In the framework of black-box optimization, I will present new algorithms, based on the stochastic estimation of the gradient via finite differences with structured directions. I will describe their convergence properties under various assumptions and show some numerical results. Speaker's website: https://www.dima.unige.it/~villa/ Zoom meeting: link
Inverse problems are pivotal in a variety of applications, such as target identification, non-destructive testing, and parameter estimation. Among these, the inverse scattering problem in inhomogeneous media poses significant challenges, as it seeks to estimate unknown parameters from available measurement data. To understand the mathematics of machine learning approaches for inverse scattering, we develop a…
Event webpage: https://sites.google.com/ncsu.edu/conf-quantum-groups-rep2024/home?pli=1 This conference is in celebration of Kailash Misra's 70th birthday.
In 1966, Mark Kac posed the famous question “Can you hear the shape of a drum?” Mathematically, this amounts to recovering the geometry of a Riemannian manifold from knowledge of its Laplace spectrum. In the case of strictly convex and smooth bounded planar domains, the problem is very much open. One technique for studying the…
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 theory, to name a few. In this talk, I will introduce this object and review some well-known results when X is a curve or a…
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 for solitons and the physical derivations of many integrable nonlocal systems. Speaker's website: https://www.math.fsu.edu/~musliman/ Zoom meeting: link
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 to address domain-specific data challenges and extract insights from scientific datasets through innovative methodological solutions. A particularly active area within SciML involves using neural operators…
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 all unit vectors \theta \in S^1, the dimension of the projection \pi_\theta (K) to \theta is min{t, 1}. To refine the question, we can replace…
(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 higher dimensions, with a focus in this talk on Calabi-Yau 3-folds. After discussing the background material in gauge theory and Calabi-Yau geometry, I will highlight…
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 these numbers is as the coefficients of the chromatic polynomial of a graph. They also appear as counting regions in the complement of a real…
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 an elastoplastic problem. Since then, it has been used to model constrained dynamical systems, nonsmooth electrical circuits, crowd motion, mechanical problems, and other applications. The…
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 of steady, unidirectional flow of fluid in a single pipeline is simple, it ceases to be so when we consider fluid flow in a large…
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 be sort of fractal). A famous and still very mysterious model is called DLA: it forms the most beautiful fractal patterns (pictures will be provided).…
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 abstract mathematics (for example, it has implications for the Novikov Conjecture) and in data science, where such embeddings are a necessary preprocessing step to traditional…
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 number of length-n binary strings that avoid 001. In this talk, I will describe a related (but to my knowledge, new) interpretation: given a length-3 binary…
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 link
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 $\mathbb R^n$, $n\geq 2$. After suitably rescaling the equation with a small phase parameter $\epsilon>0$, the rescaled solution solves a fractional Allen–Cahn equation. We show that,…