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In mathematics, my research interests lie within algebra, particularly the areas of analytic number theory and representation theory. I am interested in problems relating to modular forms, elliptic curves, and the intricate connections that lie between the two. Specifically, I have enjoyed learning about the sphere packing problem, posed by the astronomer Johannes Kepler in 1611, requiring one to arrange d-dimensional spheres as densely as possible. Progress in these problems combines the theory of lattices, linear programming, and modular forms. In 3 dimensions, a solution is given by the square-based-pyramid structure seen in a pile of oranges at a grocery store.
In mathematics, my research interests lie within algebra, particularly the areas of analytic number theory and representation theory. I am interested in problems relating to modular forms, elliptic curves, and the intricate connections that lie between the two. Specifically, I have enjoyed learning about the sphere packing problem, posed by the astronomer Johannes Kepler in 1611, requiring one to arrange d-dimensional spheres as densely as possible. Progress in these problems combines the theory of lattices, linear programming, and modular forms. In 3 dimensions, a solution is given by the square-based-pyramid structure seen in a pile of oranges at a grocery store.
This Fall, I completed my Honours Research Thesis under the supervision of Prof. Henri Darmon in the area of modular forms, Hecke operators, and Weierstrass Uniformization. I read the text "Course in Arithmetic" by J.P. Serre and excerpts of "Advanced Topics in the Arithmetic of Elliptic Curves" by Joseph Silverman.
This Fall, I completed my Honours Research Thesis under the supervision of Prof. Henri Darmon in the area of modular forms, Hecke operators, and Weierstrass Uniformization. I read the text "Course in Arithmetic" by J.P. Serre and excerpts of "Advanced Topics in the Arithmetic of Elliptic Curves" by Joseph Silverman.
Here is my research thesis:
Here is my research thesis:
I presented an introduction to these interests at Les séminaires universitaires en mathématiques à Montréal 2025 (SUMM) hosted by the Université de Montréal (UdeM). Here are my slides:
I presented an introduction to these interests at Les séminaires universitaires en mathématiques à Montréal 2025 (SUMM) hosted by the Université de Montréal (UdeM). Here are my slides:
In computer science, I am passionate about cryptography, especially lattice-based algorithms and cryptosystems. I am interested in how these structures can be applied to the creation of efficient post-quantum Fully Homomorphic Encryption (FHE) schemes. In essence, if (Gen, Enc, Dec) is a fully homomorphic encryption scheme and m1, m2 are plaintexts, then
In computer science, I am passionate about cryptography, especially lattice-based algorithms and cryptosystems. I am interested in how these structures can be applied to the creation of efficient post-quantum Fully Homomorphic Encryption (FHE) schemes. In essence, if (Gen, Enc, Dec) is a fully homomorphic encryption scheme and m1, m2 are plaintexts, then
Enc(m1+m2)=Enc(m1)+Enc(m2)
Enc(m1+m2)=Enc(m1)+Enc(m2)
Enc(m1*m2)=Enc(m1)*Enc(m2)
Enc(m1*m2)=Enc(m1)*Enc(m2)
These systems allow untrusted users of data and third parties to perform mathematical computations on encrypted data without uncovering its contents. Current systems are not efficient enough for widespread implementation and require an expensive bootstrapping step following a set number of operations to maintain the security of data. However, ongoing research seeks to produce both hardware and algorithmic improvements.
These systems allow untrusted users of data and third parties to perform mathematical computations on encrypted data without uncovering its contents. Current systems are not efficient enough for widespread implementation and require an expensive bootstrapping step following a set number of operations to maintain the security of data. However, ongoing research seeks to produce both hardware and algorithmic improvements.
Additionally, I am interested in spectral graph theory, which studies eigenvalues of adjacency and Laplacian matrices of graphs. Specifically, in the study of finite Markov chains, the transition matrix describes the probability of moving from one state to another. The largest absolute eigenvalue is 1 and corresponds to a stationary vector, i.e. a fixed point of the chain. Computations of the stationary vector led to some of the earliest versions of Google's Page Rank algorithm.
Additionally, I am interested in spectral graph theory, which studies eigenvalues of adjacency and Laplacian matrices of graphs. Specifically, in the study of finite Markov chains, the transition matrix describes the probability of moving from one state to another. The largest absolute eigenvalue is 1 and corresponds to a stationary vector, i.e. a fixed point of the chain. Computations of the stationary vector led to some of the earliest versions of Google's Page Rank algorithm.
Alongside Prof. Luc Devroye, I shortened a proof of Sinclair, equivalent to the hard direction of Cheeger's Inequality, which says that the second largest eigenvalue is no more than exp(-phi^2/8), where phi is the conductance of the Markov chain, a measure of the flow through the smallest bottleneck through the graph.
Alongside Prof. Luc Devroye, I shortened a proof of Sinclair, equivalent to the hard direction of Cheeger's Inequality, which says that the second largest eigenvalue is no more than exp(-phi^2/8), where phi is the conductance of the Markov chain, a measure of the flow through the smallest bottleneck through the graph.
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