The Hilbert scheme of d points on a smooth variety X, denoted by Hilb^d(X), is an important moduli space with connections to various fields, including combinatorics, enumerative geometry, and complexity…
In this talk I will review past and recent results pertaining to the emerging topic of integrable space-time nonlocal integrable nonlinear evolution equations. In particular, we will discuss blow-up in finite time…
Recent advancements in deep learning have led to a surge in research focused on solving scientific problems under the "AI for Science." Among these efforts, Scientific Machine Learning (SciML) aims…
Given a set K in R^2 with Hausdorff dimension t \in , what can we say about a typical orthogonal projection of K? Marstrand (1954) proved that for Lebesgue almost…
(Based on joint work with Yang Li). I will present an overview of Donaldson's program to extend the methods of gauge theory and Floer homology from 3- and 4-manifolds to…
The Whitney numbers of the first and second kind are a pair of poset invariants that are relevant in various areas of mathematics. One of the most interesting appearances of…
The sweeping process is a first-order differential inclusion involving the normal cone to a family of moving sets. It was introduced by J.J. Moreau in the early seventies to address…
Natural gas production and distribution in the US is interconnected continent-wide, and hence the simulation of fluid flow in pipeline networks is a problem of scientific interest. While the problem…
We'll study the growth of (two-dimensional) things. Think about lichen growing on a tree (tends to be sort of round). Another fun example is electricity propagating through wood (tends to…
Speaker's website: https://www.stat.berkeley.edu/~vadicgor/
Given a space X, one may want to know if it can be embedded into a vector space in a controlled way. Interestingly, this question is of interest in both…
The sum of Fibonacci numbers, i.e. the sequence 2, 4, 7, 12, 20, 33, 54, 88, ... has many combinatorial interpretations. For instance, the n-th term in this sequence is the…
During each 50-minute First Year Research Seminar, two faculty give our new graduate students a short (~20-25 minute), accessible talk about their research or research area. First Year Research Seminar…
We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls–Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in $\mathbb…
Students often perceive Multivariable Calculus as a collection of disconnected ideas and approach problem solving in the course formulaically. Students need help from instructors in developing spatial reasoning and making…
Neural operators such as Deep Operator Networks (DeepONet) and Convolutional Neural Operators (CNO) have been shown to be fairly useful in approximating an operator between two function spaces. In this…
The past decade has witnessed a remarkable surge in breakthroughs in artificial intelligence (AI), with the potential to profoundly impact various aspects of our lives. However, the fundamental mathematical principles…