Master s thesis topics Algebraic Geometry and Number Theory

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1 Master s thesis topics Algebraic Geometry and Number Theory 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. 1 Topics proposed by prof. N. Budur Character varieties Let G be a group. Consider the space of all rank n complex representations of G: Rep(G, n) = Hom(G, GL n (C)). This is an affine algebraic variety, the representation variety of G. The group GL n (C) acts by conjugation on Rep(G, n). Hence there is an action on the affine coordinate ring A of Rep(G, n). The subring of invariants B = A GLn(C) is finitely generated according to a theorem of Hilbert. Hence, it corresponds to another affine algebraic variety, called the character variety of G, which we denote by M(G, n). This is the algebraic-geometric version of the quotient Rep(G, n)/gl n (C), which as a simple set of orbits might not have any algebraic meaning. The geometry of M(G, n) is rather intricate, the points of this space parametrizing the semi-simple representations. The thesis will be dedicated the study of the character varieties and their special subvarieties, for G being a free group in a small number of generators, and n being small too. 2 Topics proposed by prof. R. Cluckers Motivic and p-adic integration Historically, integration has first been developed on the real number field, and then more generally on locally compact groups. Much more recently, a more 1

2 geometric viewpoint has been taken in the theory of motivic integration. The student will discover the theory of p-adic integration that served as inspiration to the theory of motivic integration, and will learn about relations between such theories of integration and geometry. Some knowledge of model theory, number theory, algebraic geometry and/or measure theory is welcome. Together with the student, we will choose a sub-topic in this vast domain of research. 3 Topics proposed by prof. F. Cools The specific topics proposed below comprise a mixture of commutative algebra and (mostly classical) algebraic geometry, and sometimes also a bit of combinatorics. Strassen s Conjecture Let A, B, C be finite dimensional vector spaces over the complex numbers that can be decomposed as respectively A A, B B and C C. Given tensors T A B C and T A B C, Strassen s Conjecture predicts that the rank of the tensor T = T + T A B C is equal to the sum of the ranks of T and T. Hereby, the rank of a tensor is the minimal number of elementary tensors that are needed to represent the tensor. The statement of Strassen s Conjecture can be extended to tensor products with arbitrary many factors and to symmetric tensors. In the latter case, this is related to the following so-called Waring problem: given a homogeneous form f C[x 0,..., x n ] of degree d, what is the minimal number r of linear forms l 1,..., l r C[x 0,..., x n ] such that f = l d l d r? The goal of the thesis is to give an overview of some partial answers to the conjecture. Enumerative algebraic geometry How many lines are there on a smooth cubic surface? How many bitangents does a (general) plane quartic curve have? How many conics are tangent to 5 plane conics in general position? Such questions are typical instances of problems which one studies in enumerative algebraic geometry. The goal of the thesis is to explain the solutions of some enumerative problems. Hereby, one utilizes tools in algebraic geometry such as blow-ups and intersection theory. By the way, the correct answers to the above questions are respectively 27, 28 and Topics proposed by prof. W. Veys General context Algebraic geometry and singularity theory, 2

3 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: Study a 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, Transactions American Mathematical Society 360 (2008), ] 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), ] new aspects of an evergreen in number theory, Kloosterman sums, are obtained using the Igusa zeta function of a specific polynomial. (Both papers can also be viewed on my website.) Singular points of plane curves form a rich interdisciplinary topic: geometry, analysis, algebra, topology and combinatorics pop up. The book [Singular points of plane curves, London mathematical society student texts 63, Cambridge University press (2004)] by C.T.C. Wall describes this interaction. Some of these aspects can form a nice thesis subject, together with some recent research paper on plane curve singularities. 3

4 5 Topics proposed by dr. E. Leenknegt One of the topics studied in model theory are the definable sets for a given language L. More precisely, one studies structures (M, L) (with universe M), consisting of all subsets of M n (for n > 0) that can be described using formulas in the language L. For example, one can consider semi-algebraic sets, the sets definable in the ring language L ring = (+,,, 0, 1), where M may be the field or real or p-adic numbers, or some other field. In each of these instances, the resulting structure is well-understood, and is known to have nice properties. Once a structure is studied, one may also consider expansions of it (by an expansion of a structure (M, L) we mean a structure (M, L ), where L = L {...} is a richer language than L, in the sense that it can be used to describe some sets that cannot be described using the language L). It is a natural question to ask whether properties of the base structure propagate to expansions of these structures, and also, how it can be recognized whether this is going to be the case. The idea of minimality is to impose some conditions on the L-definable subsets of the universe M of a given structure (M, L), and then see which properties of definable sets (also in higher dimensions) can already be derived from this. Probably the most famous minimality concept is o-minimality (for ordered structures, for instance over the real numbers): here the imposed condition is that the definable subsets of M should be finite unions of intervals and points. Tameness in expansions of the real field An expansion (R, L ring {...}) of the field of real numbers is o-minimal if the L-definable subsets of R are finite unions of intervals and points, and it is known that such structures are very well behaved (tame). Obviously, not every expansion of (R, L ring ) can be o-minimal. However, some of these structure still exhibit some measure of tameness. For example, there are some weaker variations of o-minimality, like weak o-minimality, which requires that every definable subset of R is a finite union of definable convex sets, or local o-minimality. In such structures, some of the nice properties of o-minimal structures will survive (though obviously not all of them!) The aim of this project is to use the paper [1] as a starting point, and then consider some of the structures mentioned in this paper in more detail, and see how their properties compare to those of the (more tame) o-minimal structures. The concept of minimality in model theory Besides o-minimality and related concepts, there exists a whole alphabet-zoo of other minimality concepts, like P -minimality, t-minimality, b-minimality, C- minimality, v-minimality and so on. A possible thesis topic would be to try to 4

5 make a survey of some of these concepts, and compare some of the motivations behind them and the possible applications of each of these. References [1] Chris Miller, Tameness in expansions of the real field, Logic Colloquium 01, Lect. Notes Log., pages , Association for Symbolic Logic, Urbana, IL, [2] Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, Cambridje University Press, Cambridge, Topics proposed by dr. Y. Liu Resonance varieties of hyperplane arrangements A hyperplane arrangement is a finite set of hyperplanes in a finite-dimensional complex vector space. An important combinatorial invariant of such an arrangement is the lattice of intersections of hyperplanes, ordered by inclusion. Given an arrangement A in C n, the study of the topology of the complement M = C n \ H A H is a very interesting topic. In [1], Orlik and Solomon gave a simple combinational description of the cohomology ring H (M, Z) in terms of the intersection lattice of A. Set A = H (M, C). For each a A 1 = H 1 (M, C), we have a 2 = 0. Then, multiplication by a defines a cochain complex: (A, a) : A 0 a A 1 The resonance varieties of M are defined by: a A 2 a. R i k (M) = {a A1 dim C H i (A, a) k} The Orlik-Solomon description mentioned above made it clear that the resonance varieties of M depend only on the intersection lattice. A basic problem is to find concrete formulas making this dependence explicit. An elegant method, called multinets, gave an explicit description for the resonance varieties R 1 k (M) for all k 1. It is developed by Falk and Yuzvinsky in [2]. The thesis will present an overview of this work Requirement: Basic algebraic topology. 5

6 References [1] P. Orlik, L. Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, [2] M. Falk, S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves, Compositio Math. 143 (2007), no. 4, Topics proposed by dr. A. Smeets I can propose thesis subjects in the field of arithmetic geometry, which studies the interaction between the arithmetic behaviour of algebraic varieties and their geometric invariants (such as their cohomology). One suggestion for such a topic can be found below, but I am happy to discuss other options; arithmetic geometry is a booming field, so the list of possibilities is endless. Insufficiency of the Brauer Manin obstruction Diophantine equations can be studied using geometric methods: a system of polynomial equations over Q defines an algebraic variety over Q, and solutions of the system correspond to rational points on the variety. Hasse proved that if a quadratic form over Q represents zero non-trivially over Q p (for all primes p) and R, then it represents zero non-trivially over Q. In other words: if a quadric hypersurface over Q has points over Q p (for all p) and over R, then it has a rational point. This is the first instance of the Hasse principle. Unfortunately, the Hasse principle does not deserve its name: if often fails. To understand when the Hasse principle can (or cannot) hold for an algebraic variety X over Q, Manin invented the Brauer Manin obstruction (1970). This is a mechanism which uses geometric input (the Brauer group of X) to produce arithmetic output more precisely, the conclusion that the Hasse principle fails for X. For a long time, all known failures of the Hasse principle could be explained by the Brauer Manin obstruction (or a refinement thereof). It is only in 2010 that Poonen was able to construct the first example of a variety over Q for which the failure of the Hasse principle cannot be explained by such an obstruction. Since then, more examples of this phenomenon (with different geometric properties) have been found. For this project, you first need to do some reading on the basics of the Brauer Manin obstruction. The goal would then be to get a detailed understanding of one of the new examples, e.g. the example of Harpaz Skorobogatov; this is the first example of an algebraic surface for which the Brauer Manin obstruction and its refinements do not explain the failure of the Hasse principle. Prerequisites: basic courses on number theory and algebraic geometry. 6

7 J.-L. Colliot-Thélène, A. Pál & A. Skorobogatov, Pathologies of the Brauer Manin obstruction. To appear in Mathematische Zeitschrift (2015). Y. Harpaz & A. Skorobogatov, Singular curves and the étale Brauer- Manin obstruction for surfaces. Annales Scientifiques de l École Normale Suprieure 47/5 (2014), B. Poonen, Insufficiency of the Brauer Manin obstruction applied to étale covers. Annals of Mathematics 171/3 (2010), Topics proposed by dr. Jean-Baptiste Teyssier D-modules and vanishing of differential forms The theory of D-modules is roughly the study of integrable systems of differential equations in several variables. Families of smooth algebraic varieties give rise to very interesting such differential equations encoding how cycles move along fibers: the Gauss-Manin connections [1]. In case of a family of smooth projective varieties, the Gauss-Manin connection underlies a richer structure related to the Hodge theory of the fibers [4]. If one relaxes the smoothness condition, more complicated differential equations occur and one enters the world of Hodge Modules [3] introduced by Morihiko Saito. The goal of this project is to understand how Hodge modules are used to prove algebraic geometry statements having a priori nothing to do with differential equations. The project will revolve more precisely around the solution by Popa-Schnell [2] of the following conjecture from Hacon-Kovacs and Luo-Zhang: Theorem 1 Let X be a smooth complex variety of general type. Then, the zero locus of a global holomorphic one-form on X is not empty. References [1] Pierre Deligne. Équations différentielles à points singuliers réguliers. Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin-New York, [2] M. Popa and C. Schnell. Kodaira dimension and zeros of holomorphic oneforms. Annals of Maths., 179, [3] C. Schnell and C. Sabbah. Mixed Hodge Module project. cmls.polytechnique.fr/perso/sabbah/mhmproject/mhm.html. [4] C. Voisin. Hodge theory and complex algebraic varieties, volume 76 of Cambridge studies in advanced mathematics. Cambridge University press,

8 9 Topics proposed by dr. J. Tuitman The effective Chabauty method Let f Q[x] be a polynomial of odd degree 2g + 1 over the rationals satisfying gcd(f, df dx ) = 1. Then the equation C : y 2 = f(x) defines a so called hyperelliptic curve over Q of genus g. We are interested in finding the set of solutions C(Q) to this equation with x, y Q, which is also called the set of rational points of X. By a famous (and very hard) theorem of Faltings, any algebraic curve (not necessarily hyperelliptic) of genus g 2 over a number field has a finite number of rational points. However, Faltings method is not effective: it does not say anything about how to find the points. There is another method by Chabauty and Coleman that does not apply to all curves, or even all hyperelliptic curves, but which is effective when it applies. We very briefly sketch this method now. There is a way to associate to the curve C its Jacobian variety Jac C, which is an Abelian variety of dimension g, i.e. a projective algebraic variety with an (Abelian) group structure on it. Instead of working with equations that define the Jacobian, one usually represents it as divisors of degree 0 on the curve, modulo divisors of functions. For hyperelliptic curves this description can be made very explicit. Now as soon as we know one point P C(Q), we get an embedding C Jac C. However, the Jacobian is an Abelian variety, so its set of rational points Jac C (Q) is a finitely generated Abelian group (by the Mordell-Weil theorem). Let the rank of this group be r (i.e. r is maximal such that Z r is contained in the group). On the other hand, for a prime number p, the sets C(Q p ) and Jac C (Q p ) are p-adic analytic manifolds. Let X denote the closure of a set X for the p-adic topology. Then Chabauty proved (long before Faltings s theorem): Theorem 1 Suppose that r < g and let p be a prime number. Then the set C(Q p ) Jac C (Q) is finite. Consequently, the set C(Q) is finite as well. Moreover, the intersection is defined by the vanishing of certain p-adic analytic integrals (called Coleman integrals). This often allows one to explicitly find the set C(Q)! The goal of this project is to understand the effective Chabauty method. For this we will first have to study (hyperelliptic) curves and their Jacobians, a bit of p-adic analytic geometry and some p-adic cohomology. Then we will try to work out some examples from the literature, and possibly do a new example ourselves. The project could either focus more on the details of the theory, or assume some of this without proof and focus more on computations and examples, depending on your interests. 8

9 References [1] W. McCallum and B. Poonen, The method of Chabauty and Coleman, Explicit methods in number theory, 99117, Panor. Synthèses, 36, Soc. Math. France, Paris, [2] J. Balakrishnan, R. Bradshaw and K. Kedlaya, Explicit Coleman integration for hyperelliptic curves, Algorithmic Number Theory (ANTS-IX), Lecture Notes in Computer Science 6197, Springer-Verlag, 2010,