Nilpotency degrees of finite-dimensional quadratic algebras carry essential information for their combinatorial and homological applications. It is known that the maximal nilpotency degree a finite-dimensional quadratic algebra with n generators can contain is at least n+1 for all n > 2. However, the optimality of this bound is still unknown. I propose a geometric visualization of the algebraic varieties of all quadratic algebras with n generators in degree d to find the true optimal bound. I then utilize this visualization to construct a linear program that deterministically determines whether a finite quadratic algebra with n generators exists that has a nilpotency degree of at least d. Thus far, I have verified that this algorithm will give the desired optimal bounds and have completed an implementation using Sage. I expect to find the true optimal bound on the maximal nilpotency degree of a finite-dimensional quadratic algebra with three generators shortly. However, the algorithm will require revisions for higher values of n due to scalability issues caused by its computational complexity. Knowing this optimal bound would solve several open problems in ring theory, including bounding the computational complexity of computing the global dimension of Koszul algebras. Finding a bound that extends to all algebras, including non-quadratic algebras, would also bound the computational complexity of determining if a finitely presented graded algebra is finite-dimensional

Fractal dimension, a measure of geometric complexity, finds application in image analysis, biology and medicine, neuroscience, geology and various other fields, yet existing methods often lack adaptability to finite data sets. Using ideas rooted in geometric measure theory, such as Hausdorff measure and Frostman’s Lemma, this research introduces a novel approach to compute fractal dimensions for finite sets, addressing limitations of traditional methods. Using Python, we developed and tested an algorithm to validate known sets such as the unit interval, square, cube, and fractal objects including the Cantor set and Sierpinski triangle. Comparative analysis was also conducted on established methods, including box-counting and correlation integral algorithms, to demonstrate the algorithm's accuracy in determining fractal dimensions. Pivoting towards data sets, we expect to use the computed fractal dimension of real data as a tool for assessing data and optimizing data compression. Our methods offer an improvement as most existing techniques use statistical methods that are limited to integer dimensions. In addition, recent studies have shown that fractal dimension values can be useful as features in machine learning. We also improve upon the calculation of the local dimension of regions in a data set, allowing for additional insights into complex data sets. This includes identifying regions of high complexity, and we expect to show that this allows for the more effective use of algorithms such as principal component analysis. All of these are increasingly important in our society due to the abundance of high-dimensional datasets in both the physical and social sciences. Overall, the benefits of studying novel ways of calculating the dimension of large data sets include efficient representation of data, improved interpretability, and decreased computational burden, as well as detecting certain features in data such as regions of high complexity.

Collision avoidance studies find important applications for motion planning of mobile robots for deployment in outer space, nuclear waste management, mobiles used for process automation, etc. Here, we integrate mobile robot simulations with mathematical modeling using Python to understand collision avoidance for mobile robotics. We used the open-source Pioneer code on the Webots platform for simulations of mobile robots which employ Kinect-based optical and IR sensors and cameras for live-tracking of objects in the environment variable and have motion controller Matlab software that provides the kinematic variables like position, velocity, and acceleration of various objects in real-time. We wrote a Python code to digitize the image matrices obtained from simulations and identified the pixels having objects that the mobile robot must avoid for collision avoidance. We calculated the instantaneous distances between the mobile robot and various objects to interpret and analyze the simulated trajectories. We used jump collision avoidance models to estimate the mobile robot trajectories in the vicinity of objects. The calculated object avoidance jump trajectory of the robot was smoothened using Gaussian data convolution methods to obtain smooth trajectories. The simulations provide attractive visualization and are useful for machine learning and testing algorithms for collision avoidance and motion planning.

The Riemann Hypothesis is an important unsolved problem in mathematics under number theory, concerning the distribution of prime numbers. It can be characterized as finding precise bounds on the partial sums of the classical M\"obius function. The M\"obius function is a technical tool that generalizes the inclusion-exclusion principal in discrete math and combinatorics. Due to the difficulty of the Riemann Hypothesis, it is common to analyze a modification of the underlying structure. This project analyzes the M\"obius function defined on the partially ordered set of triangular numbers (of the form 1, 3, 6, 10...) under the divisibility relation. My mentor and I made conjectures on the asymptotic behavior of the M\"obius and Mertens functions on the basis of experimental data. We first introduce the growth of partial sums of the triangular M\"obius function and analyze how the growth is different from the classical M\"obius function, and then analyze the partial sums of its absolute values, and how it is similar to the asymptotic in the classical case. I then created Hasse diagrams of this structure, this is a method to visualize a partially ordered set (poset). This also serves as a basis for the zeta and M\"obius matrices. Looking specifically into the poset defined by $(\mathbb{N}, \leq_{\mathcal T})$, or triangular numbers under divisibility, and applying the M\"obius function to it, we were able to create our desired matrices. Using Python libraries I created visualizations for further analysis, and was able to project previously mentioned patterns. Through which we are able to introduce two more novel conjectures bounding $\mu_{\mathcal T}(n)$ and the sums of $\frac{\mu_{\mathcal T}(i)}{i}$. We conclude the project with divisibility patterns in the Appendix.