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Series: Combinatorics Seminar

We look at two combinatorial problems which can be solvedusing careful
estimates for the distribution of smooth numbers. Thefirst is the
Ramsey-theoretic problem to determine the maximal size ofa subset of of
integers containing no 3-term geometric progressions.This problem was
first considered by Rankin, who constructed such asubset with density
about 0.719. By considering progressions among thesmooth numbers, we
demonstrate a method to effectively compute thegreatest possible upper
density of a geometric-progression-free set.Second, we consider the
problem of determining which prime numberoccurs most frequently as the
largest prime divisor on the interval[2,x], as well as the set prime
numbers which ever have this propertyfor some value of x, a problem
closely related to the analysis offactoring algorithms.

Series: Combinatorics Seminar

Series: Combinatorics Seminar

When studying the mixing of random walks on groups, information about the relative likelihoods of the elements under the walk can serve to help understand the mixing and reveal some internal structure. Starting with some elementary arguments of Diaconis and Isaacs and moving into arguments using representation theory of the symmetric group, I'll demonstrate some total and partial orders on finite groups that describe the relative likeliness under random walks. No prior knowledge is assumed.

Series: Combinatorics Seminar

In this talk we will survey some recent development on statistical properties of matchings of very large and infinite graphs. The main goal of the talk is to describe a few applications of a new concept called matching measure. These applications include new results on the number of (perfect) matchings in large girth graphs as well as simple new proofs of certain statistical physical theorems. In particular, we will sketch the proof of Friedland's Lower Matching Conjecture, and a new proof of Schrijver's and Gurvits's theorems. This talk is based on joint papers with various subsets of Miklos Abert, Peter E. Frenkel, Tamas Hubai and Gabor Kun.

Series: Combinatorics Seminar

Catalan numbers arise in many enumerative contexts as the counting sequence of combinatorial structures. We consider natural local moves on some realizations of the Catalan sequence and derive estimates of the mixing time of the corresponding Markov chains. We present a new O(n^2 log n) bound on the mixing time for the random transposition chain on Dyck paths, and raise several open problems, including the optimality of the above bound. (Joint work with Prasad Tetali and Damir Yelliusizov.)

Series: Combinatorics Seminar

Let B(d,n) denote the d-regular graph on n vertices which consists of the disjoint union of complete bipartite graphs. It follows from the results of Kahn and of Zhao that among all d-regular graphs on n vertices B(d,n) maximizes the number of independent sets. In this talk, we show a spectral stability phenomenon of this result in the following sense. The eigenvalues of (the adjacency matrix) of B(d,n) are known to be d, -d and zeroes and we show that, if the smallest eigenvalue of G is bounded away from -d, then the number of independent sets in G is exponentially smaller than that of B(d,n). Furthermore, we extend this method to study the well-known hard-core model from statistical physics. Given a d-regular bipartite graph G whose second smallest eigenvalue is bounded away from -d. Let Ind(G) denote the set of all independent sets of G. Among others, we show that in this case the random independent set I\in Ind(G), drawn from the hard-core distribution with activation parameter lambda>> (log d)/d, is essentially completely (up to o(|I|) vertices) contained in one of the partition classes of G. (This is joint work with Prasad Tetali.)

Series: Combinatorics Seminar

In the talk, an asymptotic behaviour of first order properties of the
Erdos-Renyi random graph G(n,p) will be considered. The random graph obeys
the zero-one law if for each first-order property L either its probability
tends to 0 or tends to 1. The random graph obeys the zero-one k-law if for
each property L which can be expressed by first-order formula with
quantifier depth at most k either its probability tends to 0 or tends to 1.
Zero-one laws were proved for different classes of functions p=p(n). The
class n^{-a} is at the top of interest. In 1988 S. Shelah and J.H. Spencer
proved that the random graph G(n,n^{-a}) obeys zero-one law if a is
positive and irrational. If a is rational from the interval (0,1], then
G(n,n^{-a}) does not obey the zero-one law. I obtain zero-one k-laws for
some rational a from (0,1]. For any first-order property L let us consider
the set S(L) of a from (0,1) such that a probability of G(n,n^{-a}) to
satisfy L does not converges or its limit is not zero or one. Spencer
proved that there exists L such that S(L) is infinite. Recently in the
joint work with Spencer we obtain new results on a distribution of elements
of S(L) and its limit points.

Series: Combinatorics Seminar

In a "rotor walk" the exits from each vertex follow a prescribed
periodic sequence. On an infinite Eulerian graph embedded periodically
in $\R^d$, we show that any simple rotor walk, regardless of rotor
mechanism or initial rotor configuration, visits at least on the order
of t^{d/(d+1)} distinct sites in t steps. We prove a shape theorem for the
rotor walk on the comb graph with i.i.d.\ uniform initial rotors, showing
that the range is of order t^{2/3} and the asymptotic shape of the range is a
diamond. Using a connection to the mirror model and critical percolation,
we show that rotor walk with i.i.d. uniform initial rotors is recurrent on
two different directed graphs obtained by orienting the edges of the square
grid, the Manhattan lattice and the F-lattice.
Joint work with Lionel Levine and Yuval Peres.

Series: Combinatorics Seminar

The Jacobian group Jac(G) of a finite graph G is a finite abelian group whose cardinality is the number of spanning trees of G. It is natural to wonder whether there is a canonical simply transitive action of Jac(G) on the set of spanning trees which "explains" this numerical coincidence. Surprisingly, this turns out to be related to topological embeddings: we will explain a certain precise sense in which the answer is yes if and only if G is planar. We will also explain how tropical geometry sheds an interesting new light on this picture.

Series: Combinatorics Seminar

In graph/hypergraph theory, perfect matchings are fundamental objects of study. Unlike the graph case, perfect matchings in hypergraphs have not been well understood yet. It is quite natural and desirable to
extend the classical theory on perfect matchings from graphs to hypergraphs, as many important problems
can be phrased in this framework, such as Ryser's conjecture on transversals in Latin squares and the
Existence Conjecture for block designs. I will focus on Dirac-type conditions (minimum degree
conditions) in uniform hypergraphs and discuss some recent progresses. In particular, we determine the
minimum codegree threshold for the existence of a near perfect matching in hypergraphs, which confirms a
conjecture of Rodl, Rucinski and Szemeredi, and we show that there is a polynomial-time algorithm
that can determine whether a k-uniform hypergraph with minimum codegree n/k has a perfect matching,
which solves a problem of Karpinski, Rucinski and Szymanska completely.