writing

publications and preprints

1. The Topological form is the Pfaffian form

   with Paul-Hermann Balduf. 2025.

   To appear in Documenta Mathematica.

   ( abs | arXiv )

   For a given graph $G$, Budzik, Gaiotto, Kulp, Wang, Williams, Wu, Yu, and the first author studied a topological differential form $\alpha_G$, which expresses violations of BRST-closedness of a quantum field theory along a single topological direction. In a seemingly unrelated context, Brown, Panzer, and the second author studied a Pfaffian differential form $\phi_G$, which is used to construct cohomology classes of the odd commutative graph complex. We give an explicit combinatorial proof that $\alpha_G$ coincides with $\phi_G$. We also discuss the equivalence of several properties of these forms, which had been established independently for both contexts in previous work.

2. Unstable cohomology of $\mathsf{GL}_{2n}(\mathbb{Z})$ and the odd commutative graph complex

   with Francis Brown and Erik Panzer. 2024.

   Under revision for Advances in Mathematics.

   ( abs | arXiv )

   We study a closed differential form on the symmetric space of positive definite matrices, which is defined using the Pfaffian and is $\mathsf{GL}_{2n}(\mathbb{Z})$ invariant up to a sign. It gives rise to an infinite family of unstable classes in the compactly-supported cohomology of the locally symmetric space for $\mathsf{GL}_{2n}(\mathbb{Z})$ with coefficients in the orientation bundle. Furthermore, by applying the Pfaffian forms to the dual Laplacian of graphs, and integrating them over the space of edge lengths, we construct an infinite family of cocycles for the odd commutative graph complex. By explicit computation, we show that the first such cocycle gives a non-trivial class in $H^{-6}(\mathsf{GC}_3)$.

3. Completing the $c_2$ completion conjecture for $p=2$

   with Karen Yeats.

   Commun. Number Theory Phys. 17 (2023), no. 2, pp. 343—384.

   ( abs | arXiv | doi )

   The $c_2$-invariant is an arithmetic graph invariant useful for understanding Feynman periods. Brown and Schnetz conjectured that the $c_2$-invariant has a particular symmetry known as completion invariance. This paper will prove completion invariance of the $c_2$-invariant in the $p=2$ case, extending previous work of one of us. The methods are combinatorial and enumerative involving counting certain partitions of the edges of the graph.

4. Reconstructing a bijection on the level of Le diagrams

   with Karen Yeats. 2023.

   Submitted.

   ( abs | arXiv )

   Lukowiski, Parisi, and Williams formulated the T-duality map of string theory at a purely combinatorial level as a map on decorated permutations. We combinatorially describe this map at the level of Le diagrams. This perspective makes the dimension shift under the map more transparent.

5. Further investigations into the graph theory of $\phi^4$–periods and the $c_2$ invariant

   with Oliver Schnetz, Jim Shaw, and Karen Yeats.

   Ann. Inst. Henri Poincaré D 9 (2022), no. 3, pp. 473—524.

   ( abs | arXiv | doi )

   A Feynman period is a particular residue of a scalar Feynman integral which is both physically and number theoretically interesting. Two ways in which the graph theory of the underlying Feynman graph can illuminate the Feynman period are via graph operations which are period invariant and other graph quantities which predict aspects of the Feynman period, one notable example is known as the $c_2$ invariant. We give results and computations in both these directions, proving a new period identity and computing its consequences up to 11 loops in $\phi^4$–theory, proving a $c_2$ invariant identity, and giving the results of a computational investigation of $c_2$ invariants at 11 loops.

6. A low-noise single-photon detector for long-distance free-space quantum communication

   with Elena Anisimova, Dmitri Nikulov, Mark Bourgon, Sebastian Philipp Neumann, Rupert Ursin, Thomas Jennewein, and Vadim Makarov.

   EPJ Quantum Technology 8 (2021), no. 23.

   ( abs | arXiv | doi )

   We build and test a single-photon detector based on a Si avalanche photodiode Excelitas 30902SH thermoelectrically cooled to -100 C$^{\circ}$. Our detector has dark count rate below 1 Hz, 500 um diameter photosensitive area, photon detection efficiency around 50%, afterpulsing less than 0.35%, and timing jitter under 1 ns. These characteristics make it suitable for long-distance free-space quantum communication links, which we briefly discuss. We also report an improved method that we call long-time afterpulsing analysis, used to determine and visualise long trap lifetimes at different temperatures.

theses

1. A combinatorial tale of two scattering amplitudes: see two bijections

   MMath thesis, University of Waterloo, 2021.

   ( abs | arXiv | doi | Governor General's Gold Medal )

   Received the Governor General’s Gold Medal in 2022 for highest standing in a master’s program and for research accomplishments in my dissertation, as featured in news articles here and here!

   In this thesis, we take a journey through two different but not dissimilar stories with an underlying theme of combinatorics emerging from scattering amplitudes in quantum field theories.
   
   The first part tells the tale of the $c_2$-invariant, an arithmetic invariant related to the Feynman integral in $\phi^4$–theory, which studies the zeros of the Kirchoff polynomial and related graph polynomials. Through reformulating the $c_2$-invariant as a purely combinatorial problem, we show how enumerating certain edge bipartitions through fixed-point free involutions can complete a special case of the long sought after $c_2$ completion conjecture.
   
   The second part tells the tale of the positive Grassmannian and a combinatorial T-duality map on its cells, as related to scattering amplitudes in planar $\mathcal{N}=4$ SYM theory. In particular, T-duality is a bridge between triangulations of the hypersimplex and triangulations of the amplituhedron, two objects that appear as images of the positive Grassmannian. We give an algorithm for viewing T-duality as a map on Le diagrams and characterize a nice structure to the Le diagrams (which can then be used in lieu of the algorithm). Through this Le diagram perspective on T-duality, we show how the dimensional relationship between the positroid cells on either side of the map can be directly explained.

posters

1. The Topological form is the Pfaffian form

   with Paul-Hermann Balduf.

   Presented at fpuk@King’s 2025 (Fundamental Physics UK), London, UK.

   ( poster )
2. Low-noise single-photon detector for long-distance free-space quantum communication

   with Elena Anisimova, Dmitri Nikulov, Mark Bourgon, Rupert Ursin, Thomas Jennewein, and Vadim Makarov.

   Presented at QCrypt 2015, Tokyo, Japan and Single Photon Workshop 2015, Geneva, Switzerland.

   ( qcrypt | qcrypt-abs | spw-abs )