Upcoming talks
 Tight bounds on the coefficients of partition functions via stability EUROCOMB 2017  Vienna, Austria  20170828
Research papers (see also arXiv)

Tight bounds on the coefficients of partition functions via stability
[abstract,
arXiv:1704.07784]
Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hardcore model and monomerdimer model are the independence and matching polynomials respectively.
We show how stability results follow naturally from the recently developed occupancy method for maximizing and minimizing physical observables over classes of regular graphs, and then show these stability results can be used to obtain tight extremal bounds on the individual coefficients of the corresponding partition functions.
As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of $q$colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least $5$. 
Extremes of the internal energy of the Potts model on cubic graphs
Random Structures & Algorithms (to appear)
[abstract,
arXiv:1610.08496]
We prove tight upper and lower bounds on the internal energy per particle (expected number of monochromatic edges per vertex) in the antiferromagnetic Potts model on cubic graphs at every temperature and for all $q \ge 2$.
This immediately implies corresponding tight bounds on the antiferromagnetic Potts partition function. Taking the zerotemperature 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. 
On the average size of independent sets in trianglefree graphs
Proceedings of the American Mathematical Society (to appear)
[abstract,
arXiv:1606.01043,
doi,
slides]
We prove an asymptotically tight lower bound on the expected size of an independent set in a trianglefree graph on $n$ vertices with maximum degree $d$ drawn uniformly at random, showing the average is at least $(1+o_d(1)) \frac{\log d}{d}n$. This gives an alternative proof of Shearer's upper bound on the Ramsey number $R(3,k)$. We then prove a lower bound on the total number of independent sets in such a graph which is asymptotically tight in the exponent. In both cases, tightness is exhibited by a random $d$regular graph.
Our results come from considering the hardcore model from statistical physics: a random independent set $I$ drawn from a graph with probability proportional to $\lambda^{I}$, for a fugacity parameter $\lambda>0$. We prove a general lower bound on the occupancy fraction of the hardcore model on trianglefree graphs of maximum degree $d$. The bound is asymptotically tight in $d$ for all $\lambda =O_d(1)$.
We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a trianglefree graph and give some consequences of these conjectures in Ramsey theory. 
Multicolour Ramsey numbers of paths and even cycles
European Journal of Combinatorics 63 (2017), 124–133
[abstract,
arXiv:1606.00762,
doi]
We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that the kcolour Ramsey number of both an $n$vertex path and, for even $n$, an $n$vertex cycle is between $(k1)n+o(n)$ and $kn+o(n)$.
The upper bound was recently improved by Sárközy who gave a stability version of the classical theorem of Erdős and Gallai on graphs containing no long cycles. Here we obtain the first improvement to the coefficient of the linear term in the upper bound by an absolute constant. 
Independent Sets, Matchings, and Occupancy Fractions
Jourrnal of the London Mathematical Society (to appear)
[abstract,
arXiv:1508.04675,
doi]
We give tight upper bounds on the logarithmic derivative of the independence and matching polynomials of any $d$regular graph.
For independent sets, this is a strengthening of a sequence of results of Kahn, Galvin and Tetali, and Zhao that a disjoint union of $K_{d,d}$'s maximizes the independence polynomial and total number of independent sets among all dregular graphs.
For matchings, the result implies that disjoint unions of $K_{d,d}$'s maximize the matching polynomial and total number of matchings, as well as proving the Asymptotic Upper Matching Conjecture of Friedland, Krop, Lundow, and Markström.
The proof uses an occupancy method: we show that the occupancy fraction in the hardcore model and the edge occupancy fraction in the monomerdimer model are maximized over all $d$regular graphs by $K_{d,d}$.
Conference papers

Tight bounds on the coefficients of partition functions via stability (extended abstract)
Electronic Notes in Discrete Mathematics (to appear)
[abstract,
preprint]
We show how to use the recentlydeveloped occupancy method and stability results that follow easily from the method to obtain extremal bounds on the individual coefficients of the partition functions over various classes of bounded degree graphs.

Counting in hypergraphs via regularity inheritance (extended abstract)
Electronic Notes in Discrete Mathematics 49 (2015), 413–417
[abstract,
preprint,
doi,
slides]
A new approach to the counting lemma in 3uniform hypergraphs. We develop a theory of regularity inheritance and use it to prove a slight strengthening of the counting lemma of Frankl and Rödl. Full paper to follow.
Past talks
 Tight bounds on the coefficients of partition functions via stability Two oneday Colloquia in Combinatorics (invited)  London, UK  20170511
 Extremal problems on colourings in cubic graphs via the Potts model Combinatorics Seminar  Oxford, UK  20170221
 A probabilistic approach to bounding graph polynomials Discrete Mathematics Seminar, KdVI / CWI  Amsterdam, Netherlands  20170203
 A probabilistic approach to bounding graph polynomials PhD Seminar on Combinatorics, Games and Optimisation, LSE  London, UK  20170127
 On the average size of independent sets in trianglefree graphs (slides) Student Combinatorics Day (invited)  Birmingham, UK  20160803
 Independent sets, matchings, and occupancy fractions II Seminário de Teoria da Computação, Combinatória e Otimização  São Paulo, Brazil  20160408
 Independent sets, matchings, and occupancy fractions Discrete Geometry and Combinatorics Seminar, UCL  London, UK  20151103
 Independent sets, matchings, and occupancy fractions I Mathematics Lunchtime Seminar, LSE  London, UK  20151009
 Counting in hypergraphs via regularity inheritance (slides) EUROCOMB 2015  Bergen, Norway  20150831
 Regularity inheritance in 3uniform hypergraphs Novi Sad Workshop on Foundations Of Computer Science  Novi Sad, Serbia  20150720
 Counting in hypergraphs via regularity inheritance Postgraduate Combinatorial Conference, QMUL  London, UK  20150415
 Regularity inheritance in 3uniform hypergraphs Combinatorics Seminar, Freie Universität  Berlin, Germany  20150409
 Robustness of triangle factors SUMMIT 190: Balogh, Csaba, Hajnal, and Pluhar are 190 (invited)  Szeged, Hungary  20140703
 Cycle packing Mathematics Lunchtime Seminar, LSE  London, UK  20131129