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Series: Combinatorics Seminar

A small set expander is a graph where every set of sufficiently small
size has near perfect edge expansion. This talk concerns the computational
problem of distinguishing a small set-expander, from a graph containing a
small non-expanding set of vertices. This problem henceforth referred to
as the Small-Set Expansion problem has proven to be intimately connected to
the complexity of large classes of combinatorial optimization problems.
More precisely, the small set expansion problem can be shown to be
directly related to the well-known Unique Games Conjecture -- a
conjecture that has numerous implications in approximation algorithms.
In this talk, we motivate the problem, and survey recent work consisting of
algorithms and interesting connections within graph expansion, and its
relation to Unique Games Conjecture.

Series: Combinatorics Seminar

Wiring diagrams are classical objects of combinatorics. Plabic graphs were
defined by Postnikov, to study the total positivity of the Grassmannian. We
will show how to generalize several definitions and properties of wiring
diagrams to Plabic graphs, proving a conjecture by Leclerc-Zelevinsky and
Scott on the way. We will begin with a brief introduction to total
positivity and end with connection to cluster algebras. Major part of the
talk comes from a joint work with Alexander Postnikov and David Speyer.

Series: Combinatorics Seminar

The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of Kashiwara's sl_2 lowering operator in the theory of crystal bases. The talk will include a survey of related past work on symmetric chain decomposition and unimodality by Greene-Kleitman, Griggs-Killian-Savage, Proctor, Stanley and others as well as a discussion of open questions that still remain. This is joint work with Anne Schilling.

Series: Combinatorics Seminar

Series: Combinatorics Seminar

Say that a graph with maximum degree at most $d$ is {\it $d$-bounded}. For$d>k$, we prove a sharp sparseness condition for decomposition into $k$ forestsand a $d$-bounded graph. The condition holds for every graph with fractionalarboricity at most $k+\FR d{k+d+1}$. For $k=1$, it also implies that everygraph with maximum average degree less than $2+\FR{2d}{d+2}$ decomposes intoone forest and a $d$-bounded graph, which contains several earlier results onplanar graphs.

Series: Combinatorics Seminar

Immersion is a containment relation between graphs (or digraphs) which is
defined similarly to the more familiar notion of minors, but is incomparable
to it. Of particular interest is to find conditions on a graph (or digraph)
G which guarantee that G contains a clique (or bidirected clique) of order t
as an immersion. This talk will begin with a gentle introduction, and will
then share two new results of this form, one for graphs and one for
digraphs. In the former case, we find that minimum degree 200t is
sufficient, and in the later case, we find that minimum degree t(t-1) is
sufficient, provided that G is Eulerian. These results come from joint work
with Matt DeVos, Jacob Fox, Zdenek Dvorak, Bojan Mohar and Diego Scheide.

Series: Combinatorics Seminar

The secondary polytope of a point configuration A is a polytope
whose faces are in bijection with regular subdivions of A, e.g. the secondary
polytope of the vertices of polygon is an associahedron. The resultant of a
tuple of point configurations A_1, A_2, ..., A_k in Z^n is the set of
coefficients for which the polynomials with supports A_1, A_2, ..., A_k have a
common root with no zero coordinates over complex numbers, e.g. when each A_1
is a standard simplex and k = n+1, the resultant is defined by a determinant.
The Newton polytope of a polynomial is the convex hull of the exponents, e.g.
the Newton polytope of the determinant is the perfect matching polytope.
In this talk, I will explain the close connection between secondary polytopes
and Newton polytopes of resultants, using tropical geometry, based on joint
work with Anders Jensen.

Series: Combinatorics Seminar

Ramsey theory studies the internal homogenity of mathematical structures
(i.e.
graphs, number sets), parts of which (subgraphs, number subsets) are
arbitrarily coloured. Often, the sufficient object size implies the
existence
of a monochromatic sub-object. Combinatorial games are 2-player games of
skill
with perfect information. The theory of combinatorial games studies mostly
the
questions of existence of winning or drawing strategies.
Let us consider an object that is studied by a particular Ramsey-type
theorem.
Assume two players alternately colour parts of this object by two colours
and their goal is to create certain monochromatic sub-object.
Then this is a combinatorial game.
We focus on the minimum object size such that the appropriate
Ramsey-type theorem
holds, called "Ramsey number", and on the minimum object size such that the
first player has a winning strategy in the corresponding combinatorial game,
called "game number".
In this talk, we investigate the "restricted Ramsey-type theorems".
This means, we show the existence of first player's winning strategies,
and we show that game numbers are surprisingly small, compared to
Ramsey numbers. (This is joint work with Jarek Nesetril.)

Series: Combinatorics Seminar

How large can a subset of the first N natural numbers be before it is guaranteed to contain two distinct elements which differ by a perfect square? What if I replaced "perfect square" with the image of a more general polynomial, or perhaps "one less than a prime number"? We will discuss results of this flavor, including recent improvements and generalizations.

Series: Combinatorics Seminar

The generalized flow model is a classical and widely applicable extension of network flows, where on every arc, the flow leaving the arc is a linear function of the flow entering the arc. In the talk, I will investigate a nonlinear extension of this model, with the flow leaving an arc being a concave function of the entering flow. I exhibit the first combinatorial polynomial time algorithm for solving corresponding optimization problems. This model turns out to be a common framework for solving several market equilibrium problems, such as linear Fisher markets, and immediately enables to extend them to more general settings. I will also give a survey on generalized flow algorithms and previous nonlinear flow models.