Seminars and Colloquia by Series

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.

A Tale of Two Theorems

Series
Algebra Seminar
Time
Monday, March 4, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Greg BlekhermanGeorgia Tech
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)

Log concavity of characteristic polynomials and tropical intersection theory

Series
Algebra Seminar
Time
Monday, February 18, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eric KatzWaterloo
In a recent work with June Huh, we proved the log-concavity of the characteristic polynomial of a realizable matroid by relating its coefficients to intersection numbers on an algebraic variety and applying an algebraic geometric inequality. This extended earlier work of Huh which resolved a long-standing conjecture in graph theory. In this talk, we rephrase the problem in terms of more familiar algebraic geometry, outline the proof, and discuss an approach to extending this proof to all matroids. Our approach suggests a general theory of positivity in tropical geometry.

Subdivision and Algebraic Geometry for Certified Correct Computations

Series
Algebra Seminar
Time
Monday, February 11, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Michael BurrClemson University
Many real-world problems require an approximation to an algebraic variety (e.g., determination of the roots of a polynomial). To solve such problems, the standard techniques are either symbolic or numeric. Symbolic techniques are globally correct, but they are often time consuming to compute. Numerical techniques are typically fast, but include more limited correctness statements. Recently, attention has shifted to hybrid techniques that combine symbolic and numerical techniques. In this talk, I will discuss hybrid subdivision algorithms for approximating a variety. These methods recursively subdivide an initial region into smaller and simpler domains which are easier to characterize. These algorithms are typically recursive, making them both easy to implement (in practice) and adaptive (performing more work near difficult features). There are two challenges: to develop algorithms with global correctness guarantees and to determine the efficiency of such algorithms. I will discuss solutions to these challenges by presenting two hybrid subdivision algorithms. The first algorithm computes a piecewise-linear approximation to a real planar curve. This is one of the first numerical algorithms whose output is guaranteed to be topologically correct, even in the presence of singularities. The primitives in this algorithm are numerical (i.e., they evaluate a polynomial and its derivatives), but its correctness is justified with algebraic geometry and symbolic algebra. The second algorithm isolates the real roots of a univariate polynomial. I will analyze the number of subdivisions performed by this algorithm using a new technique called continuous amortization. I will show that the number of subdivisions performed by this algorithm is nearly optimal and is comparable with standard symbolic techniques for solving this problem (e.g., Descartes' rule of signs or Sturm sequences).

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