Jimmy Dillies : On some K3 automorphisms
- Algebraic Geometry ( 171 Views )In order to construct a viable model of string theory, one seeks to build Calabi Yau threefolds with prescribed conditions. Borcea and Voisin were able to built a family of Calabi-Yau threefolds using elliptic curves and K3 surfaces admitting non symplectic involutions. We will display how the construction can be generalized by studying higher order non symplectic automorphisms on K3 surfaces
Charles Vial : On the motive of some hyperKaehler varieties
- Algebraic Geometry ( 92 Views )I will explain why it is expected that the Chow ring of hyperKaehler varieties has a similar structure as the Chow ring of abelian varieties. Examples of hyperKaehler varieties are given by K3 surfaces, and Hilbert schemes of length-n subschemes on K3 surfaces and their deformations. In fact I will introduce the notion of ``multiplicative Chow-Kuenneth decomposition'' and provide examples of varieties that can be endowed with such a decomposition. In the case of curves, or regular surfaces, this notion is intimately linked to the vanishing of a so-called "modified diagonal cycle". For example, a very general curve of genus >2 does not have vanishing modified diagonal cycle, but a result of Ben Gross and Chad Schoen establishes the vanishing of a modified diagonal cycle for hyperelliptic curves.
Nicolas Addington : Cubic fourfolds and K3 surfaces
- Algebraic Geometry ( 129 Views )Cubic fourfolds behave in many ways like K3 surfaces. Certain cubics -- conjecturally, the ones which are rational -- have K3s associated to them geometrically. Hassett has studied the cubics with K3s associated to them at the level of Hodge theory, and Kuznetsov has studied the cubics with K3s associated to them at the level of derived categories. These two notions of having an associated K3 should coincide. We prove they coincide generically. That is, Hassett's cubics form a countable union of irreducible Noether-Lefschetz divisors in moduli space, and Kuznetsov's cubics are a dense subset of these, forming a non-empty, Zariski open subset in each divisor.
Chad Schoen : Chow groups, an introduction
- Algebraic Geometry ( 124 Views )Chow groups give functors from algebraic varieties to abelian groups which are related to (co)homology. However Chow groups frequently contain more information than (co)homology. The construction of Chow groups is easy. Their computation is often difficult. This talk has two aims. First of all it will serve as an introduction to Chow groups which should be accessible to those who have taken a one semester course in Riemann surfaces, two semesters of algebraic topology, and have a passing acquaintance with affine and projective algebraic varieties. (One month in an algebraic geometry course may suffice for the latter.) Given that the next two talks in the algebraic geometry seminar will discuss various aspects of Chow groups, this talk may function as a warm up. The second aim is to introduce Bloch's conjecture on the Chow group of zero dimensional algebraic cycles on a non-singular projective surface. Throughout the talk one may assume that the base field is the complex numbers.
Paolo Stellari : Derived Torelli Theorem and Orientation
- Algebraic Geometry ( 148 Views )We will consider the problem of describing the group of autoequivalences of the derived categories of smooth K3 surfaces. After recalling the (Twisted) Derived Torelli Theorem, we will focus on its conjectural refinement, involving the preservation of the orientation of some 4-dimensional space in the total cohomology lattice. The conjecture will be proved in the generic (non-projective) case and we will discuss a few results which will possibly lead to the proof of the conjecture for smooth projective K3 surfaces. This is a joint work with D. Huybrechts and E. Macri'.
Humberto Diaz : On Chow groups of Varieties
- Algebraic Geometry ( 114 Views )For a complex algebraic variety, the Chow group is a geometric invariant which is easy to construct but often difficult to compute. In this talk, I will describe the construction of the Chow group, give some key examples and discuss some difficult open questions. I will also present a result about the Chow group of 0-cycles of the surface which parametrizes lines on a cubic 3-fold.
Alex Perry : Derived categories of cubic fourfolds and their geometric applications
- Algebraic Geometry ( 207 Views )A fundamental problem in algebraic geometry is to determine whether a given algebraic variety is birational to projective space. This is most prominently open for cubic fourfolds, i.e. hypersurfaces defined by a cubic polynomial in a five-dimensional projective space. A decade ago, Kuznetsov suggested an approach to this problem using the derived category of coherent sheaves. I will explain recent applications of this perspective to fundamental questions in hyperkahler geometry and Hodge theory, which in turn shed light on the original question about cubic fourfolds.
Pete Clark : (Postponed to a later date) Algebraic Curves Violating the Hasse Principle
- Algebraic Geometry ( 118 Views )The celebrated "Hasse Principle" holds for plane conics over a number field, but generally not for algebraic curves of positive genus. Isolated examples of curves violating the Hasse Principle go back to Lind, Reichardt and Selmer in the 1940s and 1950s. Many more examples have been found since, and it now seems likely that the Hasse principle should, in some suitable sense, most often be false. However it is challenging to make, let alone prove, a precise statement to this effect. In talk I will discuss certain "anti-Hasse principles", some which are conjectural and others (more modest) which are known to hold. In particular I will address the problem of constructing curves of any given genus g >= 1 over any global field which violate the Hasse
Franziska Hinkelmann : Analysis of discrete models of biological systems using computer algebra
- Algebraic Geometry ( 115 Views )Many biological systems are modeled qualitatively with discrete models, such as probabilistic Boolean networks, logical models, bounded Petri nets, and agent-based models. Simulation is a common practice for analyzing discrete models, but many systems are far too large to capture all the relevant dynamical features through simulation alone. We convert discrete models into algebraic models and apply tools from computational algebra to analyze their dynamics. The key feature of biological systems that is exploited by our algorithms is their sparsity: while the number of nodes in a biological network may be quite large, each node is affected only by a small number of other nodes. In our experience with models arising in systems biology and random models, this structure leads to fast computations when using algebraic models, and thus efficient analysis. All algorithms and methods are available in our package Analysis of Dynamic Algebraic Models (ADAM), a user friendly web-interface that allows for fast analysis of large models, without requiring understanding of the underlying mathematics or any software installation. ADAM is available as a web tool, so it runs platform independent on all systems.
Christopher O'Neill : Shifting numerical monoids
- Algebraic Geometry ( 97 Views )A numerical monoid is a subset of the nonnegative integers that is closed under addition. Given a numerical monoid S, consider the shifted monoid S_n obtained by adding n to each minimal generator of S. In this talk, we examine minimal relations between the generators of S_n when n is sufficiently large, culminating in a description that is periodic in the shift parameter n. We also explore several consequences, some old and some new, in the realm of factorization theory. No background in numerical monoids or factorization theory is assumed for this talk.
David Geraghty : Modularity lifting beyond the numerical coincidence of Taylor and Wiles
- Algebraic Geometry ( 90 Views )Modularity lifting theorems were introduced by Taylor and Wiles and formed a key part of the proof of Fermat's Last Theorem. Their method has been generalized successfully by a number authors but always with the restriction that the Galois representations in question have regular weight. Moreover, the sought after automorphic representation must come from a group that admits Shimura varieties. I will describe a method to overcome these restrictions, conditional on certain conjectures which themselves can be established in a number of cases. This is joint with Frank Calegari.
Laure Flapan : Hodge Groups of Hodge Structures with Hodge Numbers (n,0,...,0,n)
- Algebraic Geometry ( 116 Views )One of the main tools available for proving certain cases of the Hodge conjecture for abelian varieties is to compute the Hodge groups of the weight-1 Hodge structures associated to these abelian varieties. Thus Hodge groups of abelian varieties have been extensively investigated. In this talk, we discuss generalizing these results about Hodge groups to arbitrary-weight Hodge structures with Hodge numbers (n,0, ,0,n), particularly when n is prime or twice a prime. These generalizations yield some new results about Hodge classes of 2p-dimensional abelian varieties.
Thomas Kahle : Toric Fiber Products
- Algebraic Geometry ( 117 Views )The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same grading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We will introduce the construction, discuss its geometrical content, and give an overview over the various preserved properties. Toric fiber products have been applied most successfully to families of ideals parametrized by combinatorial objects like graphs. We will show how to use toric fiber product to prove structural theorems about classes of ideals from algebraic statistics.
Wenjing Liao : Spectral estimation on a continuum
- Algebraic Geometry ( 154 Views )The problem of spectral estimation, namely recovering the frequency contents of a signal arises in various applications, including array imaging and remote sensing. In these fields, the spectrum of natural signals is composed of a few atoms on the continuum of a bounded domain. After the emergence of compressive sensing, spectral estimation was widely explored with an emphasis on sparse measurements. However, with a few exceptions, the spectrum considered in the compressive sensing community is assumed to be located on a DFT grid, which results in a large gridding error.
In this talk, I will present the MUltiple SIgnal Classification (MUSIC) algorithm and some modified greedy algorithms, and show how the problem of gridding error can be resolved by these methods. Our work focuses on a stability analysis as well as numerical studies on the performance of these algorithms. Moreover, the MUSIC algorithm features its super-resolution effect, i.e., the capability of resolving closely spaced frequencies. We will provide some numerical experiments and theoretical justifications to show that the resolution length of MUSIC follows a power law with respect to the minimum separation of frequencies.
V. Srinivas : Etale motivic cohomology and algebraic cycles
- Algebraic Geometry ( 96 Views )This talk will report on joint work with A. Rosenschon. There are examples showing that the torsion and co-torsion of Chow groups are complicated, in general, except in the ``classical'' cases (divisors and 0-cycles, and torsion in codimension 2). Instead, we may (following Lichtenbaum) consider the etale Chow groups, which coincide with the usual ones if we use rational coefficients; we show that they have better torsion and cotorsion if we work over the complex numbers. In contrast, they can have infinite torsion in some arithmetic situations (the usual Chow groups are conjectured to be finitely generated).
Ravindra Girivaru : Lefschetz type theorems for algebraic cycles and vector bundles.
- Algebraic Geometry ( 190 Views )The Weak Lefschetz theorem (or the Lefschetz hyperplane theorem) states that for a smooth, projective variety Y and a smooth hyperplane section X in Y, the restriction map of cohomologies H^i(Y) to H^i(X) is an isomorphism for i less than dim{X}, and an injection when i equal to dim{X}. Analogues of this theorem have been conjectured for algebraic cycles. We will talk about some results in this area. We will also talk about such questions for vector bundles.
Jie Wang : The primitive cohomology of the theta divisor of an abelian fivefold
- Algebraic Geometry ( 114 Views )The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension $g$ is a Hodge structure of level $g-3$. The Hodge conjecture predicts that it is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this talk, I will explain how one can use the Prym map to show that this version of the Hodge conjecture is true for the theta divisor of a general abelian fivefold. This is joint work with Izadi and Tam\'as.
Chenglong Yu : Moduli of symmetric cubic fourfolds and nodal sextic curves
- Algebraic Geometry ( 202 Views )Period map is a powerful tool to study geometric objects related to K3 surfaces and cubic 4-folds. In this talk, we focus on moduli of cubic 4-folds and sextic curves with specified symmetries and singularities. We identify the geometric (GIT) compactifications with the Hodge theoretic (Looijenga, mostly Baily-Borel) compactifications of locally symmetric varieties. As a corollary, the algebra of GIT invariants is identified with the algebra of automorphic forms on the corresponding period domains. One of the key inputs is the functorial property of semi-toric compactifications of locally symmetric varieties. Our work generalizes results of Matsumoto-Sasaki-Yoshida, Allcock-Carlson-Toledo, Looijenga-Swierstra and Laza-Pearlstein-Zhang. This is joint work with Zhiwei Zheng.
Dick Hain : What is an algebraic group?
- Algebraic Geometry ( 112 Views )Algebraic groups are important in algebraic and arithmetic geometry. This talk will be a general introduction to them. I will discuss some basic example (elliptic curves, GLn, ...) and then introduce linear algebraic groups and affine algebraic groups. There will be lots of examples, which will help explain why they are important.
Angelo Vistoli : Fundamental gerbes
- Algebraic Geometry ( 122 Views )Let X be a connected and geometrically reduced variety over a field k, with a fixed rational point x_0 in X(k). Nori defined a profinite group scheme N(X,x_0), usually called Nori's fundamental group scheme, with the property that homomorphisms N(X,x_0) to a fixed finite group scheme G correspond to G-torsors P--> X with a fixed rational point in the inverse image of x_0 in P. If k is algebraically closed of characteristic 0 this coincides with Grothendieck's fundamental group, but is in general very different. Nori's main theorem is that if X is complete, the category of finite-dimensional representations of N(X,x_0) is equivalent to an abelian subcategory of the category of vector bundles on X, the category of essentially finite bundles. In my talk I will recall the basics of the theory of group schemes and torsors, and give a detailed description of Nori's results. Then I will explain my work in collaboration with Niels Borne, from the University of Lille, in which we extend them by removing the dependence on the base point, substituting Nori's fundamental group with a gerbe (in characteristic 0 this had already been done by Deligne), and give a simpler definition of essentially finite bundle, and a more direct and general proof of the correspondence between representations and essentially finite bundles.
Sam Grushevsky : Stable cohomology of compactifications of the moduli spaces of abelian varieties
- Algebraic Geometry ( 123 Views )Cohomology of A_g, the moduli space of principally polarized complex g-dimensional abelian varieties, is the same as the cohomology of Sp(2g,Z). By Borel's result on group homology it turns out that for g>k the cohomology H^k(A_g) is independent of g - it is then called the stable cohomology of A_g. Similarly, the stable cohomology of the moduli space of curves was the subject of Mumford's conjecture, proven by Madsen and Weiss by topological methods. In a joint work with Klaus Hulek and Orsola Tommasi we show that the cohomology of the perfect cone toroidal compactification of A_g stabilizes, and compute some of this stable cohomology using algebro-geometric methods.
Christine Berkesch Zamaere : Torus actions and holonomic D-modules
- Algebraic Geometry ( 88 Views )Just as algebraic varieties with group actions admit quotients, we provide a quotient construction for D-modules with torus actions that is with several important properties in algebraic analysis. As an application, we apply tools from toric geometry to obtain new information about hypergeometric systems of PDEs studied by Gauss, Appell, and Lauricella, among others. In particular, we determine when such "Horn systems" are regular holonomic. This is joint work with Laura Felicia Matusevich and Uli Walther.
Giulia Sacca : Compact Hyperkahler manifolds in algebraic geometry
- Algebraic Geometry ( 116 Views )Hyperkahler (HK) manifolds appear in many fields of mathematics, such as differential geometry, mathematical physics, representation theory, and algebraic geometry. Compact HK manifolds are one of the building blocks for algebraic varieties with trivial first Chern class and their role in algebraic geometry has grown immensely over the last 20 year. In this talk I will give an overview of the theory of compact HK manifolds and then focus on some of my work, including a recent joint work with R. Laza and C. Voisin.
Greg Pearlstein : Boundary components of Mumford-Tate domains
- Algebraic Geometry ( 194 Views )Mumford-Tate groups arise as the natural symmetry groups of Hodge structures and their variations. I describe recent work with Matt Kerr on computing the Mumford-Tate group of the Kato-Usui boundary components of a degeneration of Hodge structure.
Izzet Coskun : Brill-Noether Theorems for moduli spaces of sheaves on surfaces
- Algebraic Geometry ( 172 Views )In this talk, I will discuss the problem of computing the cohomology of the general sheaf in a moduli space of sheaves on a surface. I will concentrate on the case of rational and K3 surfaces. The case of rational surfaces is joint work with Jack Huizenga and the case of K3 surfaces is joint work with Howard Nuer and Kota Yoshioka.
Emanuele Macri : MMP for moduli spaces of sheaves on K3 surfaces and Cone Conjectures
- Algebraic Geometry ( 133 Views )We report on joint work with A. Bayer on how one can use wall-crossing techniques to study the birational geometry of a moduli space M of Gieseker-stable sheaves on a K3 surface X. In particular: (--) We will give a "modular interpretation" for all minimal models of M. (--) We will describe the nef cone, the movable cone, and the effective cone of M in terms of the algebraic Mukai lattice of X. (--) We will establish the so called Tyurin/Bogomolov/Hassett-Tschinkel/Huybrechts/Sawon Conjecture on the existence of Lagrangian fibrations on M.