Fall 2022

Fall 2022

Title: TBA

Abstract: TBA

Title: Extending Group Actions on Metric Spaces 

  Title: Elastic instability of cylindrical vessels immersed in fluid

Abstract: 

We develop a numerical model to study the deformation of growing neo-Hookean elastic cylindrical vessels immersed in an incompressible fluid. The vessel is treated as a two-dimensional shell embedded in a three-dimensional space and the fluid-structure interaction is described using the Immersed-Boundary formulation. In our simulation, the shell grows in a confined space, that is, its surface area increases over time while the vessel itself is restricted from lengthening due to the periodic boundary condition. To accommodate for the new surface area, the shell is under axial compression and must alter its original geometry; hence, it buckles. We recover the two well-known modes of buckling: bending and barreling. We also observed other buckling modes such as kinking, twisting, or lumen collapsing. As an outlook for our project, we will add flow to our model and study how buckling affects the fluid flow within the shell. To strike for a more realistic model of a biological vessel, we also consider using a different non-linear elastic model such as the exponential Fung model. 

Title: Exceptional Collections in Type $\tilde{A}_n$

Abstract:

For Q a quiver of type $\tilde{A}$, we will associate to each exceptional collection of kQ modules a unique combinatorial object  which is a collection of arcs on an annulus. We will see that these diagrams can be placed into parametrized families and in the case when all  arrows of Q but one point in the same direction, the number of families is counted by a generalization of the Catalan numbers. We will finish by  providing a bijection between the exceptional collections in this case and certain lattice paths in $\mathbb{R}^2$.

Title: Cyclic Group Actions and Seiberg-Witten Floer K-Theory 

Abstract:

Given a spin 3-manifold Y satisfying b_1=0 and equipped with a Z/m-action preserving the spin structure, I will outline how to use the Seiberg-Witten equations to define a family of invariants which lie in a lattice constructed from the complex representation ring of a twisted product of Pin(2) and Z/m. These invariants give rise to equivariant relative 10/8-ths type inequalities for Z/m equivariant spin cobordisms. I will explain how these inequalities provide applications to knot concordance, give obstructions to extending cyclic group actions to spin fillings, and via taking branched covers give us genus bounds for knots in punctured 4-manifolds. In some cases these bounds are strong enough to determine the genus for a large class of knots in certain 4-manifolds.  

Title: Numerical Methods for the Reynolds Equation 

Title: Double Descent and the Neural Tangent Kernel 

Abstract:

Suppose you're fitting ten datapoints with a polynomial. You can fit a line to them reasonably well, or you can choose a degree-10 polynomial that fits them perfectly. In most cases, the line is the better choice: while it doesn't fit the observed data precisely, we'd expect it to generalize better to data we haven't seen yet. This is the idea behind the *bias-variance tradeoff* in statistics: you need to carefully choose how many parameters to include in a model to prevent both overfitting and underfitting.


In the world of deep learning, however, an unexpected phenomenon appears: past a certain level of overparameterization, overfitting stops, and generalization error starts to decrease! This phenomenon is known as "Double Descent" (Belkin et al 2019). This is thought to be caused by an implicit regularization in the learning algorithm. One approach to describing such a regularization is the Neural Tangent Kernel (NTK) (Jacot et al 2019), which describes neural networks in the infinite-width limit.


In this talk, I will explain the NTK, how it relates to the bias-variance tradeoff, and what else it can teach us about the wild, overparameterized world of deep neural networks. 

Title: TBA

Abstract: TBA