Seminars and Colloquia by Series

Effective Chabauty for Sym^2

Series
Algebra Seminar
Time
Monday, April 29, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jennifer ParkMIT
While we know by Faltings' theorem that curves of genus at least 2 have finitely many rational points, his theorem is not effective. In 1985, R. Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is small, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. In this talk, we draw ideas from tropical geometry to show that we can also give an effective bound on the number of rational points of Sym^2(X) that are not parametrized by a projective line or an elliptic curve, where X is a (hyperelliptic) curve of genus g > 2, when the Mordell-Weil rank of the Jacobian of the curve is at most g-2.

p-adic heights and integral points on hyperelliptic curves

Series
Algebra Seminar
Time
Monday, April 22, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jennifer BalakrishnanHarvard University
We give a Chabauty-like method for finding p-adic approximations to integral points on hyperelliptic curves when the Mordell-Weil rank of the Jacobian equals the genus. The method uses an interpretation ofthe component at p of the p-adic height pairing in terms of iterated Coleman integrals. This is joint work with Amnon Besser and Steffen Mueller.

Athens-Atlanta number theory seminar 2 - Arithmetic statistics over function fields

Series
Algebra Seminar
Time
Tuesday, April 16, 2013 - 17:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jordan EllenbergUniversity of Wisconsin
What is the probability that a random integer is squarefree? Prime? How many number fields of degree d are there with discriminant at most X? What does the class group of a random quadratic field look like? These questions, and many more like them, are part of the very active subject of arithmetic statistics. Many aspects of the subject are well-understood, but many more remain the subject of conjectures, by Cohen-Lenstra, Malle, Bhargava, Batyrev-Manin, and others. In this talk, I explain what arithmetic statistics looks like when we start from the field Fq(x) of rational functions over a finite field instead of the field Q of rational numbers. The analogy between function fields and number fields has been a rich source of insights throughout the modern history of number theory. In this setting, the analogy reveals a surprising relationship between conjectures in number theory and conjectures in topology about stable cohomology of moduli spaces, especially spaces related to Artin's braid group. I will discuss some recent work in this area, in which new theorems about the topology of moduli spaces lead to proofs of arithmetic conjectures over function fields, and to new, topologically motivated questions about counting arithmetic objects.

Athens-Atlanta number theory seminar 1 - The arithmetic of hyperelliptic curves

Series
Algebra Seminar
Time
Tuesday, April 16, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dick GrossHarvard University
Hyperelliptic curves over Q have equations of the form y^2 = F(x), where F(x) is a polynomial with rational coefficients which has simple roots over the complex numbers. When the degree of F(x) is at least 5, the genus of the hyperelliptic curve is at least 2 and Faltings has proved that there are only finitely many rational solutions. In this talk, I will describe methods which Manjul Bhargava and I have developed to quantify this result, on average.

Stark-Heegner/Darmon points on elliptic curves over totally real fields

Series
Algebra Seminar
Time
Monday, April 15, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Amod AgasheFlorida State University
The classical theory of complex multiplication predicts the existence of certain points called Heegner points defined over quadratic imaginary fields on elliptic curves (the curves themselves are defined over the rational numbers). Henri Darmon observed that under certain conditions, the Birch and Swinnerton-Dyer conjecture predicts the existence of points of infinite order defined over real quadratic fields on elliptic curves, and under such conditions, came up with a conjectural construction of such points, which he called Stark-Heegner points. Later, he and others (especially Greenberg and Gartner) extended this construction to many other number fields, and the points constructed have often been called Darmon points. We will outline a general construction of Stark-Heegner/Darmon points defined over quadratic extensions of totally real fields (subject to some mild restrictions) that combines past constructions; this is joint work with Mak Trifkovic.

Geometric perspectives on phylogenetics

Series
Algebra Seminar
Time
Monday, April 8, 2013 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Seth SullivantNorth Carolina State University
I will discuss two problems in phylogenetics where a geometric perspective provides substantial insight. The first is the identifiability problem for phylogenetic mixture models, where the main problem is to determine which circumstances make it possible to recover the model parameters (e.g. the tree) from data. Here tools from algebraic geometry prove useful for deriving the current best results on the identifiability of these models. The second problem concerns the performance of distance-based phylogenetic algorithms, which take approximations to distances between species and attempt to reconstruct a tree. A classical result of Atteson gives guarantees on the reconstruction, if the data is not too far from a tree metric, all of whose edge lengths are bounded away from zero. But what happens when the true tree metric is very near a polytomy? Polyhedral geometry provides tools for addressing this question with some surprising answers.

Rota's conjecture, the missing axiom, and the tropical Laplacian

Series
Algebra Seminar
Time
Monday, April 8, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
June HuhUniversity of Michigan
Rota's conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. I will talk about Rota's conjecture and several related topics: the proof of the conjecture for representable matroids, a relation to the missing axiom, and a search for a new discrete Riemannian geometry based on the tropical Laplacian. This is an ongoing joint effort with Eric Katz.

The distribution of rational points on curves over a finite field on average

Series
Algebra Seminar
Time
Monday, April 1, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kit-Ho MakGeorgia Tech
Let p be a prime, let C/F_p be an absolutely irreducible curve inside the affine plane. Identify the plane with D=[0,p-1]^2. In this talk, we consider the problem of how often a box B in D will contain the expected number of points. In particular, we give a lower bound on the volume of B that guarantees almost all translations of B contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense. This is joint work with Alexandru Zaharescu.

Matroids over rings

Series
Algebra Seminar
Time
Monday, March 25, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex FinkN.C. State
Matroids are widely used objects in combinatorics; they arise naturally in many situations featuring vector configurations over a field. But in some contexts the natural data are elements in a module over some other ring, and there is more than simply a matroid to be extracted. In joint work with Luca Moci, we have defined the notion of matroid over a ring to fill this niche. I will discuss two examples of situations producing these enriched objects, one relating to subtorus arrangements producing matroids over the integers, and one related to tropical geometry producing matroids over a valuation ring. Time permitting, I'll also discuss the analogue of the Tutte invariant.

Tropical complexes

Series
Algebra Seminar
Time
Monday, March 11, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dustin CartwrightYale University
A tropical complex is a Delta-complex together with some additional numerical data, which come from a semistable degeneration of a variety. Tropical complexes generalize to higher dimensions some of the analogies between curves and graphs. I will introduce tropical complexes and explain how they relate to classical algebraic geometry.

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