Seminar

Abstract:

Integration by parts is one of the most time-consuming methods of integration that we learn in calculus, second only to the dreaded trig-substitution. However, in certain settings, we can speed up the process by using tools from linear algebra. By considering functions as vectors, we can find a basis for a subspace that allows us to compute integrals via a simple matrix multiplica-tion. Will this discovery lead to a paradigm shift in integration techniques? No! But it is an interesting intersection of ideas. Food and drinks will be provided!

Curves are dimension $1$, surfaces have dimension $2$, but fractals have non-integer dimensions. But what is this ``dimension''? What does it really mean for the West coast of Britain to have dimension 1.25? After defining the Hausdorff dimension, I will compute that of some household fractals such as Cantor's set, Koch snowflake, and Sierpinski carpet. Expect to get your black-and-white copies of some fractals!

Logic is heavily used in mathematics as the basis for proofs. Some believe that logic is an independent discipline which is not a part of mathematics and the last is based on logic. Others think that logic is just one of mathematical subjects. Our goal is to find out which one of these two arguments is right. In this talk, we will make an attempt to show that logic can be built by using mathematical tools and objects and therefore is a subset of mathematics. This also implies that mathematics is self-sufficient.

Over the last century, there has been tremendous progress in understanding the transmission of infectious diseases in human populations. However, most historical outbreaks have not been studied in detail because data that might permit such investigations are buried in thousands of handwritten records.  In recent years, my research group has digitized a large number of these historical records and begun to dissect the epidemic patterns.

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.

Abstract:

We will discuss how the ideas of twistor theory apply to the analysis and geometry of space-time.

Abstract:

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