Tuesday, April 16, 2013 - 17:15 , Location: Skiles 006 , Jordan Ellenberg , University of Wisconsin , Organizer:
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.
Tuesday, April 16, 2013 - 16:00 , Location: Skiles 006 , Dick Gross , Harvard University , Organizer:
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.
Monday, April 15, 2013 - 15:05 , Location: Skiles 005 , Amod Agashe , Florida State University , firstname.lastname@example.org , Organizer:
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.
Monday, April 8, 2013 - 17:00 , Location: Skiles 006 , Seth Sullivant , North Carolina State University , Organizer: Anton Leykin
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.
Monday, April 8, 2013 - 16:05 , Location: Skiles 006 , June Huh , University of Michigan , Organizer: Matt Baker
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.
Monday, April 1, 2013 - 15:00 , Location: Skiles 006 , Kit-Ho Mak , Georgia Tech , Organizer: Anton Leykin
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.
Monday, March 25, 2013 - 15:05 , Location: Skiles 005 , Alex Fink , N.C. State , Organizer: Matt Baker
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.
Monday, March 11, 2013 - 16:05 , Location: Skiles 006 , Dustin Cartwright , Yale University , Organizer: Josephine Yu
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.
Monday, March 4, 2013 - 15:00 , Location: Skiles 005 , Greg Blekherman , Georgia Tech , Organizer: Greg Blekherman
I will explain and draw connections between the following two theorems: (1) Classification of varieties of minimal degree by Del Pezzo and Bertini and (2) Hilbert's theorem on nonnegative polynomials and sums of squares. This will result in the classification of all varieties on which nonnegative polynomials are equal to sums of squares. (Joint work with Greg Smith and Mauricio Velasco)
Monday, February 25, 2013 - 15:05 , Location: Skiles 005 , Julio Andrade , ICERM , Organizer: