Counting the number of ways a gas can fill a room

Ewan Davies is a PhD student in the Department of Mathematics. His research is on graph theory, the study of connected systems of abstract ‘things’ which we call graphs. In this example the ‘things’ in the graph are particles of a gas or atoms in a molecule, and he developed a new method for understanding mathematical models in these graphs. More information about Ewan’s research is available on his website

Counting in hypergraphs via regularity inheritance

We develop a theory of regularity inheritance in 3-uniform hypergraphs. As a simple consequence we deduce a strengthening of a counting lemma of Frankl and Rödl. We believe that the approach is sufficiently flexible and general to permit extensions of our results in the direction of a hypergraph blow-up lemma

Extremes of the internal energy of the Potts model on cubic graphs

We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the anti-ferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$. This immediately implies corresponding tight bounds on the anti-ferromagnetic Potts partition function. Taking the zero-temperature limit gives new results in extremal combinatorics: the number of $q$-colorings of a $3$-regular graph, for any $q \ge 2$, is maximized by a union of $K_{3,3}$'s. This proves the $d=3$ case of a conjecture of Galvin and Tetali

Business schools inside the academy: What are the prospects for interdepartmental research collaboration?

Established literature about the role of business schools tends towards more parochial concerns, such as their need for a more pluralist and socially reflexive mode of knowledge production (Starkey and Tiratsoo 2007; Starkey et al 2009) or the failure of management’s professionalism project expressed through the business school movement (Khurana 2007). When casting their gaze otherwise, academic commentators examine business schools’ weakening links with management practice (Bennis and O’Toole 2005). Our theme makes a novel contribution to the business school literature through exploring prospects for research collaborations with other university departments. We draw upon the case of UK business schools, which are typically university-based (unlike some of their European counterparts), and provide illustrations relating to collaboration with medical schools to make our analytical points. We might expect that business schools and medical schools effectively collaborate given their similar vocational underpinnings, but at the same time, there are significant differences, such as differing paradigms of research and the extent to which the practice fields are professionalised. This means collaboration may prove challenging. In short, the case of collaboration between business schools and medical schools is likely to illuminate the challenges for business schools ‘reaching out’ to other university departments

Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs

We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density ?_c(?) and provide (i) for ? ?_c(?). The critical density is the occupancy fraction of hard core model on the clique K_{?+1} at the uniqueness threshold on the infinite ?-regular tree, giving ?_c(?) ~ e/(1+e)1/(?) as ? ? ?

A proof of the Upper Matching Conjecture for large graphs

We prove that the `Upper Matching Conjecture' of Friedland, Krop, and Markstr\"om and the analogous conjecture of Kahn for independent sets in regular graphs hold for all large enough graphs as a function of the degree. That is, for every $d$ and every large enough $n$ divisible by $2d$, a union of $n/(2d)$ copies of the complete $d$-regular bipartite graph maximizes the number of independent sets and matchings of size $k$ for each $k$ over all $d$-regular graphs on $n$ vertices. To prove this we utilize the cluster expansion for the canonical ensemble of a statistical physics spin model, and we give some further applications of this method to maximizing and minimizing the number of independent sets and matchings of a given size in regular graphs of a given minimum girth

A Call for University-Based Business Schools to “Lower Their Walls:” Collaborating With Other Academic Departments in Pursuit of Social Value

The walls around many business schools remain high, eroding interdisciplinary education and research collaboration that might address some grand challenges facing society. In response, we adopt a public interest perspective and argue business schools should lower their walls to engage with other academic departments to address such grand challenges in a way that engenders social value. We identify forces for lower and higher walls that surround business schools and influence prospects for interdisciplinary collaboration. We highlight examples of successful relationships between business schools and other academic departments, which offer some optimism for a reimagined public interest mission for business schools. Finally, we draw out some boundary conditions to take a more contingent view of possibilities for such interdisciplinary collaboration encompassing business schools