between the
canonical model and the moduli space of stable curves?
Can these be interpretted as moduli spaces in their own right?
Can one count the number of rational
points of moduli spaces of curves, bounded
with respect to various heights? What are the natural
functions (effective divisors) one might use to specify these
heights?
What are the naturally defined strata
in the deformation space of a plane curve singularity?
How can one interpret blow-ups along such strata?

Algebraic geometry is the study of the solutions
to systems of polynomial equations. Geometric
aspects of the solution sets translate into algebraic
properties of the equations, and vice versa. This
approach has the advantage that computers can
manipulate equations very efficiently. Promising
hypotheses can thus be checked on explicit examples.
Increasingly, computational approaches to geometry
are transforming the education of undergraduate
and graduate students, both in pure mathematics
and in related fields where mathematics is applied.
This emphasis on concrete examples and explicit computation
creates new research opportunities for young people,
especially those who have only just started to learn
the technical intricacies of the subject.

This project addresses the geometry of spaces of rational curves
on smooth projective varieties, with a view toward understanding the
structure of rational points for varieties defined over function
fields. Consider a rationally-connected variety: Which homology
classes contain free rational curves? Very free rational curves? Is
the space of such curves connected? Irreducible? Rationally
connected? Of general type? Is there a workable notion of `rational
simple connectedness' and is this a birational property? How can we
distinguish unirational varieties as a subclass of rationally-connected
varieties? These questions are related to fundamental problems in Diophantine
geometry over function fields: Does a rationally-connected variety
over C(t) satisfy weak approximation? Can the hypothesis of the Tsen/Lang
Theorem over C(s,t) be formulated geometrically? For rationally-connected
varieties over C(s,t), to what extent do cohomological obstructions
govern the existence of rational points?


This award will support research on systems of polynomial
equations with coefficents varying in parameters. Our goal is to
solve these equations with rational functions that depend on these
parameters. The case of a single equation (or of several independent
equations) was addressed in the mid 20th century; the feasibility
of finding a solution depends on the degree of the equation, the number
of free variables, and the number of varying parameters. Recently,
a comprehensive geometric approach was developed when there is just
one varying parameter. However, for multiple (not necessarily
independent) equations in two varying parameters much remains to be
understood. This work will also have broader impacts on the education of
graduate students and postdoctoral fellows, the development of web-based
collaboration tools, and the promotion of robust academic networks
linking universities across the country.

This project addresses three central problems in algebraic geometry:
Can one compute the ample cone of a polarized holomorphic-symplectic variety from its Hodge structure? What is the right functorial definition for compact moduli spaces of higher-dimensional varieties (and what might they be good for)? To what extent does the birational geometry of moduli spaces govern the behavior of related Geometric Invariant Theory problems, and vice versa? These questions are intertwined in intricate and beautiful ways: Intersection-theoretic constructions govern curve classes on both the moduli space of stable curves and holomorphic-symplectic varieties. The Torelli Theorem for K3 surfaces is the starting point for their moduli theory; the lack of such a result for higher dimensional holomorphic-symplectic manifolds is a major impetus for analyzing their ample cones. The elusive dream of a geometric compactification for the moduli space of K3 surfaces animates work on the interplay between Geometric Invariant Theory and moduli spaces.


Algebraic geometry is the study of geometric objects defined by polynomial equations, which are called varieties. Examples of varieties include circles, ellipses, parabolas, spheres, etc. A fundamental problem is to classify all the varieties of a given type. One approach is to analyze all the varieties defined by polynomials of given degree, e.g., the conic sections studied in high school analytic geometry. Here the type of the variety is expressed in algebraic terms. Alternately, one can study all the varieties sharing common geometric characteristics, e.g., those with given numerical invariants. This project addresses the interplay between the algebraic and geometric quantities, and how these govern the behavior of families of varieties.

This project addresses the theory of rational curves on K3 surfaces, as a prototype and model for investigations of rational curves on higher-dimensional varieties of Fano and intermediate type. The key problems concern the existence of infinitely many rational curves on K3 surfaces over countable fields, techniques for the generation of such curves and computation of their numerical invariants, Brill-Noether loci and enumerative geometry of rational curves, aspects of mixed-characteristic deformation theory, Galois representations, and Brauer groups.


The term `K3 surface' was coined by A. Weil in the 1950's, and honors the seminal contributions of Kummer, Kaehler, and Kodaira to their structure. These surfaces have been central to complex geometry for decades, but recently their arithmetic properties have received increasing attention. This project addresses problems at the interface of complex, algebraic, and arithmetic geometry. In particular, what is the structure of the curves on a K3 surface? Can they be constructed explicitly? And how do they reflect symmetries of the ambient surface?

The Institute for Computational and Experimental Research in
Mathematics (ICERM) is a new Institute. Its mission is to support
and encourage research in the mathematical sciences with a
thematic focus on the fruitful interplay between mathematics and
computers, explored and developed through computation and
experimentation. ICERM will accomplish these goals through a
variety of programs and activities. The Institute will run one
large research program per semester, with each program focusing
on a particular subtopic that aligns with the general
theme. Program participants will range from graduate students to
senior researchers. During the summers, the Institute will host
additional programs for postdoctoral scholars and early career
researchers, as well as for undergraduates selected from a
national pool.

ICERM will be a novel institute in an area of growing
importance. An Institute in computational and experimental
mathematics has the potential to cut across disciplines within
and beyond mathematics and statistics. The Institute programs
will strongly emphasize training and retaining graduate students
and postdoctoral scholars in the mathematical sciences community.
ICERM's leadership has substantial experience in supporting and
working with underrepresented groups in the mathematical
sciences, and Institute programming will provide new
opportunities for underrepresented students and scholars. ICERM
has already initiated partnerships with leading research intense
companies and national research laboratories, and effective plans
to nurture these partnerships will help to sustain a rich and
deep interaction with the broader research community.

Rice University has a dynamic geometry group, with senior faculty in low- dimensional topology; quasi-crystals, spectral theory, and mathematical physics; geometric measure theory; algebraic and complex geometry; and Teichmueller theory and minimal surfaces. This project will support a Research Training Group led by these faculty, involving undergraduate students, graduate students, and postdoctoral fellows. Our main objective is to increase the number of students and postdocs pursuing independent research in geometry, as well as related areas of topology and analysis.

Geometric problems stimulate progress across almost every subfield of mathematics. The study of conic sections initiated by the ancient Greeks spurred the development of coordinate systems and polynomial algebra in the 17th century, leading to the modern field of algebraic geometry. The desire to find curves and surfaces minimizing the energy of physical systems motivated the development of calculus. More recently, geometric questions about knots and links have led to numerous advances in modern algebra. Thus geometric examples offer common ground where specialists in different areas can exchange ideas and techniques. The Geometry Group at Rice builds on this common ground to train students and postdoctoral fellows in the methods of mathematical research.

Diophantine geometry is the study of integer solutions of polynomial equations, seen through the prism of the geometry of their solutions over the complex numbers. Structural characteristics of the integer solutions are sometimes governed by subtle geometric phenomena. We rely on results on the geometry of complex surfaces, especially those defined by equations of small degree. Examples include equations of degree three or four in three variables. Our hope is to discern larger patterns governing the behavior of large classes of problems sharing common characteristics.

This project addresses problems at the interface of algebraic and Diophantine geometry arising from fundamental questions about the behavior of rational points on algebraic varieties. In the simplest situations, these touch on beautiful constructions from classical algebraic geometry. Beyond these, one quickly encounters deep geometric problems not accessible through classical techniques. Specific research questions will include: How to interpret moduli spaces of K3 surfaces with level structure geometrically? To what extent can these be organized using notions of derived equivalence for K3 surfaces and their twisted analogs? Can these techniques be used to evaluate Brauer-Manin obstructions explicitly? Given two derived-equivalent K3 surfaces, how are their Diophantine properties related, especially over local fields? For del Pezzo fibrations over curves, how are spaces of rational curves governed by cohomological invariants? As the numerical invariants of the fibrations vary, what inductive structures are shown by the spaces of rational curves? These will be addressed using deformation-theoretic properties of rational curves, structural descriptions of cones of effective curves arising from Bridgeland stability conditions, and classifications of degenerate fibers of K3 fibrations.

The American Mathematical Society Summer Research Institute in Algebraic Geometry will be held July 13-31, 2015 on the campus of the University of Utah in Salt Lake City. Since the 1960s, the US algebraic geometry community has organized decennial meetings under the auspices of the American Mathematical Society. These extended summer conferences summarized progress in the field and pointed to future developments. The resulting proceedings volumes have been valuable references for generations of algebraic geometers. As the field has grown, these meetings help different communities within algebraic geometry keep in contact and current on the broader trends in the field. This award supports travel and local expenses for approximately 200 of the anticipated 600 participants in the 2015 summer research institute.

The last decade has seen major advances in arithmetic geometry, birational geometry, derived algebraic geometry, derived categories of coherent sheaves, enumerative geometry, geometric representation theory, Hodge theory, Kaehler-Einstein metrics and stability, and p-adic geometry. The institute will offer morning plenary talks by top experts summarizing recent progress, and afternoon parallel seminars with more detailed discussion of relevant techniques. The institute will support participation by many young researchers, as their exposure to the breadth of modern algebraic geometry will set the research agenda for the coming decades. More details about the institute can found at the following two websites:

http://www.ams.org/meetings/amsconf/summerinst-2015

https://sites.google.com/site/2015summerinstitute/

When can we write down all the solutions of a polynomial equation? We seek equations that can be parametrized with rational functions. These are used in mapmaking (stereographic projection), computer graphics, and modeling problems. Indeed, parametrizations are often the most efficient way to render geometric objects as screen images. Mathematicians have developed a rich theory for deciding when such parametrizations are possible. This project will advance this theory.

An algebraic variety is rational if it can be obtained from projective space by a sequence of algebraic modifications. Given a family of smooth complex projective varieties, it is difficult to say which members are rational or irrational, or even to formulate qualitative results about the locus of rational members. The PI will use the technique of decomposition of the diagonal, and related tools from deformation theory, Hodge theory and classical geometry, to shed light on this question. The PI and his collaborators have recently shown that rationality is not a deformation invariant property: There are families of smooth complex projective varieties with both rational and irrational members. Despite examples, like cubic fourfolds, that have been extensively studied, there are no cases where the loci of rational and irrational members have been precisely described. The PI will refine and make effective the decomposition of the diagonal technique to clarify which members of a family are irrational - can they be expressed in Hodge-theoretic terms? At the same time, the PI will advance constructive techniques to exhibit rational and unirational parametrizations with prescribed properties.