Seminars and Colloquia by Series

TBA by Jeffrey Rosenthal

Series
Combinatorics Seminar
Time
Friday, April 17, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jeffrey RosenthalUniversity of Toronto

TBA (joint with Stochastics Seminar)

Counting extensions revisited

Series
Combinatorics Seminar
Time
Friday, March 27, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Lutz WarnkeGeorgia Tech

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. 
In 1989, Joel Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices.  
For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary. 
We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open. 

Graph fractal dimension and structure of fractal networks: a combinatorial perspective

Series
Combinatorics Seminar
Time
Friday, March 6, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pavel SkumsGeorgia State University

We study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product (or Prague or Nešetřil-Rödl) dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colorings and graph Kolmogorov complexity, and analyze the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework to evolutionary studies by revealing the growth of self-organization of heterogeneous viral populations over the course of their intra-host evolution, thus suggesting mechanisms of their gradual adaptation to the host's environment. As far as the authors know, the proposed approach is the first theoretical framework for study of network fractality within the combinatorial paradigm. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.

Based on joint work with Leonid Bunimovich

Scalefree hardness of the Euclidean TSP

Series
Combinatorics Seminar
Time
Friday, February 7, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Wesley PegdenCarnegie Mellon University

We  show  that  if  $P\neq NP$,  then  a  wide  class  of  TSP heuristics fail to approximate the length of the TSP to asymptotic 
optimality, even for random Euclidean instances.  Previously, this result was not even known for any heuristics (greedy, etc) used in practice.  As an application, we show that when  using  a  heuristic from  this  class,  a  natural  class  of  branch-and-bound algorithms takes exponential time to find an optimal tour (again, even on a random point-set),  regardless  of  the  particular  branching  strategy  or lower-bound algorithm used.

Sets without 4APs but with many 3APs

Series
Combinatorics Seminar
Time
Friday, January 31, 2020 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andrei (Cosmin) PohoataCalifornia Inst. of Technology, Pasadena, CA

 It is a classical theorem of Roth that every dense subset of $\left\{1,\ldots,N\right\}$ contains a nontrivial three-term arithmetic progression. Quantitatively, results of Sanders, Bloom, and Bloom-Sisask tell us that subsets of relative density at least $1/(\log N)^{1-\epsilon}$ already have this property. In this talk, we will discuss some sets of $N$ integers which unlike $\left\{1,\ldots,N\right\}$ do not contain nontrivial four-term arithmetic progressions, but which still have the property that all of their subsets of density at least $1/(\log N)^{1-\epsilon}$ must contain a three-term arithmetic progression. Perhaps a bit surprisingly, these sets turn out not to have as many three-term progressions as one might be inclined to guess, so we will also address the question of how many three-term progressions can a four-term progression free set may have. Finally, we will also discuss about some related results over $\mathbb{F}_{q}^n$. Based on joint works with Jacob Fox and Oliver Roche-Newton.

Fast uniform generation of random graphs with given degree sequences

Series
Combinatorics Seminar
Time
Friday, January 24, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Andrii ArmanEmory University

In this talk I will discuss algorithms for a uniform generation of random graphs with a given degree sequence. Let $M$ be the sum of all degrees and $\Delta$ be the maximum degree of a given degree sequence. McKay and Wormald described a switching based algorithm for the generation of graphs with given degrees that had expected runtime $O(M^2\Delta^2)$, under the assumption $\Delta^4=O(M)$. I will present a modification of the McKay-Wormald algorithm that incorporates a new rejection scheme and uses the same switching operation. A new algorithm has expected running time linear in $M$, under the same assumptions.

I will also describe how a new rejection scheme can be integrated into other graph generation algorithms to significantly reduce expected runtime, as well as how it can be used to generate contingency tables with given marginals uniformly at random.

This talk is based on the joint work with Jane Gao and Nick Wormald.

Non-concentration of the chromatic number of a random graph

Series
Combinatorics Seminar
Time
Friday, January 10, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 202
Speaker
Lutz Warnke

We shall discuss the recent breakthrough of  Annika Heckel on the chromatic number of the binomial random graph G(n,1/2),  showing that it is not concentrated on any sequence of intervals of length n^{1/4-o(1)}.

To put this into context, in 1992 Erdos (and also Bollobás in 2004) asked for any non-trivial results asserting a lack of concentration, pointing out that even the weakest such results would be of interest.  
Until recently this seemed completely out of reach, in part because there seemed to be no obvious approach/strategy how to get one's foot in the door. 
Annika Heckel has now found such an approach, based on a clever coupling idea that compares the chromatic number of G(n,1/2) for different n. 
In this informal talk we shall try to say a few words about her insightful proof approach from https://arxiv.org/abs/1906.11808

Please note the unusual room (Skiles 202)

Thresholds versus fractional expectation-thresholds

Series
Combinatorics Seminar
Time
Friday, December 6, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jinyoung ParkRutgers University

(This is a joint event of the Combinatorics Seminar Series and the ACO Student Seminar.)

In this talk we will prove a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kalai and Kahn. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). Our approach builds on recent breakthrough work of Alweiss, Lovett, Wu, and Zhang on the Erdos-Rado “Sunflower Conjecture.” 

This is joint work with Keith Frankston, Jeff Kahn, and Bhargav Narayanan.

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