Instantons appear in many problems of theoretical physics. Quiver gauge theories are a rising subject due to its natural emergence in perturbative open string theories. The talk will discuss features of these two topics and how one can use quiver gauge theories to solve the long standing problem of finding the moduli space of k G instantons on C^2, where G is any simple gauge group, including the non-simply laced cases.

The majority of epidemic models fall into two categories: 1) deterministic models represented by differential equations and 2) stochastic models which can be evaluated by simulation. In this presentation I will discuss the precise connection between these models. Until recently, exact correspondence was only established in situations exhibiting large degrees of symmetry or for infinite populations. I will consider SIR dynamics on finite static contact networks. I will give an overview of two provably exact deterministic representations of the underlying stochastic model for tree-like networks. These are the message passing description of Karrer and Newman and my pair-based moment closure representation. I will discuss the relationship between the two representations and the relative merits of both.

I will discuss the three-point correlator of two protected scalar operators and one higher spin twist-two operator in N = 4 SYM, in the limit of large spin. This structure constant can be extracted from the OPE of the four-point correlator of protected scalar operators. Based on the OPE structure, symmetry arguments and intuition from the available perturbative results, it is possible to predict the structure constant at all loops in perturbation theory. Furthermore, this allows also to propose an expression for the all loops four-point correlator in a particular limit.

In 2010, Eguchi--Ooguri--Tachikawa observed an unexpected relation between K3 elliptic genus and the sporadic group M24. In this talk I'll briefly review the recent developments on the topic of moonshine. In particular I will describe a more general relation between mock modular forms and finite groups, using Niemeier lattices as the starting point and including the M24 observation as a special case. I will also discuss various approaches in attempting to understand these mysterious relations, focusing on the study of compactification of heterotic strings on K3 surfaces. This talk will be based on joint work with Duncan--Harvey and with Dong--Harrison--Kachru--Whalen--Wrase.

Characterizing how we explore abstract spaces is key to understand our (ir)rational behavior and decision making. While some light has been shed on the navigation of semantic networks, however, little is known about the mental exploration of metric spaces, such as the one dimensional line of numbers, prices, etc. Here we address this issue by investigating the behavior of users exploring the “bid space” in online auctions. We find that they systematically perform Lévy flights, i.e., random walks whose step lengths follow a power-law distribution. Interestingly, this is the best strategy that can be adopted by a random searcher looking for a target in an unknown environment, and has been observed in the foraging patterns of many species. In the case of online auctions, we measure the power-law scaling over several decades, providing the neatest observation of Lévy flights reported so far. We also show that the histogram describing single individual exponents is well peaked, pointing out the existence of an almost universal behaviour. Furthermore, a simple model reveals that the observed exponents are nearly optimal, and represent a Nash equilibrium. We rationalize these findings through a simple evolutionary process, showing that the observed behavior is robust against invasion of alternative strategies. Our results show that humans share with the other animals universal patterns in general searching processes, and raise fundamental issues in cognitive, behavioural and evolutionary sciences.

In this talk we review recent developments in the study of $U(N)$ Chern Simons theories coupled to matter in the fundamental. The theories we study are not necessarily supersymmetric. These theories are interesting for several reasons. First, it turns out to be possible to exactly compute several quantities in these models in the t'Hooft large N limit at all values of the t'Hooft coupling $\lambda$. Second, these exact results strongly suggest that previously known Giveon-Kutasov type strong weak coupling dualities for supersymmetric theories extend to nonsupersymmetric theories. Third, several of these theories have conjectured bulk dual descriptions governed by Vasiliev's equations. Finally, theories with two gauge groups and matter in the bifundamental representation effectively reduce to fundamental models in the limit that one of the two gauge groups is much larger than the other, and so also have known Vasiliev dual descriptions. In the context of the N=6 ABJ theory, this observation allows us to find a Vasiliev dual of a field theory with a known string dual, and so realize Vasiliev theory as a limit of string theory.

In this talk we review recent developments in the study of $U(N)$ Chern Simons theories coupled to matter in the fundamental. The theories we study are not necessarily supersymmetric. These theories are interesting for several reasons. First, it turns out to be possible to exactly compute several quantities in these models in the t'Hooft large N limit at all values of the t'Hooft coupling $\lambda$. Second, these exact results strongly suggest that previously known Giveon-Kutasov type strong weak coupling dualities for supersymmetric theories extend to nonsupersymmetric theories. Third, several of these theories have conjectured bulk dual descriptions governed by Vasiliev's equations. Finally, theories with two gauge groups and matter in the bifundamental representation effectively reduce to fundamental models in the limit that one of the two gauge groups is much larger than the other, and so also have known Vasiliev dual descriptions. In the context of the N=6 ABJ theory, this observation allows us to find a Vasiliev dual of a field theory with a known string dual, and so realize Vasiliev theory as a limit of string theory.

Given a (nonempty) set $A$ of integers, two of the most obvious things to do with it are to form the sumset $A+A=\{a+b:\,a,b\in A\}$ and the difference set $A-A=\{a-b:\,a,b\in A\}$. One might also wish to consider the restricted sumset $A\hat{+}A=\{a+b:\,a,b\in A,\,a\neq b\}$. One can then ask various obvious questions about the relationships between the sizes of various of these sets and what this implies about structure, and I shall discuss some known results on this, including generalisations to more general contexts, e.g. in group theory. An intuition one might have is that the sumset/restricted sumset will be smaller than the difference set as addition is commutative but subtraction isn't: I shall survey various known results showing that this intuition is non-trivially wrong. At the end I shall discuss some recent constructions of sets $A$ which give new record large values of $\log(|A+A|)/\log(|A-A|)$. The original part of the talk is based on joint work with my research student Matthew Wells.