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Series: Combinatorics Seminar

Various parameters of many models of random rooted trees are fairly
well understood if they relate to a near-root part of the tree or to global tree
structure. In recent years there has been a growing interest in the analysis
of the random tree fringe, that is, the part of the tree that is close to the
leaves. Distance from the closest leaf can be viewed as the protection level of
a vertex, or the seniority of a vertex within a network.
In this talk we will review a few recent results of this kind for a number of
tree varieties, as well as indicate the challenges one encounters when trying
to generalize the existing results. One tree variety, that of decreasing binary
trees, will be related to permutations, another one, phylogenetic trees, is
frequent in applications in molecular biology.

Series: Combinatorics Seminar

The fundamental EKR theorem states that, when n≥2r, no pairwise intersecting family of r-subsets of {1,2,...,n} is larger than the family of all r-subsets that each contain some fixed x (star at x), and that a star is strictly largest when n>2r. We will discuss conjectures and theorems relating to a generalization to graphs, in which only independent sets of a graph are allowed. In joint work with Kamat, we give a new proof of EKR that is injective, and also provide results on a special class of trees called spiders.

Series: Combinatorics Seminar

The flow polytope associated to an acyclic graph is the set of all
nonnegative flows on the edges of the graph with a fixed netflow at each
vertex. We will examine flow polytopes arising from permutation matrices,
alternating sign matrices and Tesler matrices. Our inspiration is the
Chan-Robins-Yuen polytope (a face of the polytope of doubly-stochastic
matrices), whose volume is equal to the product of the first n Catalan
numbers (although there is no known combinatorial proof of this fact!). The
volumes of the polytopes we study all have nice product formulas.

Series: Combinatorics Seminar

The theme of this talk is walks in a random environment of "signposts"
altered by the walker. I'll focus on three related examples:
1. Rotor walk on Z^2. Your initial signposts are independent with the
uniform distribution on {North,East,South,West}. At each step you rotate
the signpost at your current location clockwise 90 degrees and then follow
it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps
you will visit order n^{2/3} distinct sites. I'll outline an elementary
proof of a lower bound of this order. The upper bound, which is still open,
is related to a famous question about the path of a light ray in a grid of
randomly oriented mirrors. This part is joint work with Laura Florescu and
Yuval Peres.
2. p-rotor walk on Z. In this walk you flip the signpost at your current
location with probability 1-p and then follow it. I'll explain why your
scaling limit will be a Brownian motion perturbed at its extrema. This part
is joint work with Wilfried Huss and Ecaterina Sava-Huss.
3. p-rotor walk on Z^2. Rotate the signpost at your current location
clockwise with probability p and counterclockwise with probability 1-p, and
then follow it. This walk “organizes” its environment of signposts. The
stationary environment is an orientation of the uniform spanning forest,
plus one additional edge. This part is joint work with Swee Hong Chan, Lila
Greco and Boyao Li.

Series: Combinatorics Seminar

I will talk about the problem of computing the number of integer partitions
into parts lying in some integer sequence. We prove that for certain
classes of infinite sequences the number of associated partitions of an
input N can be computed in time polynomial in its bit size, log N. Special
cases include binary partitions (i.e. partitions into powers of two) that
have a key connection with Cayley compositions and polytopes. Some
questions related to algebraic differential equations for partition
sequences will also be discussed.
(This is joint work with Igor Pak.)

Series: Combinatorics Seminar

Researchers here at Georgia Tech initiated a "Ramsey Theory" on binary trees and used the resulting tools to show that the local dimension of a poset is not bounded in terms of the tree-width of its cover graph. Subsequently, in collaboration with colleagues in Germany and Poland, we extended these Ramsey theoretic tools to solve a problem posed by Seymour. In particular, we showed that there is an infinite sequence of graphs with bounded tree-chromatic number and unbounded path-chromatic number. An interesting detail is that our research showed that a family conjectured by Seymour to have this property did not. However, the insights gained in this work pointed out how an appropriate modification worked as intended.
The Atlanta team consists of Fidel Barrera-Cruz, Heather Smith, Libby Taylor and Tom Trotter The European colleagues are Stefan Felsner, Tamas Meszaros, and Piotr Micek.

Series: Combinatorics Seminar

In the talk we state, explain, comment, and finally prove a
theorem (proved jointly with Yuval Peled) on the size and the structure
of certain homology groups of random simplicial complexes. The main
purpose of this presentation is to demonstrate that, despite topological
setting, the result can be viewed as a statement on Z-flows in certain
model of random hypergraphs, which can be shown using elementary
algebraic and combinatorial tools.

Series: Combinatorics Seminar

In enumerative combinatorics, it is quite common to have in hand a number of known initial terms of a combinatorial sequence whose behavior you'd like to study. In this talk we'll describe two techniques that can be used to shed some light on the nature of a sequence using only some known initial terms. While these methods are, on the face of it, experimental, they often lead to rigorous proofs. As we talk about these two techniques -- automated conjecturing of generating functions, and the method of differential approximation -- we'll exhibit their usefulness through a variety of combinatorial topics, including matchings, permutation classes, and inversion sequences.

Series: Combinatorics Seminar

Many classical hard algorithmic problems on graphs, like coloring, clique number, or the Hamiltonian cycle problem can be sped up for planar graphs resulting in algorithms of time complexity $2^{O(\sqrt{n})}$. We study the coloring problem of unit disk intersection graphs, where the number of colors is part of the input. We conclude that, assuming the Exponential Time Hypothesis, no such speedup is possible. In fact we prove a series of lower bounds depending on further restrictions on the number of colors. Generalizations for other shapes and higher dimensions were also achieved. Joint work with E. Bonnet, D. Marx, T. Miltzow, and P Rzazewski.

Series: Combinatorics Seminar

A class of graphs is *χ-bounded* if the chromatic number of all graphs in
the class is bounded by some function of their clique number. *String
graphs* are intersection graphs of curves in the plane. Significant
research in combinatorial geometry has been devoted to understanding the
classes of string graphs that are *χ*-bounded. In particular, it is known
since 2012 that the class of all string graphs is not *χ*-bounded. We prove
that for every integer *t*≥1, the class of intersection graphs of curves in
the plane each of which crosses a fixed curve *c* in at least one and at
most *t* points is *χ*-bounded. This result is best possible in several
aspects; for example, the upper bound *t* on the number of crossings of
each curve with *c* cannot be dropped. As a corollary, we obtain new upper
bounds on the number of edges in so-called *k*-quasi-planar topological
graphs. This is joint work with Alexandre Rok.