Abstracts

Abelian Sandpile Models  (Dr. Youngsu Kim)

In this talk, we introduce abelian sandpile groups and their connections to graph theory and algebra. Sandpile models were invented by Bak, Tang, and Wiesenfeld to study self-organized criticality in dynamical systems. Its algebraic interpretation, called critical or Picard groups, can be defined by the chip-firing game played on certain graphs. We provide explicit forms of critical groups for several classes of directed graphs. This is joint work with J. Jun and M. Pisano.

Mirror Symmetry, Polytopes and More (by Cristian Rodriguez Avila)

Algebraic Geometry studies shapes that are defined as the set of common zeros of a system of polynomial equations.  In the 1980s, theoretical physicists discovered a “weird” phenomenon called mirror symmetry  which suggests that some types of objects come in pairs X and Y. And according to mirror symmetry we expect that geometric properties of X correspond to different geometric properties of Y, and vice versa. This discovery surprised mathematicians, leading to many amazing discoveries in algebraic geometry and is now actively being studied by researchers from different branches of mathematics. On another note, a convex polytope is a generalization of a convex polygon such as a triangle, a convex quadrilateral, etc. Polytopes have applications in lots of different sciences as well as in “real” life problems, for example, they can be used for optimization. In particular, they also have applications in algebraic geometry and mirror symmetry! In this talk we will learn how convex polytopes can be useful in algebraic geometry, how they have helped to understand predictions from mirror symmetry and how I used them in my research to attempt to understand some open problems coming from mirror symmetry. No background knowledge will be assumed!

Error-Correcting Codes and Algebraic Coding Theory (Dr. Angelynn Alvarez)

Cloud and distributed storage systems have reached a massive scale in today's society, making data reliability against storage failures crucial. One way to repair a result of a storage failure is to use error-correcting codes that efficiently recover lost data with minimum overhead. An \textit{error-correcting code} is an algorithm for expressing a sequence of elements where any errors caused by lost data can be detected and possibly corrected. These codes have many real-world applications, such as in cell phones, credit cards,  computer RAM,  and photography from spacecrafts. A \textit{locally recoverable code} is a special kind of error-correcting code where any symbol of a codeword can be recovered using a set of other symbols.  In this talk, I will discuss the basic construction of different types of error-correcting codes and  share some recent developments of locally recoverable codes that use algebraic code constructions.

Classical Developments of Compressible Fluid Mechanics  (Dr. Abbrescia, Leonardo)

The flow of compressible fluids is governed by the Euler equations, and understanding the dynamics for large times is an outstanding open problem whose full resolution is unlikely to happen in our lifetimes. The main source of difficulty is that any global-in-time theory must incorporate singularities in the PDEs, a fact we have known even in one spatial dimension since Riemann’s 1860 work. In this 1D setting, mathematicians have successfully spent the past 160 years painting a nearly-full picture of fluid dynamics that incorporates singularities.  There is a monumental gap in our understanding of compressible fluids in the physical 3D setting compared to the 1D case. This is due in large to the (provable) inaccessibility of the technical PDE tools used in 1D when quantifying the dynamics in 3D. Nevertheless, Christodoulou’s 2007 celebrated breakthrough on shock singularities for the Euler equation has sparked a dramatic wave of results and ideas in multiple space dimensions that have the potential to make the first meaningful dent in the global-in-time theory of compressible fluids. Roughly, shocks are a form of singularity where the fluid solution remains regular but certain first derivatives blow up.

In this talk I will discuss the recent culmination of the wave of results initiated by Christodoulou: my work on the maximal classical development (MCD) for compressible fluids, joint with J. Speck. Roughly speaking, the MCD describes the largest region of spacetime where the Euler equations admit a classical solution. For an open set of smooth data, my work reveals the intimate relationship between shock singularity formation and the full structure of the MCD. This fully solves the 162 year old open problem of extending Riemann’s historic 1D result to 3D without symmetry assumptions. In addition to the mathematical contribution, the geo-analytic information of the MCD is precisely the correct “initial data” needed to physically describe the fluid “past” the initial shock singularity in a weak sense. I will also briefly discuss the countless open problems in the field, all of which can be viewed as “building blocks” which will shine the first lights onto the outstanding global-in-time open problem of fluids.