40 DM16 Abstracts IP1 [email protected] Accuracy, Privacy, and Validity: When Right is Wrong and Wrong is Right IP4 In 2008 a simple pen-and-paper privacy attack on aggre- Mathematical Models: Uses, Abuses, and Non-uses gate allele frequency statistics in a Genome-Wide Asso- ciation Study rocked the world of genomics research and Models are indispensable, but have to be used with caution. resulted in a change in access policy for aggregate statistics Some early quantitative models, drawn from the early his- in studies funded by the US National Institutes of Health. tory of British railways and related to the ubiquitous grav- After describing the original attack and summarizing re- ity models of transportation, urban planning, spacial eco- cent advances in attack strategies, we shift to the defense, nomics, and related areas, will be presented. They demon- discussing differential privacy, a notion of privacy tailored strate how even clearly false models can be useful, and how to statistical analysis of large datasets. Signal properties of sometimes they are misused or tragically not used. differential privacy include its resilience to arbitrary side in- formation and the ability to understand cumulative privacy Andrew M. Odlyzko loss over multiple statistical analyses. Finally, we describe University of Minnesota a tight connection between differential privacy and statis- [email protected] tical validity under adaptive (exploratory) data analysis. IP5 Cynthia Dwork Microsoft Research, USA Tangles and the Mona Lisa: Connectivity Versus [email protected] Tree Structure Tangles, first introduced by Robertson and Seymour in IP2 their work on graph minors, are a radically new way to define regions of high connectivity in a graph. The idea From Algorithm to Theorem (in Probabilistic is that, whatever that highly connected region might ‘be’, Combinatorics) low-order separations of the graph cannot cut through it, and so it will orient them: towards the side of the separa- The general question is let X be a set of interesting ”things” tion on which it lies. A tangle, thus, is simply a consistent (permutations, graphs, partitions, ...). Pick x in X at way of orienting all the low-order separations in a graph. random; what does x ”look like”? There are a host of The new paradigm this brings to connectivity theory is that results for taking theorems(e.g., the Gale-Ryser theorem) such consistent orientations of all the low-order separations and turning them into algorithms for efficient generation. may, in themselves, be thought of as highly connected re- This talk goes in the opposite direction: given an algorithm gions: rather than asking exactly which vertices or edges for random generation, what (limit) theorems does it im- belong to such a region, we only ask where it is, collect- ply? One key example, drawn from joint work with Chern, ing pointers to it from all sides. Pixellated images share Kane, and Rhoades—there is a clever algorithm for gener- this property: we cannot tell exactly which pixels belong ating a random set partition due to Stam. This allowed us to the Mona Lisa’s nose, rather than her cheek, but we can to prove the limiting normality of the number of crossings identify ‘low-order’ separations of the picture that do not (and many other functionals), a long-open problem. cut right through such features, and which can therefore be used collectively to delineate them. This talk will outline Persi Diaconis a general theory of tangles that applies not only to graphs Stanford University and matroids but to a broad range of discrete structures. [email protected] Including, perhaps, the pixellated Mona Lisa. Reinhard Diestel IP3 Univeritat Hamburg [email protected] Stabilisation in Algebra, Geometry, and Combina- torics IP6 Throughout mathematics, one encounters sequences of al- gebraic varieties—geometric structures defined by polyno- Induced Matchings, Arithmetic Progressions and mial equations. As the dimension of the variety grows, Communication typically so does its complexity, measured, for instance, by the degrees of its defining equations. And yet, many se- Extremal combinatorics is one of the central branches of quences stabilise in the sense that from some member of the discrete mathematics that deals with the problem of es- sequence on, all complexity is inherited from the smaller timating the maximum possible size of a combinatorial members by applying symmetries. I will present several structure which satisfies certain restrictions. Often, such examples of this, as yet, only partially understood phe- problems also have applications to other areas including nomenon. Beautiful combinatorics of well-quasi-ordered theoretical computer science, additive number theory and sets plays a key role in the proofs. The hope is that, con- information theory. In his talk, we will illustrate this fact versely, algebraic stabilisation may in the future also shed using several closely related examples, focusing on the re- new light on well-quasi-orders. cent works with Alon, Fox, Huang and Moitra. Jan Draisma Benny Sudakov Eindhoven University of Technology, The Netherlands ETH Zurich Department of Mathematics and Computer Science D-Math DM16 Abstracts 41 [email protected] [email protected] IP7 SP2 Quasirandomness, Sidorenko’s Conjecture and Hot Topics Session: Graph Isomorphism in Graph Norms Quasipolynomial Time - Part I of II Using the theory of quasirandomness as an underlying In the first talk I will sketch the main ingredients of the theme, we will discuss recent progress on a number of prob- algorithm and indicate how they lead to quasipolynomial lems in extremal graph theory, including Sidorenko’s con- recurrence. Familiarity with basic concepts of group theory jecture and a question of Lov´asz asking for a classification (such as kernel of a homomorphism) will be assumed. of graphs that define norms. L´aszl´o Babai David Conlon University of Chicago University of Oxford [email protected] [email protected] CP1 IP8 The Family of Plane Graphs with Face Sizes 3 or 4 Excluded Grid Theorem: Improved and Simplified A reduction theorem will be proved for the family of plane One of the key results in Robertson and Seymour’s seminal graphs with faces sizes 3 or 4. Each reduction produces work on graph minors is the Excluded Grid Theorem. The large proper minors in the same family. Each reduction is theorem states that there is a function f, such that for every shown to be necessary. positive integer g, every graph whose treewidth is at least f(g) contains the (gxg)-grid as a minor. This theorem has Sheng Bau found many applications in graph theory and algorithms. University of Natal, Pietermaritzburg An important open question is establishing tight bounds on Private Bag X01, Scottsville 3209 f(g) for which the theorem holds. Robertson and Seymour [email protected] showed that f(g) ≥ Ω(g2logg), and this remains the best current lower bound on f(g). Until recently, the best upper bound was super-exponential in g. In this talk, we will give CP1 an overview of a recent sequence of results, that has lead to Bijections to Split Graphs the best current upper bound of f(g)=O(g19polylog(g)). AgraphG is a split graph if its vertices can be partitioned into a clique and a stable set. We call a split graph balanced Julia Chuzhoy if its partition is unique and unbalanced otherwise. In this Toyota Technological Institute at Chicago talk, we present several proofs, using natural bijections, [email protected] that the number of unbalanced split graphs on n vertices is equal to the number of split graphs on n−1vertices.These proofs demonstrate connections between split graphs and SP0 other well-known combinatorial families. Hot Topics Session: Graph Isomorphism in Quasipolynomial Time - Part II of II Karen Collins Wesleyan University I will explain the core “Local Certificates’ algorithm in de- [email protected] tail and sketch the aggregation of the local certificates. Familiarity with basic concepts of group theory (such as Ann N. Trenk kernel of a homomorphism) will be assumed. Wellesley College L´aszl´o Babai [email protected] University of Chicago [email protected] Christine T. Cheng University of Wisconsin-Milwaukee [email protected] SP1 2016 Dnes Knig Prize Lecture - Phase Transitions CP1 in Random Graph Processes Unhinging Cycles: An Approach to Universal Cy- One of the most interesting features of Erd˝os-R´enyi ran- cles Under Equivalence Relations dom graphs is the ‘percolation phase transition’, where the global structure intuitively changes from only small compo- For all prime numbers p, we investigate universal cycles nents to a single giant component plus small ones. In this of string equivalence classes of length-p under equivalence talk, we discuss the percolation phase transition of Achliop- relations based on non-transitive group actions. We offer tas processes, which are a class of time-evolving variants of two types of equivalences which admit a universal cycle Erd˝os-R´enyi random graphs that (i) can exhibit somewhat of all length-p equivalence classes: first any equivalence surprising phenomena, and (ii) are difficult to analyze due with p greater than a constant dependent on the group to dependencies between the edges. action, second for any p and an equivalence satisfying cer- tain orbit space conditions. We will also introduce a new Lutz Warnke method, called unhinging, to construct the De Bruijn-like University of Cambridge graph that is used to prove the existence of universal cycles. 42 DM16 Abstracts haviour in Grids Melinda Lanius The Prisoner’s Dilemma is played on a graph by assigning Department of Mathematics to each vertex a strategy and the sum of the payoffs of a University of Illinois at Urbana-Champaign single round of the game played with each of the vertex’s [email protected] neighbours.