Master's thesis topics Algebraic Geometry and Number Theory 2018-2019 - KU Leuven
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Master’s thesis topics
Algebraic Geometry and Number Theory
2018-2019
This is the list of possible topics for a master’s thesis proposed by the staff
members of the research group Algebraic Geometry and Number theory. Every
topic comes with a short description. In case you are interested in or have questions
about one of the topics, please contact the corresponding staff member. Anyway,
before putting a topic from our research group on your preference list, we expect you
to contact us.
1 Topics proposed by prof. N. Budur
Formality in deformation theory
This project is about deformation theory in (basic) algebraic geometry. Understand-
ing deformations of an object is a fundamental problem in mathematics, since it
amounts to a partial, local classification. In geometry, the deformation theory of
complex manifolds treated by Kodaira and Spencer was a first step. Since then,
mathematicians have been successful at deforming and understanding all kinds of
other objects: vector bundles, group representations, etc.
A principle of Deligne says that every deformation problem over a field of char-
acteristic zero is controlled by a differential graded Lie algebra. So deformation
theory boils down to algebra. Unfortunately, the differential graded Lie algebras one
is faced with are usually infinite-dimensional. This is the case for example for the
de Rham complex of global smooth forms on a smooth manifold which controls the
deformations of the trivial representation of the fundamental group of the manifold.
The principle of Deligne has been fortified and put on steroids through work of
Drinfeld, Fukaya, Kontsevich, and others: every deformation problem in character-
istic zero is controlled by a typically finite-dimensional L-infinity algebra. In the
1above example, the de Rham complex gets replaced by the cohomology algebra of
the manifold, but with an algebra structure given not only by cup products, but also
by higher-order Massey products.
The simplest kind of deformation theory occurs when one has ”formality”, that
is, when the higher-order Massey products are zero. In the presence of formality,
there could be an even stronger reason for it, ”intrinsic formality”, that is, when the
algebra permits no higher-order products at all. One measures this with Hochschild
cohomology.
In this project, the student will do the following concrete exercise. Let A be
the cohomology algebra of the complement of an arrangement of hyperplanes. The
algebra A is a combinatorial object, the so-called Orlik-Solomon algebra. It can be
easily written down in terms of the intersection lattice of the arrangement. It is well-
known that A is formal, following the work of Deligne-Grifitths-Morgan-Sullivan
relating formality with purity of the mixed Hodge structure on cohomology. The
question that the student will try to answer is: is A intrinsically formal?
To answer this question, the student will compute the Hochschild cohomology of
the algebra A. This will be an exercise in homological algebra. To the best of my
knowledge, the answer to the question is not known.
2 Topics proposed by prof. R. Cluckers
The world of p-adic numbers
On the p-adic numbers, much of analysis and geometry works analogously to how it
works on the real numbers. These themes on p-adic numbers have moreover many
close connections to number theory, and some of these are used in cryptography. In
this thesis, the student may learn about integration on the field of p-adic numbers,
and how it is related to certain counting and summing problems. This indeed relates
to number theory, geometry, algebra, and analysis and is since long a central tool
in the research of the KU Leuven Algebra Section. If time allows, we may go from
the study of p-adic to motivic integration. On the p-adics, there is no fundamental
theorem of calculus (relating derivatives with integrals), and also, there is no relation
between the integral of f and the ’area’ under the graph of f. Rather, these p-adic
integrals relate to finite counting of points on some geometric objects, and, to finite
exponential sums like Gauss sums. The p-adic numbers are also central in the work
of the recent Fields medaille winner Scholtze, who develops rich geometry with them.
Also the concept of o-minimality on the reals has analogues on the p-adic numbers,
with much ongoing research in all of these directions.
23 Topics proposed by prof. J. Nicaise
Toric geometry and mirror symmetry
Background
In string theory, the universe is modelled by means of a 10-dimensional real manifold:
the 4 dimensions of space-time, and 6 additional dimensions for the strings to vibrate.
These additional dimensions come in the form of a complex Calabi-Yau threefold, a
type of algebraic variety that was extensively studied by geometers for independent
reasons. Physicists realized that the Calabi-Yau threefold is not uniquely determined
by the physical theory; these threefolds rather seemed to come in mirror pairs, giving
rise to equivalent theories.
Spectacular applications of these ideas to enumerative geometry forced algebraic
geometers to take them very seriously. An important challenge was to provide rig-
orous foundations for the theory of mirror symmetry: how can we define mirror
pairs, and how can we construct them? An early source of mirror constructions
was provided by toric geometry, in pioneering work by V. Batyrev around 1993. In
toric geometry, one constructs a rich and interesting class of algebraic varieties from
combinatorial data such as lattice polytopes. Batyrev identified a class of lattice
polytopes to which Calabi-Yau manifolds can be associated, as well as a duality
between these polytopes giving rise to mirror pairs of Calabi-Yau varieties.
Aim of the project
We will study the basic theory of toric geometry and Calabi-Yau varieties, and how
they are connected in the theory of mirror symmetry. We will also work out some
interesting explicit examples.
References
• D. Cox and S. Katz. Mirror symmetry and algebraic geometry. Mathematical
Surveys and Monographs, 68. American Mathematical Society, Providence,
RI, 1999.
• W. Fulton. Introduction to toric varieties. Annals of Mathematics Studies,
131. Princeton University Press, Princeton, NJ, 1993.
34 Topics proposed by prof. W. Veys
General context: Algebraic geometry and singularity theory,
links with number theory.
The objects of study in algebraic geometry are the algebraic varieties, namely the
solution sets of one or more polynomial equations over R or C, or over finite or more
exotic fields. One studies their geometric properties using algebraic techniques.
Singularity theory treats the singular or non-smooth points of algebraic varieties
(and of real or complex analytic varieties). This is a very broad domain involving
geometry, algebra, analysis and topology. For instance in our research group we use
intensively resolution of singularities, a classical technique to study a singular variety
by means of a closely related variety without singular points.
A number theoretical topic related to singularities is Igusa zeta functions. Here
one studies the number of solutions of polynomial congruences modulo a power r
of a prime number p. How are these numbers of solutions varying in terms of r?
Resolution of singularities is an important technique to attack the problem.
More concretely
I prefer to fix a concrete subject only after discussing with the student. Do you like
a combination of mathematical disciplines, or not? Do you prefer abstract or (more
or less) concrete stuff? Are you maybe interested in doing a Ph.D. later? ...
Some suggestions:
• Research paper on Igusa zeta functions or their more geometric/topological
variants topological zeta functions.
(a) For instance [Veys and Zuniga-Galindo, Zeta functions for polynomial
mappings, log-principalization of ideals, and Newton polyhedra, Transac-
tions American Mathematical Society 360 (2008), 2205-2227] treats such
zeta functions associated to several polynomials instead of (classically)
one polynomial. This paper involves number theory, algebraic geometry
and combinatorics.
(b) In the paper [Gong, Veys and Wan, Power moments of Kloosterman sums,
Journal of Number Theory 164 (2016), 103-126] new aspects of an ever-
green in number theory, Kloosterman sums, are obtained using the Igusa
zeta function of a specific polynomial.
4(Both papers can also be viewed on my website.)
• What is the genus? This question is the title of a recent book by P. Popescu-
Pampu [History of Mathematics Subseries 2162, Springer (2016)]. From the
book’s website: It presents, through the works of the pioneers, the sophisti-
cated evolution of one of the most important notions of geometry and topology.
Exploring several of the evolutionary branches of the mathematical notion of
genus, this book traces the idea from its prehistory in problems of integration,
through algebraic curves and their associated Riemann surfaces, into algebraic
surfaces, and finally into higher dimensions. Its importance in analysis, al-
gebraic geometry, number theory and topology is emphasized through many
theorems. Almost every chapter is organized around excerpts from a research
paper in which a new perspective was brought on the genus or on one of the
objects to which this notion applies.
Exploring and developing in detail some parts of this book should form a nice
thesis subject, maybe together with some recent research paper related to the
subject.
5 Topics proposed by Dr. E. Leenknegt
p-adic dynamics and applications of p-adic numbers in other
sciences
There are different completions possible for the field of rational numbers, depending
on the norm that is used. Completion with respect to the standard, euclidean norm
gives rise to the familiar field of real numbers. Completion with respect to p-adic
norms gives rise to the p-adic fields Qp . Because of the properties of these non-
archimedean norms, these fields have very different topologies compared to the field
of real numbers. It is probably for this reason that applications in other areas of
science are now beginning to surface, as this allows for alternative descriptions of
reality compared to what can be achieved using real numbers.
The aim of this thesis would be to study the underlying mathematical theory
for these applications, which is mainly p-adic analysis and p-adic ergodic theory
(this is a branch of mathematics that studies dynamical systems with an invariant
measure and related problems), and then also look at how this is applied outside
of mathematics. Currently there are applications in areas like computer science,
genetics, quantum physics,... (Note: if you are interested in applications in physics,
it would be advisable that you already have a good background in physics.)
56 Topics proposed by Dr. K. Nguyen
6.1 Theory of height functions
The theory of heights is a very beautiful theory with deep applications in many
central problems of arithmetic and its related topics. The basic idea behind the height
is that we don’t know a common method to solve arbitrary Diophantine equations
(In fact this method does not exist, it is the negative solution of Hilbert’s tenth
problem given by Matiyasevich.) Hence, to find solutions of Diophantine equations,
it is good to know that this equation has a solution in a bounded interval and verify
any integer point in this set.
The problem becomes more difficult if we consider the solutions in the rational
field Q, as there are infinitely many rational points in a bounded interval. To use
the above idea, we can look at the complexity of a rational number ab where a and b
are coprime: it is good to look at any solution of the form ab with 0 ≤ |a|, |b| ≤ X.
In general, for an algebraic number α (a root of a polynomial in Z), the complexity
of α is given by the coefficients of the minimal polynomial Pα of α over Z. The height
of α is the maximum of the absolute values of coefficients of Pα , which measures
the complexity of α. Generally, height functions measure the complexity of points
in algebraic varieties over global fields, local fields or fields with absolute value in
general.
Studying the properties of height functions is one of the main problems in arith-
metic geometry. It concerns to many big conjectures of arithmetic as Mordell con-
jecture, Vojta conjecture, abc conjecture, André-Oort conjecture, et cetera. Height
functions also relate to many other topics in mathematics as approximation of al-
gebraic number, Arakelov theory, Diophantine approximation, arithmetic dynamics,
Nevanlinna theory, Hyperbolic complex spaces, et cetera.
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