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Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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.

Series: Algebra Seminar

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)

Series: Algebra Seminar

Series: Algebra Seminar

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.

Series: Algebra Seminar

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).

Series: Algebra Seminar

The Witt group of a scheme is a globalization to schemes of the classical Witt group of a field. It is a part of a cohomology theory for schemes
called the derived Witt groups. In this talk, we introduce two problems about the derived Witt groups, the Gersten conjecture and a
finite generation question for arithmetic schemes, and explain recent progress on them.