Let X be an affine variety and let H be a hyperplane. An interesting question is to consider how the algebraic invariants of I_X change as we pass to X \cap H considered as a subvariety of H. For example, the codimension, projective dimension and Betti numbers are all likely to change, but it is often hard to predict how, even if the free resolution of I_X is well understood. In this talk I'll focus on a simple but illustrative example - the variety of maximal minors of a 2 by n generic matrix and see how the free resolution changes as we set variables equal to zero. This talk should be accessible to all grad students and will feature the phrase "Slashy Method".

In 2003 Dohmen, Ponitz and Tittmann introduced a 2-variable, generalized chromatic polynomial. We will discuss some of the more interesting properties of this polynomial and how it can be extended to signed graphs. We will then use it to define a matching on signed graphs and give the unexpected complete classification of all matchings.

Many popular polytopes arise as deformations of different types of permutohedra. We will give examples of such polytopes, we'll make the notion of a deformation precise, and we'll discuss some of the perks of being a deformed permutohedron.

We will discuss the relationship between metric ribbon graphs, moduli spaces of Riemann surfaces, and Teichmuller space. In particular, the graph perspective provides a cell structure which is extremely useful in studying cohomological aspects of moduli space and the mapping class groups.

This will be an expository talk on a paper by Fomin and Zelevinksy. We will describe the double Bruhat cell decomposition of GL_n (and say a little about how things carry over to the general case of semisimple groups). Using planar networks, a "twist map", and double wiring diagrams, we will see how to a) parametrize the double Bruhat cells, giving explicit formulas for "factorization schemes", and b) test whether a matrix is totally nonnegative by checking a minimal number of subdeterminants.

The central curve of a linear program is an algebraic curve specified by the associated hyperplane arrangement and cost vector. This curve is the union of the various central paths for minimizing or maximizing the cost function over any region in this hyperplane arrangement. Here we will discuss the algebraic properties of this curve and its beautiful global geometry. In the process, we'll need to study the corresponding matroid of the hyperplane arrangement. This will let us give a refined bound on the total curvature of the central curve, a quantity relevant for interior point methods. This is joint work with Jesus De Loera and Bernd Sturmfels appearing in arXiv:1012.3978.

Permutation Polytopes are convex hulls of subgroups of the group of n x n permutation matrices. We discuss early developments on these polytopes, which includes work on the Birkhoff polytope. We then present new results on volumes of certain families of permutation polytopes, as detailed in joint work with Katherine Burggraf and Jesus De Loera.

My goal is to make this talk a nice introduction to tropical geometry and some of its applications. I will describe what tropical linear spaces are and how they can be used to get a handle on classical A-discriminants. Also, I will describe an effective algorithm for computing them. I will show several examples, pictures, and even software demonstrations. No previous knowledge of tropical geometry (or discriminants) will be assumed.

Multiview geometry is the study of projections of a three-dimensional scene into multiple two-dimensional images. This is the framework in which one studies computer vision. We will describe the projective multiview variety, over which we'll formulate the optimization problem of triangulating a point in space from its view in multiple images. The multigraded Hilbert scheme will be introduced as a tool for studying this variety and its associated initial ideals. This is joint work with Rekha Thomas and Bernd Sturmfels.

We give a brief introduction to tropical max-plus eigenvalue-eigenvector of skew-symmetric matrices, prove a formula for the number of combinatorial types of tropical eigenvector, discuss its applications to pairwise ranking, and entertain the audience with some interesting open problems.

The classical problem, considered by Harer and Zagier, was to count the number of ways to obtain a surface of genus g by gluing completely the sides of a 2k-gon. Dually, this is the same as counting 1-face graphs on the surface, i.e. such that by cutting the surface along the graph one obtains a single disc. A natural generalization is to count incomplete gluings -- when some of the sides of the polygon remain unglued. In this case one obtains genus g surfaces with multiple polygonal boundaries. Remarkably, the generalized problem is in a sence simpler than the original one, and admits an explicit exact solution. If time permits, I will also talk about applications, which are interesting in their own right.

Coloring the vertices of a graph with no adjacent vertices in the same color, is an old problem in combinatorics. Most sufficient conditions are of local character, for example three colors in enough for a cycle since the maximal degree is two. But necessary conditions tend to be global: we need to know if the cycle have an even or odd number of vertices to determine if two or three colors are necessary. For graphs with more involved symmetries and parity questions than in cycles, the most efficient tool is a translation from the graph theory into algebra where the best tools for that are. I will first sketch this setup and describe how it was used by Lovasz to find the chromatic numbers for Kneser graphs. Then I will, in a very speculative manner, describe a program for how to use Groebner bases and toric geometry to attack coloring questions. Simple examples are demonstrated, and accessible research questions for grad students in combinatorics and commutative algebra are presented.

Toric varieties are geometrical objects that can be described combinatorially by polyhedral fans. Often interesting toric varieties come from G representations where G is a semisimple group with maximal torus T. As a motivating example I'll discuss De concini and Procesi's 'wonderful compactification.' Then I'll discuss loop groups and show how representations of loop groups give rise to toric varieties whose fan are given by Voronoi cells associated to a symmetric positive definite form on the Lie algebra of T. No background in representation theory or loop groups is necessary.

Geometric Decomposition Theory is concerned with the question what objects we can build given a set of geometrical objects and a set of rules to compose them. Classical examples include Hilberts 3rd Problem (solved by Dehn), the Banach Tarski Paradox and Tarski's Circle-Squaring Problem (solved by Laczkovich). We take up a different question: Can a convex set S be tiled into convex subsets such that arbitrarily many tiles are similar to S? We will investigate this, and answer a question of Laczkovich along the way.

We will examine total positivity for matrices and for Grassmannians, with a view towards how these examples fit into a broader notion of positivity, framed in terms of cluster algebras. The cluster variables play the role that minors play in the matrix setting and that Pluecker coordinates play in the Grassmannian setting; the totally positive part of a space carrying a cluster algebra structure consists of the points at which all cluster variables take positive values.

No cluster algebra or total positivity background is assumed. These two examples have very nice combinatorial characterizations, using enumeration of paths in certain weighted graphs.

Plane quartic curves span many different fields and over 150 years of mathematics. This theory began with the beautiful combinatorics of the 28 bitangents of a quartic curve. A classical theorem of Hilbert states that every positive ternary quartic is a sum of squares. Using their bitangents, we will represent positive quartics as the sum of three squares in 8 different ways and start to understand the 6-dimensional spectrahedron of all sums of squares representations.

Consider an n by n square matrix where n is large. For each entry, flip a fair coin, making the entry +1 if the coin comes up heads, and -1 if the coin comes up tails. What is the probability that the matrix has determinant equal to zero? Though simple to state, this question is surprisingly hard to answer, and is still open. This talk will discuss some of the ideas in combinatorics, probability theory, and additive combinatorics that have been used to approach this question, with a focus on the breakthrough approach of Terence Tao and Van Vu in 2007. The talk will also discuss joint work with Jean Bourgain and Van Vu.

Given a complete simplicial fan in R^n, the Gelfand-Kapranov-Zelevinsky cone parametrizes piecewise linear functions from the fan to R up to addition of a linear function. Oda and Park showed how to give a decomposition of the of the Gelfand-Kapranov-Zelevinsky cone based on how close the corresponding functions are to being convex. While their results are very useful in algebraic geometry, no prior knowledge will be assumed, and there will be many examples.

Two polytopes are combinatorially distinct if their face posets are nonisomorphic. The problems of counting and classifying the distinct combinatorial types of n-facet polytopes have been of interest since antiquity, but only in the last 50 years have effective techniques been developed to attack this problem in dimensions higher than 3, such as the beneath-beyond method popularized by Altshuler and Shemer. We'll develop some new approaches to this problem based on facet splitting, cutting planes, and Pachner moves, and mention some associated problems. We'll also establish a relationship between the enumeration problem and the problem of proving properties on the edge-vertex graphs of polytopes.

The talk is self-contained; no background in the theory of polytopes is assumed.

In this expository talk I'll describe some of the finite dimensional representation theory of the general linear group by way of the highest weight representations of the Lie algebra sl_n(C). The first part of the talk will be about sl_2(C), but I will later indicate how some of the ideas translate to sl_n(C). I hope to give the listener a good idea of how Schur functions arise as the irreducible characters of the general linear group.

A common theme of enumerative combinatorics are counting functions given by polynomials that are evaluated at positive integers. For example, one proves in any introductory graph theory course that the number of proper k-colorings of a given graph G is a polynomial in k, the "chromatic polynomial" of G. Combinatorial reciprocity theorems give interpretations of these polynomials at negative integers. For example, when we evaluate the chromatic polynomial of G at -1, we obtain (up to a sign) the number of acyclic orientations of G, that is, those orientations of G that do not contain a coherent cycle.

Combinatorics is abundant with polynomials that count something when evaluated at positive integers, and many of these polynomials have a completely different interpretation when evaluated at negative integers. We follow a common thread of chromatic and flow polynomials of graphs, Ehrhart polynomials enumerating integer points in polytopes, and characteristic polynomials of hyperplane arrangements.

Voronoi's (second) reduction theory is a classical way to construct a nice polyhedral fundamental domain for the cone of positive semidefinite matrices under the action of the group GL(n,Z). It is based on Delone subdivsions of lattices. In the talk I will explain this theory and I will give a geometric application: How to find optimal lattice sphere coverings of a given dimension? Then I will give a combinatorial application: Some parts of the fundamental domain are closely related to regular matroids. This connection was recently used by Brannetti, Melo, Viviani for the tropical Schottky problem.

A sequence of finite graphs is said to converge if the densities of the different kinds of subgraphs converge. But what does the sequence converge to? It turns out that the answer has to do with a useful and very general way of generating random graphs, exchangeable random graphs. We will define what this means, give some examples and go through some of the basic theory of graph limits developed by Lovasz, Szegedy and others.

Cluster algebras are a class of combinatorially defined rings that provide a unifying structure for phenomena in a variety of algebraic and geometric contexts. A partial list of related areas includes quiver representations, total positivity, statistical physics, and Teichmuller theory. In this talk I will give an elementary introduction to cluster algebras: I will explain what they are, and why you might care.

This talk can be viewed as a 50-minute commercial break for the course Math 274 (Topics in algebra: cluster algebras) which I will give in Spring 2011.

A graph G is called `bipartite' if can be properly colored with two colors (the chromatic number of G is at most 2). Consider two alternative characterizations for this notion:

Probabilistic: G is bipartite if for any random walk W on G, the position of W at arbitrarily large time t restricts the possible starting position of W.

Topological: G is bipartite if in the `space of directed edges' of G, there's no way to walk from an edge E to the same edge with the reverse orientation.

Brightwell and Winkler introduced a generalization of the first property which they called the 'warmth' of a graph, and showed that the warmth was a lower bound for chromatic number. This was somewhat surprising since the warmth a graph G is defined in terms of homomorphisms *from* some fixed graph (whereas chromatic number is about homomorphisms *into* complete graphs) . In this sense warmth also looks like the Hom complexes first introduced by Lovasz, where the topological connectivity of a `space of directed edges' provides a lower bound for chromatic number. In addition, warmth and Hom complexes behave similarly with respect to certain graph operations including `foldings'. However, no direct connection has been established (as far as I know). We'll discuss these concepts and present a conjectural inequality relating warmth and the topology of the Hom complexes.

The moduli space of conics passing through these three generic points in $\mathbb{TP}^2$ is in fact a generic tropical plane in $\mathbb{TP}^5$. There are seven combinatorial types of generic planes in $\mathbb{TP}^5$. The tropical grassmannian $\mathcal{G}_{t}(3,6)$ is the moduli space of tropical planes in $\mathbb{TP}^5$. The purpose of this talk is to discuss the image of the map $\chi:(\mathbb{TP}^2)^3\to\mathcal{G}_{t}(3,6)$, which maps generic points in $\mathbb{TP}^2$ to the moduli space of conics passing through those three points, and show which of seven combinatorial types of generic planes in $\mathbb{TP}^5$ are realized as a moduli space of tropical quadratic curves passing through three generic points in $\mathbb{TP}^2$. This is joint work with Bernd Sturmfels.

We consider the homogeneous components U_r of the map on R = k[x,y,z]/(x^A, y^B, z^C) that multiplies by x + y + z. We give a bijective proof of the identity proven by J. Li and F. Zanello equating the determinant of the middle component U_r when (A, B, C) = (a + b, a + c, b + c) to the number of plane partitions in an a by b by c box. We also prove that, for certain vector subspaces of R, similar identities hold relating determinants to symmetry classes of plane partitions, in particular classes 3, 6, and 8. Joint work with A. Guo, X. Jin, and G. Liu.

I will explain a combinatorial version of a conjecture of Garcia, Stillman and Sturmfels describing the equations of the space of tensors of rank 2.

How the brain computes is still largely a mystery. Among the many open questions facing neuroscience is the communication bottleneck problem. In recent years, evidence has been accumulating that large populations of neurons (e.g. in the visual cortex) tease apart structure from a much smaller number of signals (e.g. fibers emanating from the retina). These computations (reflected in binary spike trains) must then be communicated to other regions of the brain, and evidence from neuroanatomy suggests that the number of fiber projections leaving a region (the bottleneck) is typically an order of magnitude smaller than the number of neurons in a local population. Thus, some form of compression (other than JPEG!) is taking place in the brain. We explain a principle called Adaptive Compressive Sampling (ACS) that can explain how the brain performs this communication. In addition to being neurologically plausible, our scheme is unsupervised, robust, independent of wiring, and achieves optimal compression. Surprisingly, a crucial ingredient of our theory is an abstract theorem in (combinatorial) Ramsey theory. Since this is a math audience, we can go into the gory (and beautiful) details of the proofs. There will also be computer simulations and open problems! This is joint work with G. Isely and F. Sommer.