Pete Clark : (Postponed to a later date) Algebraic Curves Violating the Hasse Principle
- Algebraic Geometry ( 119 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
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.
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.
Chad Schoen : A family of surfaces constructed from genus 2 curves
- Algebraic Geometry ( 176 Views )This talk is about complex analytic geometry, the field of mathematics concerned with complex manifolds and more generally with complex analytic spaces. The "curves" of the title are compact Riemann surfaces and the "surfaces" in the title are compact complex manifolds of dimension 2 over the complex numbers (and hence dimension 4 over the real numbers). The talk will explore the problem of constructing two dimensional complex manifolds by deforming known complex analytic spaces. It will focus on a single example. The talk should be quasi-accessible to anyone who has courses in Riemann surfaces and algebraic topology.
Humberto Diaz : The Rost nilpotence principle
- Algebraic Geometry ( 150 Views )In this talk, I will discuss Chow motives and the Rost nilpotence principle, which played a role in Voevodsky's celebrated proof of the Milnor conjecture. Conjectural in general, this principle is useful in determining when motivic decompositions obey Galois descent. After covering some preliminaries, I will give an overview of a new proof of this principle for surfaces over a perfect field.
Parker Lowrey : Virtual Grothendieck-Riemann-Roch via derived schemes
- Algebraic Geometry ( 145 Views )The usefulness of the various Riemann-Roch formulas as computational tools is well documented in literature. Grothendieck-Riemann-Roch is a commutative diagram relating pullback in K-theory to the pullback of associated Chow invariants for locally complete intersection (l.c.i.) morphisms. We extend this notion to quasi-smooth morphisms between derived schemes, this is the "derived" analog of l.c.i. morphisms and it encompasses relative perfect obstruction theories. We will concentrate on the naturality of the construction from the standpoint of pure intersection theory and how it interacts with the virtual Gysin homomorphism defined by Behrend-Fantechi. Time permitting we will discuss the relationship with existing formulas, i.e., Ciocan-Fonanine, Kapranov, Fantechi, and Goettsche.
Sarah J Frei : Moduli spaces of sheaves on K3 surfaces and Galois representations
- Algebraic Geometry ( 245 Views )Moduli spaces of sheaves on K3 surfaces have been well-studied when defined over the complex numbers, because they are one of the known families of hyperkaehler varieties. However, many of their arithmetic properties when defined over an arbitrary field are still unknown. In this talk, I will tell you about a new result in this direction: two such moduli spaces of the same dimension, when defined over a finite field, have the same number of points defined over every finite field extension of the base field, which is surprising when the moduli spaces are not birational. The way to get at this result is to study the cohomology groups of the moduli spaces as Galois representations. Over an arbitrary field, we find that all of the cohomology groups are isomorphic as Galois representations.
Sam Payne : Boundary complexes and weight filtrations
- Algebraic Geometry ( 116 Views )The boundary complex of an algebraic variety is the dual complex of the boundary divisor in a compactification of a log resolution. I will present recent work showing that the homotopy type of this complex is independent of the choice of resolution and compactification, and give relations between these complexes and Deligne's weight filtration on singular cohomology.
Chris Hall : Sequences of curves with growing gonality
- Algebraic Geometry ( 120 Views )Given a smooth irreducible complex curve $C$, there are several isomorphism invariants one can attach to $C$. One invariant is the genus of $C$, that is, the number of handles in the corresponding Riemann surface. A subtler invariant is the gonality of $C$, that is, the minimal degree of a dominant map from $C$ of $\mathbb{P}^1$. A lower bound for either invariant has diophantine consequences when $C$ can be defined over a number field, but the ability to give non-trivial lower bounds depends on how $C$ is presented. In this talk we will consider a sequence $C_1,C_2,\ldots$ of finite unramified covers of $C$ and give spectral criteria for the gonality of $C_n$ to tend to infinity.
Seth Baldwin : Positivity in T-equivariant K-theory of flag varieties associated to Kac-Moody groups
- Algebraic Geometry ( 122 Views )The cohomology rings of flag varieties have long been known to exhibit positivity properties. One such property is that the structure constants of the Schubert basis with respect to the cup product are non-negative. Brion (2002) and Anderson-Griffeth-Miller (2011) have shown that positivity extends to K-theory and T-equivariant K-theory, respectively. In this talk I will discuss recent work (joint with Shrawan Kumar) which generalizes these results to the case of Kac-Moody groups.
Arend Bayer : Stability conditions on the local P2 revisited
- Algebraic Geometry ( 127 Views )We will give a description of the space of Bridgeland stability conditions on the derived category of sheaves on P2 sitting inside a compact Calabi-Yau threefold. We will discuss its fractal-like boundary, its relation with the group of auto-equivalences, with mirror symmetry, and with counting invariants for both P2 and the quotient stack [C3/Z_3]. This is joint work with E. Macri.
Thomas Lam : First steps in affine Schubert calculus
- Algebraic Geometry ( 144 Views )I will explain some attempts to develop a theory of Schubert calculus on the affine Grassmannian. I will begin with the different descriptions of the (co)homology rings due to Bott, Kostant and Kumar, and Ginzburg. Then I will discuss the problems of finding polynomial representatives for Schubert classes and the explicit determination of structure constants in (co)homology.
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).
Michael Griffin : On the distribution of Moonshine and the Umbral Moonshine conjectures.
- Algebraic Geometry ( 102 Views )Monstrous Moonshine asserts that the coefficients of the modular j-function are given in terms of ''dimensions'' of virtual character for the Monster group. There are 194 irreducible representations of the Monster, the largest of the sporadic simple groups, and it has been a longstanding open problem to determine their distribution in Moonshine. Witten and others have demonstrated deep connections between Monstrous Moonshine and quantum physics. The distributions of the Monster representations in Moonshine can be interpreted as the distributions of black hole states in 3 dimensional quantum gravity. In joint work with Ono and Duncan, we obtain exact formulas for these distributions. Moonshine type-phenomena have been observed for other finite simple groups besides the Monster. The Umbral Moonshine conjectures of Cheng, Duncan, and Harvey asserts that the Moonshine extends to 24 isomorphism classes of even unimodular positive-definite rank 24 lattices. Monstrous Moonshine can be regarded as the case of the Leech lattice. In 2013, Gannon proved the case for the Mathieu group M24. We offer a method of proof for the remaining 22 cases.
Jayce Getz : Hilbert modular generating functions with coefficients in intersection homology
- Algebraic Geometry ( 148 Views )In a seminal Inventiones 1976 paper, Hirzebruch and Zagier produced a set of cycles on certain Hilbert modular surfaces whose intersection numbers are the Fourier coefficients of elliptic modular forms with nebentypus. Their result can be viewed as a geometric manifestation of the Naganuma lift from elliptic modular forms to Hilbert modular forms. We discuss a general analogue of this result where the real quadratic extension is replaced by an arbitrary quadratic extension of totally real fields. Our result can be viewed as a geometric manifestation of quadratic base change for GL_2 over totally real fields. (joint work with Mark Goresky).
Nils Bruin : Prym varieties of genus four curves
- Algebraic Geometry ( 232 Views )Many arithmetic properties of hyperbolic curves become apparent from embeddings into abelian varieties, in particular their Jacobians. For special curves, particularly those that arise as unramified double covers of another curve (of genus g), the Jacobian variety itself is decomposable. This leads to Prym varieties. These are principally polarized abelian varieties of dimension g-1. Having an explicit description of these varieties is an essential ingredient in many computational methods. We discuss an explicit construction for g equal to 4. This is joint work with Emre Can Sertoz.
Leonardo Mihalcea : Quantum-K theory of the Grassmannians
- Algebraic Geometry ( 171 Views )If X is a Grassmannian (or an arbitrary homogeneous space) the 3-point, genus 0, Gromov-Witten invariants count rational curves of degree d satisfying certain incidence conditions - if this number is expected to be finite. If the number is infinite, Givental and Lee defined the K-theoretic Gromov-Witten invariants, which compute the sheaf Euler characteristic of the space of rational curves in question, embedded in Kontsevich's moduli space of stable maps. The resulting quantum cohomology theory - the quantum K-theory algebra - encodes the associativity relations satisfied by the K-theoretic Gromov-Witten invariants. In joint work with Anders Buch, we shown that the (equivariant) K-theoretic Gromov-Witten invariants for Grassmannians are equal to structure constants of the ordinary (equivariant) K-theory of certain two-step flag manifolds. We therefore extended - and also reproved - the "quantum=classical" phenomenon earlier discovered by Buch-Kresch-Tamvakis in the case of the usual Gromov-Witten invariants. Further, we obtained a Pieri and a Giambelli rule, which yield an effective algorithm to multiply any two classes in the quantum K algebra.
Andrew Critch : Causality and Algebraic Geometry
- Algebraic Geometry ( 146 Views )Abstract: Science, and perhaps all learning, is the problem of inferring causal relationships from observations. It turns out that algebraic geometry can provide powerful intuition and methods applicable to causal inference. The relevant theory of graphical causal models is a major entry point to the budding field of algebraic statistics, where algebraic geometry meets statistical modeling, and this talk will give an introduction to it from the perspective of an algebraic geometer. I'll introduce some conceptual tools and methods that are peculiar to algebraic statistics, and work through an example such causal inference computation using the commutative algebra software Macaulay2. At the end I'll review some of my research on hidden Markov models and varieties, and their close connection to matrix product state models of quantum-entangled qubits.
Jason Polak : Exposing relative endoscopy
- Algebraic Geometry ( 131 Views )For a reductive algebraic group G with Lie algebra g and involution \theta we define relative orbital integrals with respect to G acting on the -1 eigenspace of \theta on g. We prove some fundamental lemmas relating these orbital integrals to relative orbital integrals on smaller groups, providing the first example of a theory of relative endoscopy in our setting
Greg Pearlstein : Boundary components of Mumford-Tate domains
- Algebraic Geometry ( 196 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.
Sam Grushevsky : Stable cohomology of compactifications of the moduli spaces of abelian varieties
- Algebraic Geometry ( 124 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.
Jie Wang : The primitive cohomology of the theta divisor of an abelian fivefold
- Algebraic Geometry ( 116 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.
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.