Research and Talks

Title: Enriched model categories and the Dold-Kan correspondence

Supervisor(s): Prof. Martin Frankland and Prof. Donald Stanley

Institution: University of Regina

Year(s): 2021 - 2024

Abstract: The work we present in this thesis is an application of the monoidal properties of the Dold–Kan correspondence and is constituted of two main parts. In the first one, we observe that by a theorem of Christensen and Hovey, the category of non- negatively graded chain complexes of left R-modules has a model structure, called the Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. Hence, the Dold–Kan correspondence induces a model structure on the category of simplicial left R-modules and some properties, notably it is monoidal. In the second part, we observe that changing the enrichment of an enriched, tensored and cotensored category along the Dold–Kan correspondence does not preserve the tensoring nor the cotensoring. Thus, we generalize this observation to any weak monoidal Quillen adjunction and we give an insight of which properties are preserved and which are weakened after changing the enrichment of an enriched model category along a right weak monoidal Quillen adjoint.

Title: Applications of the Fundamental to Cell Complexes

Supervisor(s): Prof. Paul Arnaud Songhafouo Tsopmené and Prof. Donlad Stanley

Institution: African Institute for Mathematical Sciences - AIMS Cameroon

Year(s): 2018

Abstract: In this work, we discuss about some applications of the notion of fundamental group to cell complexes. We prove that the fundamental group of a space X does not change after we have attached n-cells to it with n > 2. When n = 2, it does and we have an explicit relationship between the fundamental groups involved. As a consequence of that result, we show that the fundamental group of a cell complex X is reduced to that of its 2-skeleton X2, that is, π1(X, x0) ∼= π1(X2, x0). As application, we show that for two given closed and path-connected surfaces Sg and Sh both either orientable or nonorientable and of genus g and h respectively, a necessary condition for them to be homeomorphic is that their genera must be equal, that is, g = h. Furthermore, we prove that considering the presentation hgα|rβi of a group G, there exists a space XG such that π1(XG) ∼= G. That space is a 2-dimensional cell complex whose 1-skeleton is the wedge sum X1 = ∨αS 1α as many copies of the unit circle S1 as generators gα’s, and we complete the construction of XG by attaching 2-cells e2β to X1 along the loops associated to the relators rβ. We end the work with a description of the space XG when G = ha|an i, n ≥ 1. We prove that XG is homeomorphic to a simple space, and we then notice that it is a surface only when n = 1, 2. Key words: Fundamental group, cell complex, closed and path-connected surface.

Title: On NTRU Encryption and Signature Schemes

Supervisor(s): Prof. Emmanuel Fouotsa

Institution: Higher Teacher Training College of Bambili - University of Bamenda

Year(s): 2017

Abstract: In this report, we describe the NTRU public key cryptosystem, notably its Encryption Scheme (NES) and its Signature Scheme (NSS). Those two schemes are based on ring of convolution polynomials of rank N denoted

R = Z[x]/(xN − 1).

The security of NES and NSS is mainly based on the difficulty to find two polynomials f(x) and g(x) with small coefficients satisfying the modular equation 

f(x)*h(x) ≡ g(x) (mod q), 

for a given polynomial h(x) and a prime integer q. Coppersmith and Shamir showed that this problem is equivalent to the Shortest Vector Problem (SVP) in a special lattice determined by the integers parameters N, q and the public key h of the system and hence that recovering the secret key (f, g) from the public key h is reduced to finding a shortest vector of the lattice. This is possible when the choice of the parameters is not discriminating and this while using algorithms of lattices reduction as LLL. Hence, Coppersmith and Shamir described an attack on the NTRU cryptosystem based on lattices that we develop in our work.

Key Words: NTRU, NES, NSS, Lattice, SVP, CVP, LLL Algorithm, Cryptanalysis, Post-Quantum Cryptography.

Title: Cryptanalyse d'un cryptosystème basé sur les courbes elliptiques et RSA

Supervisor(s): Prof. Emmanuel Fouotsa and Prof. Celestin Lélé

Institution: Faculty of Sciences - University of Dschang

Year(s): 2014 - 2016

Abstract: In this report, we present the Nitaj’s attack on the KMOV public key cryptosystem. This cryptosystem is based on the elliptic curves on the ring ZN where N = pq is an RSA modulus. Here KMOV, whose security is mainly based on the difficulty of factoring the whole N, is considered respectively with the public and private keys (N, e) and (p, q, d) with

pgcd(e; (p + 1)(q + 1)) = 1 and d ≡ e−1 (mod (p + 1)(q + 1)).

Using Diophantine approximations and lattice reduction techniques, we show that KMOV

is insecure when the public key e satisfies the general equation ex − (p + 1)(q + 1)y = z

where the parameters x, y and z are as follow

pgcd(x, y) = 1, |z| < N 1 4 y and xy < √ 2 √ N 12,

because N is factorisable while proceeding as follows :

The Diophantine approximations allows us to recover y/x among the convergents of the continued fraction expansion of eN. After finding x and y, one gets an approximation p ̃ of p verifying |p − p ̃| < N 14 . Finally, this approximation leads to the factorization of N by using Coppersmith’s algorithm based on lattice reduction technics.

Proceedings: Toric Topology and Polyhedral Products

Date: February 2024

Abstract: In this paper we present unpublished work by David Stone on polyhedral smash products. He proved that the polyhedral smash product of the CW-pair (D^2, S^1) over a simplicial complex K is homeomorphic to an iterated suspension of the geometric realization of K. Here we generalize his technique to the CW-pair (D^{k+1},S^k), for an arbitrary k. We generalize the result further to a set of disks and spheres of different dimensions.

Journal: Journal of Homotopy and Related Structures

Date: October 2024

Abstract: By a theorem of Christensen and Hovey, the category of non-negatively graded chain complexes has a model structure, called the h-model structure or Hurewicz model structure, where the weak equivalences are the chain homotopy equivalences. The Dold–Kan correspondence induces a model structure on the category of simplicial modules. In this paper, we give a description of the two model categories and some of their properties, notably the fact that both are monoidal.

Organizer: Pacific Institute for Mathematical Sciences - PIMS

Propozed by: Serious:Labs

Co-authors: Maksym Neyra-Nesterenko, Moumita Shau, Bahar Mousazadeh (Team 9)

Date: 2021 

Abstract: Real-time simulation of various complex physical phenomena is a challenging task, since one must meet the demands of limited computing time for high frame rates and accurate, reproducible physical simulation. One approach to balance efficient and accurate simulation is to model the dynamics with position- based dynamics. In this report, we investigate how one can adapt position-based dynamics to incorporate hydraulic pressures to simulate heavy equipment. The key impact of this work is to lay groundwork for developing standardized virtual training and evaluation of heavy equipment operation and safety.

Organizer: Pacific Institute for Mathematical Sciences - PIMS

Propozed by: Awesense

Co-authors: Santanil Jana, Thaddeus Janisse, Youssouph Cissokho

Date: 2022 

Abstract: How many electric vehicle (EV) chargers can we fit on a given grid infrastructure? This question depends on many factors: the structure of the grid, what part of the grid is of interest, what kinds of chargers we want to fit, what measures we want to use to fit the chargers, and the load data for the grid. In this paper we come up with an application programming interface (API) that can take in live grid data and produce clear and usable answers to this question.