Seminars and Colloquia by Series

The structure of space curve arrangements with many incidences

Series
Combinatorics Seminar
Time
Tuesday, March 10, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Joshua Zahl – MIT –
In 2010, Guth and Katz proved that if a collection of N lines in R^3 contained more than N^{3/2} 2-rich points, then many of these lines must lie on planes or reguli. I will discuss some generalizations of this result to space curves in three dimensional vector spaces. This is joint work with Larry Guth.

Two combinatorial applications of smooth numbers

Series
Combinatorics Seminar
Time
Wednesday, February 18, 2015 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan McNew – Dartmouth College
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.

Graph Fourientations and the Tutte Polynomial

Series
Combinatorics Seminar
Time
Monday, February 16, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Spencer Backman – University of Rome
A fourientation of a graph is a choice for each edge of whether to orient it in either direction, bidirect it, or leave it unoriented. I will present joint work with Sam Hopkins where we describe classes of fourientations defined by properties of cuts and cycles whose cardinalities are given by generalized Tutte polynomial evaluations of the form: (k+l)^{n-1}(k+m)^g T (\frac{\alpha k + \beta l +m}{k+l}, \frac{\gamma k +l + \delta m}{k+m}) for \alpha,\gamma \in {0,1,2} and \beta, \delta \in {0,1}. We also investigate classes of 4-edge colorings defined via generalized notions of internal and external activity, and we show that their enumerations agree with those of the fourientation classes. We put forth the problem of finding a bijection between fourientations and 4-edge-colorings which respects all of the given classes. Our work unifies and extends earlier results for fourientations due to myself, Gessel and Sagan, and Hopkins and Perkinson, as well as classical results for full orientations due to Stanley, Las Vergnas, Greene and Zaslavsky, Gioan, Bernardi and others.

Two combinatorial applications of smooth numbers

Series
Combinatorics Seminar
Time
Tuesday, February 10, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan McNew – Dartmouth College
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.

Two combinatorial applications of smooth numbers

Series
Combinatorics Seminar
Time
Tuesday, February 3, 2015 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan McNew – Dartmouth College
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.

Likelihood Orders for Random Walks on Groups

Series
Combinatorics Seminar
Time
Tuesday, January 27, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Megan Bernstein – Stanford University –
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.

Statistical matching theory

Series
Combinatorics Seminar
Time
Thursday, January 22, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Peter Csikvari – MIT –
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.

Catalan Shuffles

Series
Combinatorics Seminar
Time
Tuesday, January 13, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Emma Cohen – Georgia Tech
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.)

Independent sets in regular graphs: spectral stabilty and the hard-core model

Series
Combinatorics Seminar
Time
Tuesday, January 6, 2015 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hiep Han – Emory University and University of Sao Paulo, Brazil –
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.)

First-order properties of Erdos-Renyi random graphs

Series
Combinatorics Seminar
Time
Tuesday, December 9, 2014 - 13:35 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Maksim Zhukovskii – MIPT, Moscow, Russia
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.

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