Abstract: We discuss recent work in algebraic geometry that was motivated by the study of rough paths in stochastic analysis. Every path in a real vector space is encoded in a signature tensor whose entries are iterated integrals. As the path varies over a nice family, we obtain an algebraic variety with interesting properties. Lyndon words and free Lie algebras make a prominent appearance.
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We will discuss how the ideas of twistor theory apply to the analysis and geometry of space-time.
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The Borel-Weil-Bott theorem is a very famous result in representation theory with a strong connection to geometry. We discuss the statement and proof of the theorem with examples.
Consider the number 61,917,364,224. What's so special about it? Nothing really comes to mind. But this exact number was crucial towards a two-sentence published paper that gave a counterexample to one of Euler's famous conjectures that tried to generalize Fermat's Last Theorem. I will discuss why we can't just assume conjectures from famous people are "obviously'' true.
Geometric measure theory (GMT), born in the 1960’s, is a generalization of calculus and differential geometry to the realm of “non-smooth” geometric objects. GMT helped solve Plateau’s Problem, posed in 1760(!) by Lagrange. I will tell you about the basic tools of GMT, its other success stories, its current avenues as a hot research area, and what doing research in GMT will look like for a young mathematician like you! Some amazing lemmas and theorems on the fly, but no proofs or technical discussions expected
Recently, the fractional Laplacian has received great attention in modeling complex phenomena that involve long-range interactions. However, the nonlocality of the fractional Laplacian introduces considerable challenges in both analysis and simulations. In this talk, I will present numerical methods to discretize the fractional Laplacian as well as error estimates. Compared to other existing methods, our methods are more accurate and simpler to implement, and moreover they closely resembles the central difference scheme for the classical Laplace operator.
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Oscillatory and excitable cells are just two of many types of cells in your body. They interact with each other by sending pulses or voltage spikes. An excitable cell amplifies any input it receives, as long as the input is large enough while an oscillatory cell needs no input and generates a spontaneous rhythm. One can connect these cells and model how they interact using differential equations; in this talk, I will discuss what we have found.
Thack 627