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It has long been observed that for practically any computational problem that has been intensely studied, different instances are best solved using different algorithms. This is particularly pronounced for computationally hard problems,…
Since Artificial Intelligence (AI) software uses techniques like deep lookahead search and stochastic optimization of huge neural networks to fit mammoth datasets, it often results in complex behavior that is difficult for people to…
In this paper we study the problem of maximizing the distance to a given point over an intersection of balls. It was already known that this problem can be solved in polynomial time and space if the given point is not in the convex hull of…
Nobel laureate Philip Anderson and Elihu Abrahams once stated that, "even if machines did contribute to normal science, we see no mechanism by which they could create a Kuhnian revolution and thereby establish a new physical law." In this…
Recent years have seen an explosion of work and interest in text-to-3D shape generation. Much of the progress is driven by advances in 3D representations, large-scale pretraining and representation learning for text and image data enabling…
For many computational problems involving randomness, intricate geometric features of the solution space have been used to rigorously rule out powerful classes of algorithms. This is often accomplished through the lens of the multi Overlap…
The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence…
Despite the promises of data-driven artificial intelligence (AI), little is known about how we can bridge the gulf between traditional physician-driven diagnosis and a plausible future of medicine automated by AI. Specifically, how can we…
Since Alan Turing envisioned Artificial Intelligence (AI) [1], a major driving force behind technical progress has been competition with human cognition. Historical milestones have been frequently associated with computers matching or…
This paper develops asymptotic methods to count faces of random high-dimensional polytopes. Beyond its intrinsic interest, our conclusions have surprising implications - in statistics, probability, information theory, and signal processing…
We present two algorithms that compute the Newton polytope of a polynomial defining a hypersurface H in C^n using numerical computation. The first algorithm assumes that we may only compute values of f - this may occur if f is given as a…
In this paper a new algorithm has been proposed which can fix the problem of Weiler Atherton algorithm. The problem of Weiler Atherton algorithm lies in clipping self intersecting polygon. Clipping self intersecting polygon is not…
Geometry theorem proving forms a major and challenging component in the K-12 mathematics curriculum. A particular difficult task is to add auxiliary constructions (i.e, additional lines or points) to aid proof discovery. Although there…
Artificial Expert Intelligence (AEI) seeks to transcend the limitations of both Artificial General Intelligence (AGI) and narrow AI by integrating domain-specific expertise with critical, precise reasoning capabilities akin to those of top…
Artificial Intelligence (AI) is one of today's most relevant emergent technologies. In view thereof, this paper proposes that more attention should be paid to the philosophical aspects of AI technology and its use. It is argued that this…
This overview article highlights the critical role of mathematics in artificial intelligence (AI), emphasizing that mathematics provides tools to better understand and enhance AI systems. Conversely, AI raises new problems and drives the…
Today's AI models learn primarily through mimicry and refining, so it is not surprising that they struggle to solve problems beyond the limits set by existing data. To solve novel problems, agents should acquire skills for exploring and…
Output from generative AI such as ChatGPT, can be repetitive and biased. But more worrying is that this output can mysteriously tip mid-response from good (correct) to bad (misleading or wrong) without the user noticing. In 2024 alone, this…
Research has a long history of discussing what is superior in predicting certain outcomes: statistical methods or the human brain. This debate has repeatedly been sparked off by the remarkable technological advances in the field of…
Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…