This is a seminar series intended to involve people in the Front Range interested in Algebra, GeoMEtry and Number Theory. The seminar will meet either at CSU (green) or at CU Boulder (gold); look on this page for specific information. For more information, or to provide a speaker, contact Renzo, ## Schedule of Talks:
## · CU Boulder Campus map## · CSU Campus map## Abstracts:August 21: David Zureick-Brown
Serre proved in 1972 that the image of the adelic Galois representation associated to an elliptic curve E without complex multiplication has open image; moreover, he also proved that for an elliptic curve over Q the index of the image is always divisible by 2 (and in particular never surjective). More recently, Greicius in his thesis gave criteria for surjectivity and gave an explicit example of an elliptic curve E over a number field K with surjective adelic representation. Soon after, Zywina, building on earlier work of Duke, Jones, and others, proved that the adelic image `random' elliptic curve is maximal. In this talk I will explain recent joint work with David Zywina in which we generalize these theorems and prove that a random abelian variety in a family with big monodromy has maximal image of Galois. I'll explain what big monodromy and maximal mean an explain the analytic and geometric techniques used in previous work and the new geometric ideas -- in particular, Nori's method of semistable approximation-- needed to generalized to higher dimension. August 30: Chris HallLet $A$ be a principally-polarized abelian variety of dimension $g$ over a field $K$. The field extensions $K_\ell=K(A[\ell])$ obtained by adjoining $\ell$-torsion for each prime $\ell$, and we say $A$ has {\it big monodromy} if the Galois group of $K_\ell/K$ contains $\Sp_{2g}(\F_\ell)$ for almost all $\ell$. For example, if $f(x)$ is a square-free polynomial with complex coefficients and of degree $2g$, then we will show that the Jacobian of the hyperelliptic curve $y^2=f(x)(x-t)$ over the function field $\C(t)$ has big monodromy. The first half of this talk should be accessible to graduate students. September 13: Dusty RossThe Gopakumar-Marino-Vafa formula, proven almost 10 years ago, evaluates certain triple Hodge integrals on the moduli space ofcurves in terms of Schur functions. It has since been realized the the GMV formula is a special case of the Gromov-Witten/Donaldson-Thomas correspondence. We prove an orbifold generalization of the GMV formula and use it to deduce the orbifold GW/DT correspondence for local Z_n gerbes over the projective line. In the first hour of this talk, I will give a broad overview of curve counting invariants and the GW/DT correspondence, paying particular attention to the toric case as that is where our motivation lies. I will introduce the main objects which appear in the gerby GMV formula: loop Schur functions, abelian Hodge integrals, and generalized symmetric groups, and give a statement of the theorem. Given time, I will conclude by showing how the formula can be used to deduce the GW/DT correspondence on a certain class of orbifolds. This hour should be accessible to a broad audience. In the second hour I will discuss some of the main ideas used in the proof of the formula. September 18: Nicholas ProudfootIf you ask somebody what a toric variety is, that person might say "it's a variety that you associate to a polyhedron", or perhaps "it's a variety that you associate to a fan". I will explain what's behind these two answers and how they are related. I'll also talk about hypertoric varieties, in which polyhedra are replaced by hyperplane arrangements and fans are (conjecturally) replaced by pointed oriented matroids. The last part is work in progress with Matthew Arbo. September 27: Takashi Kimuragroup G (e.g. G is a finite group action on X) then the equivariant cohomology $H^*_G(IX)$ of the inertia variety has an orbifold product introduced by Chen-Ruan as a kind of classical limit of orbifold Gromov-Witten theory of $[X/G]$. The orbifold product is the
prototypical example of an inertial product on $H^*_G(IX)$. We will explain how, under certain conditions, inertial products on equivariant K-theory have compatible inertial versions of power operations, Chern classes, and the
Chern character. We then use these power operations to provide a simple presentation of the virtual K-theory ring of P(1,2) and P(1,3) and compare the results with the ordinary K-theory ring of a crepant resolution. The first hour will give a presentation of the main ideas and constructions aimed at beginning graduate students. October 4: Paul JohnsonEnumerative geometry counts geometric things satisfying certain conditions. For example, Schubert calculus counts subspaces of vector spaces; Hurwitz numbers count ramified covers of Riemann surfaces, or alternatively certain sets of permutations. What is of interest is patterns among these counts, which often manifest themselves as nice properties of their generating functions. For example, if X is a variety with Euler characteristic zero, and F(g,X) is the generating function ``counting genus g curves in X'', then it is a general principle that F(g,X) should be a quasimodular form. We verify this principle when $X$ is an orbifold curve and the counting problem is Hurwitz theory. The main tool in the proof is the representation theory of the symmetric group. The majority of the talk will be background and context; the whole talk should be accessible to graduate students. October 11: Aaron BertramAlgebraic geometry begins to get interesting when one considers nine points on an elliptic curve and asks whether or not the curve is the unique elliptic curve to pass through those points. The answer depends on whether the sum of the points is ``trivial'' in the Jacobian. What happens tropically? Somewhat disappointingly, but interestingly, the condition for nine (general) points on a tropical elliptic curve to determine the curve is topological. This is based on joint work with Dylan Zwick, Lance Miller and Drew Johnson. October 18: Julia GordonThis talk is about a method, based on model theory, that can be used to get some uniform in "p" estimates for integrals over p-adic fields in the cases when it is hard to see directly how such integrals behave for different places p. One example of such a situation is orbital integrals on p-adic groups. Let $G$ be a reductive p-adic group (such as the group $GL_n(\Q_p)$). Harish-Chandra introduced the notion of an orbital integral on $G$, and proved that the orbital integrals, normalized by the discriminant, are bounded, in a certain sense. However, it is not easy to see how this bound behaves if we let the $p$-adic field vary (for example, if the group $G$ is defined over a number field $F$, and we consider the family of groups $G_v=G(F_v)$, as v runs over the set of finite places of $F$). Using a method based on model theory and motivic integration, we prove that the bound on orbital integrals can be taken to be a fixed power (depending on $G$) of the cardinality of the residue field. This statement has an application to the recent work of S.-W. Shin and N. Templier on counting zeroes of L-functions. This project is joint work with R. Cluckers and I. Halupczok. October 18: Jonathan WiseI'll describe some categories arising from deformation theory that behave in many ways like commutative rings. Then we'll discuss how to axiomatize the situation to arrive at a definition of "higher commutative rings". At the end, we'll talk about what this has to do with "derived algebraic geometry". I'll assume some familiarity with commutative algebra and basic category theory, but not much else. October 23: David AyalaVarious interesting examples of invariants of 3-manifolds and links satisfy a local-to-global expression. In this talk, we will axiomatize invariants satisfying such an expression and classify them in terms of algebraic data. Through this language there is a refinement of Poincar\'e duality which leads to some interesting conclusions. A few examples will be explicated. This is a report on joint work with John Francis and Hiro Tanaka. November 6: David JensenThe Hassett-Keel program aims to give modular interpretations of log canonical models for the moduli spaces of curves. The program, while relatively new, has attracted the attention of a number of researchers, and has rapidly become one of the most active areas of research concerning the moduli of curves. In the genus four case, we construct several of these models as GIT quotients of a single, elementary space. After discussing our construction, we will consider how we expect these ideas to generalize to the general genus case. November 8: Olivia DumitrescuHarbourne-Hirshowitz's conjecture for $\bar M_{0,n}$. In particular, we reduce the F-nef conjecture for $\bar M_{g}$ to a combinatorial question. November 13: Katherine StangeThe curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i]^2, inspired by, and intimately related to, the `sensual quadratic form' of Conway. November 15: Korben RusekTalk 1 $A$-Discriminant Amoebae and the Horn Uniformization Let $A\subset\mathbb{Z}^n$ be a collection of $n+m+1$ exponent vectors. The $A$-discriminant variety is the collection of polynomials with support $A$ with degenerate roots. The $A$-discriminant amoeba is the component-wise logarithm of this variety. This variety (as well as the resulting amoeba) admits an elegant and computationally efficient parametrization called the Horn uniformization map. We also give various examples of recent applications of the $A$-discriminant variety from real polynomial solving to satellite orbits. Talk 2 The $A$-Discriminant Amoeba Structure We discuss the structure of $A$-discriminant amoebae in more depth. We focus on the case where $A$ is a collection of $n+3$ points in $\mathbb{Z}^n$ in general position. We will see how these discriminant amoebae have surprisingly simple structure. A few simple observations will lead to new and tight lower and upper bounds on the number of connected components in the complement of the amoeba. We will also discuss tight bounds on the number of cusps in such Amoebae. Both talks should be accessible to graduate students.
This talk is about some aspects of linear algebra seen from a non-linear
algebra perspective. Eigenvectors and generalized eigenvectors to
square matrices with complex entries can be packed into subschemes of
projective space in a natural way. These schemes have an interesting
geometry reflecting the underlying linear algebra. In this talk I will
explain an application of this formalism, namely a numerical method to
compute the Jordan normal form of matrices. The method operates via
numerical homotopies and their associated deformations, which provides a
link to intersection theory in algebraic geometry. Joint work with
Chris Peterson.
Title: Quadratic twists of elliptic curves (Bo-Hae Im, Chung-Ang University / Indiana University) Abstract:
First of all, if we have three elliptic curves $E_i$ over a number
field $K$, we prove that there exists a number field $L$ over $K$ there
exist infinitely many $d\in L$ such that the rank of quadratic twists by
$d$ of each $E_i$ are positive.
Secondly, as a joint work with Michael Larsen, we talk about
elliptic curves over a number field $K$ with the property that for any
products $D$ of $d_i$ in a finite set of $d_i$ in $K$, the rank of the
quadratic twist by $D$ of $E$ is positive. December 4: Bhargav Bhatt A basic theorem in Hodge theory is the isomorphism between de Rham and Betti cohomology for complex manifolds; this follows directly from the Poincare lemma. The p-adic analogue of this comparison lies deeper, and was the subject of a series of extremely influential conjectures made by Fontaine in the early 80s (which have since been established by various mathematicians). In my talk, I will first discuss the geometric motivation behind Fontaine’s conjectures, and then explain a simple new proof based on general principles in derived algebraic geometry — specifically, derived de Rham cohomology — and some classical geometry with curve fibrations. This work builds on ideas of Beilinson who proved the de Rham comparison conjecture this way. In previous semesters the seminar page was maintained by Renzo Cavalieri, Rachel Pries, Jeff Achter and Yano Casalaina. You can find the Fall 09 page here, and the Spring 10 here, Fall 10 here and Spring 11 here! Then there is Fall 11 and Spring 12. |