Academic Explorations

A Resounding Yes: Advancing Scientific Exploration

In the history of mathematical and scientific discovery, few figures are as enigmatic and inspiring as Alexander Grothendieck.

Berlin

Indeed, his early life was marked by adversity and displacement. Since his parents were Russian-Jewish , his childhood was overshadowed by the rise of Nazism, the Jews being prosecuted. His family's flight from Nazi Germany to France and their involvement in the Spanish Civil War set the stage for a chain of dramatic events, and soon he would find himself challenged by both intellectual pursuit and political activism he needed to sustain for obvious reasons. The internment of Grothendieck and his mother in the Rieucros concentration camp during World War II, coupled with the tragic death of his father in Auschwitz,did shape his perspectives on humanity, justice, and the role of intellectuals in society. Life would bring luck, apart from the obstructions he had to endure and overcome, once he managed to nullify some of these adversities, and his struggle would bear fruits, Grothendieck's passion for mathematics not only would endure but flourish.

Paris

Innovation

Decyphering Grothendieck his notes.

   

Cubicles

Alexander Grothendieck his scribbles were a mystery at first. He passed not that long ago and, someone has to look into them, someone has to do it. After his passing, he left behind a vast collection of these notes, scribbles, and manuscripts and drawings, but what do they contain? There are some dozens of pages. The prospect of exploring these has felt daunting at first but eventually i thought i could cope so I decided to take the plunge. I've begun the process of trying to decipher some of these posthumous notes. By now i can browse through the pages, understanding a good bulk of them, i suspect eventually i might be able to translate most of them if not all. It started small, focusing on just the initial few pages, the first scribbles I could access. However, wading into these first few pages has already shifted my perspective. My initial impression is that these specific notes weren't necessarily left behind with the direct intention of "advancing science" in the way a polished paper or a finished theorem might (his habit throughout his carreer). They feel different. They seem much more like intermediate structures, snapshots of a fine mind at work, building blocks within his own intricate thought processes rather than finished edifices intended for immediate public consumption. Does this make them less valuable? Seeing them as part of a process doesn't diminish their importance in my eyes. There's immense value in witnessing (even retrospectively and partially) how such a keen intellect worked through problems, the scaffolding he built before reaching his known peaks. They deserve appreciation for what they are: intimate glimpses into a remarkable intellectual journey. Based on these initial attempts, I feel optimistic. The language, though dense and characteristic, feels approachable. I have a growing confidence that, given time, I could translate the rest too. Interestingly, this dive into pure, abstract mathematics runs parallel to my own ongoing work. I've also made some small, but steady, progress in my photonics research recently. The ultimate goal there remains crystallizing an idea, perhaps pointing out a photonic law more properly, or at least illuminating its potential more clearly. And perhaps these two threads – the historical deciphering and the future-facing physics – aren't entirely separate. I have this strange, persistent impression that these deep mathematical structures, the kind Grothendieck explored, will somehow loop back into my life and work, perhaps even geographically. It's a bit whimsical, but I can almost picture myself puzzling over these notes, or finally formulating that photonic principle, while living amidst the beauty of the Philippines, since i have been enjoying the calm of its coral beaches, and building my business there along my spouse. For now, though, it's one step at a time. One page of Grothendieck deciphered, one small advancement in photonics. It's a journey, and I'm excited to see where both paths lead. The first i will cover here is a cube.

   

Grothendieck was exploring dimensions here related to category theory. The cube represent several nodes, and morphisms (he used arrows to draw these), the morphisms he created between them. The cube structure was designed to organize the morphism. The framework is a scaffold, you can see the vertices and edges drawn, forming the outline and a fainter simpler cube is visible atop. The digraph (atop) Is there partial to sustain vertices (think of them as nodes) and they point along the edges and within the volume (faces) hence serving as true nodes. He wrote down some maths near each node, using M as his point of reference: Here M is meant to point out states and variable within the system he was modelling. The system is an Arc, because the nodes connect with lines by arrows representing directed edges. They are indicate relationships, helping him draw the transitions and hence transformations (or implications) between those nodes. So you could say he was drawing multiple dimensions ,since he was interested in certain dimensions his cube would clarify for him: the cube axes would point them out. Note that there are multiple edge types, hence he used different colors for each and he changed his arrows too (thick black, or thin black and dashed blue and solid red) These represent the categories (or relationships if you may) and paths within his structure. Keep staring and you will notice the Arc. Perhaps later on i will cover more of his notes.

Regarding my own work, i do not have much to brag about right now. There are certain obstructions that remain unresolved for now. The photon's motion occurs only within boundaries permitted by dimensionally sustainable adjustments in my work and i need to use functors to solve part of the problem. So far i was able to draw the spacetime manifold, since it must adapt its structure in infinitesimal steps to maintain the photon’s categorical invariance. I am using Mirzakhani her systems, to create my approach. (the rate of change of the photon soul with respect to the system's evolution is identically zero) I have a difficult time solving this problem, but i think i will eventually, providing i invest enough time and energy.

It is not the photon's energy, mass, or spin alone that enforces this behavior — it is the preservation of its categorical soul structure. I suspect i could create some sort of diagram to point out photonic evolution, but as of now, it is unclear to me how.

Artificial General Intelligence (AGI) has been recently in the press. Most people have heard the term AI; however, AGI is where we are heading. AGI represents a great kind of progress regarding artificial intelligence research, since we are aiming to create machines capable of performing any intellectual task that a human can. While current AI systems excel in specific domains, AGI aspires to achieve a level of cognitive flexibility and understanding akin to human intelligence. Understanding the fundamental building blocks of biological intelligence—neurons—to envisioning the complex networks that could underpin AGI, is part of this process, because through the lens of molecular biology AI can be understood more thoroughly. The molecular reality of the brain consists partially of neurons, the Principal Units Neurons are the principal cells responsible for processing and transmitting information within the brain. When we investigate deeper, we notice each neuron consists of a cell body (soma), dendrites, and an axon. We know these dendrites are the ones receiving signals from other neurons, while the axon transmits signals to other neurons. Along the synapse, a specialized junction between neurons, the transmission of these signals takes place through neurotransmitters and hence we observe at the molecular level, how neurons communicate through a complex interplay of electrical and chemical signals. These lessons can be applied to aid us in our dreams, developing AI. Ion channels, neurotransmitter receptors, and various signalling molecules play crucial roles in the process and understanding these molecular mechanisms is essential for replicating neural functions in artificial systems.

Neural networks:

Let us consider neural networks. Biological Neural Networks located in the brain, consist of neurons because they relate to cognition. These neuronal structures are organized into sophisticated networks that facilitate complex cognitive functions since these networks are characterized by their connectivity patterns, synaptic strengths, and plasticity. Synapses have the most curious ability to strengthen or weaken over time, and their ability is a key mechanism underlying learning and memory. The computational side, artificial neural networks (ANNs) express our model. We design them to mimic the brain's structure and function. The vast potential of these ANNs is obvious, and they consist of layers of interconnected nodes (analogous to neurons) that process information through weighted connections (analogous to synapses). Training these networks involves adjusting the weights based on input data, a process akin to synaptic plasticity. Extensive advancements have been achieved allowing for commercial models such as the models OpenAI has released, and while current AI systems are constrained by their narrow focus and lack of generalization, they do excel in specific tasks but struggle with tasks outside their training data. This limitation arises from the rigid and predefined architectures of existing AI models. Through a Molecular Lens achieve AGI will be no doubt achieved more rapidly since researchers are exploring the molecular underpinnings of intelligence. By our efforts studying the molecular biology of neurons and neural networks, we aim to uncover principles that can be applied to artificial systems. Such challenging, daunting task involves integrating knowledge from neuroscience, molecular biology, and computer science to create more flexible and adaptive AI models. Next-generation molecular computing, which leverages the principles of molecular biology to perform computations allow us to reflect and finetune our products. Molecular computing systems use molecules, (including DNA and proteins), in order to store first, and second process certain data. These systems offer the potential for massive parallelism and energy efficiency, and hence these are key attributes for sustaining AGI. Neuromorphic engineering is another approach that seeks to emulate the brain's architecture and functionality in hardware. By designing circuits that mimic neural processes at the molecular level, researchers aim to create more efficient and powerful AI systems. Neuromorphic chips, for example, use spiking neural networks to replicate the brain's communication patterns.

Molecular computing:

There are several aspects of neurology currently of interest for the development of advanced large language models, especially Neurogenesis (Neurogenesis being the process of generating new neurons), along with synaptic plasticity, are fundamental to learning and memory and it is no surprise recent studies have shown that neurogenesis continues into adulthood, especially in the hippocampus, a region associated with memory formation. Understanding these processes at a molecular level can provide insights into creating more adaptive and resilient AI systems. At the cellular level, Glial Cells including the astrocytes and oligodendrocytes, play crucial roles in supporting neurons. Astrocytes regulate neurotransmitter levels and maintain the crucial blood-brain barrier (in turn protecting the brain), while oligodendrocytes insulate axons with myelin. An interesting remark to make is how Incorporating the functions of glial cells into AI models could indeed enhance their efficiency and robustness. Neurotransmitters such as dopamine, serotonin, and glutamate are essential for modulating neural activity. (Dopamine, being involved in reward processing and motivation) By modelling neurotransmitter dynamics, AI systems will no doubt improve, as we design them to be better, they will mimic human decision-making and emotional responses.

Brain-Computer Interfaces (BCIs) enable direct communication between the brain and external devices. These peculiar advances in BCI technology have the potential to enhance human-AI interaction, allowing for more intuitive and effortless integration of AI into daily life. Because AI is important for molecular medicine, the medical community will support and contribute the development of AGI and understanding the molecular basis of neural signalling can improve the development of BCIs. One productive approach to help develop AI is through the study of Artificial Neural Networks (ANNs), since these have experienced significant advancements, especially with the advent of deep learning techniques. These networks, composed of multiple layers, can learn complex patterns from large datasets. The most useful language right now to approach AGI is Python, and Python proves to be both simple and complex though we will likely end up developing more specialized languages. Python has become a standard tool for developing and training ANNs due to its extensive libraries such as TensorFlow and PyTorch. The Supercomputers we have developed today, will progress into AI Supercomputers and related hardware play a critical role in advancing AI research. Since they provide the computational power necessary to train large-scale neural networks and simulate complex biological processes, they use plenty of energy, hence the switch towards using more green setups is of essence. Recently google has released AlphaFold 3, the latest iteration of their project called DeepMind, and AlphaFold 3 is all about DeepMind's protein structure prediction capabilities, through this exact prediction model, google has revolutionized drug discovery. By accurately predicting protein structures, AlphaFold 3 enables researchers to understand disease mechanisms and identify potential drug targets more effective. Google their breakthrough does demonstrate the renewed potential of AI in transforming biomedical research. We are now capable of analysing vast amounts of biological data, we can identify novel drug candidates, and optimize our drug design process. Machine learning algorithms can predict the efficacy and safety of new drugs, significantly reducing the time and cost associated with standard drug development processes. By understanding and replicating the fundamental principles of biological intelligence, we can envision a future where machines possess the cognitive flexibility and understanding of humans. This endeavour not only advances our hi-tech capabilities but also deepens our understanding of the nature of intelligence itself. BNNs interconnected with robots can exhibit clever behaviours such as learning and memory, which are crucial for developing AGI systems. Our normalization Strategies involves parallels between artificial and biological neural networks, which are essential for stabilizing network activity and enhancing learning efficiency, hence our vision regarding AGI has proven to be somewhat functional. The development of AGI necessitates addressing certain societal implications, including the impact on employment, privacy, and security. For example, the use of high-density CMOS microelectrode arrays in engineered biological neural networks can significantly enhance the spatiotemporal resolution, aiding in the development of more sophisticated AI models, and since these models are rapidly evolving, we need to control them while we allow them to empower our research. Drug Discovery AI technologies, especially AlphaFold 3, have revolutionized drug discovery by accurately predicting protein structures, only contributing to the field of medicine, demonstrating the transformative potential of AI in biomedical research. While a variety of open source LLM’s have been released so far, privacy and security regulation will become more of essence in the future. A new hi-tech saga begins.