Champlain Math minors Peter Orzell (CSI), Scott Barrett (EGP), and David Johnston (EGP) will be presenting this year at the Hudson River Undergraduate Mathematics Conference.
The Hudson River Undergraduate Mathematics Conference is designed for students to do some independent research in a math topic of their choosing and then present what they learned for their peers. This is not original research; rather the talks are expository on topics not always covered in the math curriculum. HRUMC rotates around different schools in the Northeast, and this year’s will be held at Saint Michael’s College on Saturday, April 2.
These students, who did their work under the supervision of Dr. Melanie Brown and Dr. Scott Stevens, will be Champlain’s first-ever presenters at HRUMC. All three are senior math minors and will be fantastic ambassadors at this conference. All three are presenting in the afternoon, so even if you are helping at Admitted Student Day in the morning, you could still hop over to SMC after lunch. The invited address this year is from Scott Aaronson at MIT on “The P vs. NP Problem,” which also might interest some of you.
Peter will be presenting on “Semantic and Contextual Insight from Word Vectorization.” Extracting meaning and context from written text has long proved to be a difficult task in computing. Manipulation of word data mapped to a vector space, known as a word embedding, is a recent strategy that has been used to gain an enormous amount of insight into the meaning behind text. This scheme has largely been brought to community attention by word2vec and its authors, T. Mikolov, K. Chen, G. Corrado, and J. Dean, in their 2013 ICLR paper, Efficient Estimation of Word Representations in Vector Space. In their described method, the cosine distance between mapped word nodes can be used as a determiner of conceptual similarity. One completely emergent benefit of the system is that a “conceptual algebra” can even be performed by preserving a relationship between two concept nodes as a difference vector. This difference vector can then be applied as a transformation to another concept node to yield a conceptually analogous result, e.g. \(vector(king) – vector(man) + vector(woman) \ approx vector(queen))\. In this presentation, Peter will briefly examine the mathematics behind creating aword2vec word embedding, methods for navigating the embedding mathematically, and a few of the potential insights that can be obtained from working with such an embedding.
Scott will be presenting on “Lambda Calculus: A Computational System of Mathematics.” Lambda calculus is a system formalized by Alonzo Church as a way of expressing computation to solve any mathematical problem. It was formalized around the same time as its better-known cousin, the Turing machine, and was invented to solve the same problem in mathematics. Church’s paper was released a year before Turing’s. Lambda calculus is an extremely elegant system that is logically equivalent to Turing machines, but is less intuitive to grasp. Both formal systems have inspired many influential programming languages, but the languages that spawned from each school of thought are as radically different as the systems themselves. This talk will explain the basics of lambda calculus and explore how it relates to modern-day computer science.
David will be presenting on “Programmatic Art Through Ray Marching Primitives.” Ray marching is a graphics programming technique that is used primarily in games and graphics demos. While ray marching can be used for general rendering, it is more commonly used for cooking up specific effects such as displacement mapping, soft shadows, and sub-surface scattering. In contrast to classic ray casting, ray marching is simpler to implement, parallelizes well on modern graphics processing units (GPUs), and can be used on shapes that do not have analytic intersection functions. This presentation will focus on ray marching primitives using the OpenGL Shading Language (GLSL), with a little dip into shading those primitives if time allows.
It promises to be an exciting day filled with interesting mathematical ideas and applications!