Banff |

Godsil65 |

Banff |

IPAM: Lake Arrowhead Retreat. |

Paley's Grave in Banff |

drawing by Kathabela Wilson |

WilsonFest |

WilsonFest |

Banff |

AWM 50 |

- Beautiful Combinatorics, Linear Algebra Methods in Combinatorics, and Algebraic Graph Theory.
- Extremal set theory questions and their analogues in other structures.

- Postdoctoral Mentors: Jeff Kahn, Benny Sudakov, and Po-Shen Loh
- Ph.D. Advisor: Jacques Verstraete
- MMath Advisor: Chris Godsil
- Undergraduate Mentors: Rick Wilson and Ada Chan

- A. Chowdhury, Inclusion Matrices and the MDS Conjecture, Electr. J. Combin. 23(4) (2016), #P4.29
Let

*F_q*be a finite field of order*q*with characteristic*p*. An arc is an ordered family of vectors in*(F_q)^k*in which every subfamily of size*k*is a basis of*(F_q)^k*. The MDS conjecture, which was posed by Segre in 1955, states that if*k*<=*q*, then an arc in*(F_q)^k*has size at most*q+1*, unless*q*is even and*k*=3 or*k*=*q-1*, in which case it has size at most*q+2*.We propose a conjecture which would imply that the MDS conjecture is true for almost all values of

*k*when*q*is odd. We prove our conjecture in two cases and thus give simpler proofs of the MDS conjecture when*k*<=*p*, and if*q*is not prime, for*k*<=*2p-2*. To accomplish this, given an arc*G*of*(F_q)^k*and a nonnegative integer*n*, we construct a matrix M_G^{\uparrow n}, which is related to an inclusion matrix, a well-studied object in combinatorics. Our main results relate algebraic properties of the matrix M_G^{\uparrow n} to properties of the arc*G*and may provide new tools in the computational classification of large arcs. - A. Chowdhury, G. Sarkis, and S. Shahriari, The Manickam-Miklos-Singhi Conjectures for Sets
and Vector Spaces, J. Combin Theory Ser. A, 128:84-103, 2014.
- Beamer Slides
- A New Quadratic Bound for the Manickam-Miklos-Singhi Conjecture (full calculations for the case of sets)

More than twenty-five years ago, Manickam, Miklos, and Singhi conjectured that for positive integers

*n,k*with*n*at least*4k*, every set of*n*real numbers with nonnegative sum has at least k-element subsets whose sum is also nonnegative. We verify this conjecture when*n*is at least*8k^{2}*, which simultaneously improves and simplifies a bound of Alon, Huang, and Sudakov and also a bound of Pokrovskiy when*k*is at most 10^{45}.Moreover, our arguments resolve the vector space analogue of this conjecture. Let

*V*be an*n*-dimensional vector space over a finite field. Assign a real-valued weight to each 1-dimensional subspace in*V*so that the sum of all weights is zero. Define the weight of a subspace*S*of*V*to be the sum of the weights of all the 1-dimensional subspaces it contains. We prove that if*n*is at least*3k*, then the number of*k*-dimensional subspaces in*V*with nonnegative weight is at least the number of*k*-dimensional subspaces in*V*that contain a fixed 1-dimensional subspace. This result verifies a conjecture of Manickam and Singhi from 1988. - A. Chowdhury, A Note on the
Manickam-Miklos-Singhi Conjecture, European J. Combin., 35:131-140,
2014.
For a positive integer

*k*, let*f(k)*be the minimum integer*N*such that whenever*n*is at least*N*, every set of*n*real numbers with nonnegative sum has at least*k*-element subsets whose sum is also nonnegative. In 1988, Manickam, Miklos, and Singhi proved that*f(k)*exists and conjectured that*f(k)*is at most*4k*. In this note, we apply the Kruskal-Katona theorem to prove that*f(3)*equals 11, that*f(4)*is at most 24, and that*f(5)*is at most 40, which improves previous upper bounds in these cases. Moreover, we show how our method could potentially yield a quadratic upper bound on*f(k)*. We end by discussing how our methods apply to a vector space analogue of the Manickam-Miklos-Singhi conjecture. - A. Chowdhury, On a Conjecture of
Frankl and Furedi, Electron. J. Combin. 18(1) (2011), P56.
- Extended Abstract in ENDM. 38 (2011), 259-263.
- Overhead Slides

Inspired by Fisher's inequality, Frankl and Furedi conjectured in 1991 that if is a nontrivial λ-intersecting family of subsets of

*X*and has size*m*, then the number of pairs*{x,y}*in contained in some set F in is at least . We verify this conjecture in some special cases, focusing especially on the case where is additionally required to be*k*-uniform and λ is small. - A. Chowdhury and B. Patkos, Shadows
and Intersections in Vector Spaces, J. Combin. Theory Ser. A
117(8): 1095-1106, 2010.
We prove a vector space analogue of a version of the Kruskal-Katona theorem due to Lovasz. We apply this result to extend Frankl's theorem on

*r*-wise intersecting sets to vector spaces. In particular, we obtain a short new proof of the Erdos-Ko-Rado theorem for vector spaces. - A. Blokhuis, A.E. Brouwer, A. Chowdhury, P. Frankl, T. Mussche, B.
Patkos, and T. Szonyi,
A Hilton-Milner Theorem for Vector Spaces,
Electron. J. Combin. 17(1) (2010), R71.
We prove a vector space analogue of the Hilton-Milner theorem. As an application, we determine the chromatic number of the

*q*-Kneser graph, which is the vector space analogue of the Kneser graph. - A. Chowdhury, C. Godsil, and G. Royle,
Colouring Lines in Projective Space, J. Combin. Theory Ser. A 113(1):
39-52, 2006.
The

*q*-Kneser graph is the vector space analogue of the Kneser graph. Determining the chromatic number of the Kneser graph was a longstanding open problem, whose solution by Lovasz and Barany involved a novel use of algebraic topology. Motivated by this, we determine the chromatic number of the*q*-Kneser graph in some special cases.

- Shadows and Intersections, Ph.D. Thesis, UC San Diego, 2012.
- Colouring Subspaces, Masters Thesis, University of Waterloo, 2005.