Department of Mathematics, Duke University

• Monday, April 19, 2021, 11:00am, Zoom link, Undergraduate Senior Thesis
  Schur Polynomials and Crystal Graphs
  Lucas Fagan (Duke University, Mathematics)

   

  Schur polynomials are fundamental objects in representation theory and combinatorics, and are connected to lots of different areas of math. Most simply, they are a set of symmetric polynomials that form the basis for the space of all symmetric polynomials. In the context of this paper, Schur polynomials arise in the representation theory of general linear groups by giving characters of representations, which is important because characters determine representations up to isomorphism. As fundamental as Schur polynomials are, they are still interesting to study. The goal of this project is to produce new formulas for Schur polynomials.

• Monday, April 19, 2021, 1:00pm, Zoom link, Undergraduate Senior Thesis
  Resolving Simpson’s Paradox in NC Public School Grading System
  Atsushi Hu (Duke University, Mathematics)

   

  In this paper we further develop from my research from PRUV2020 on analyzing potential problems with the grading systems of public elementary schools quantitatively and developing alternative grading metrics. We mainly focus on deducing the distribution of scores based on the available score data from a Bayesian approach, through methods such as Bayesian linear regression, Gaussian Process Regression and Gaussian Mixture Model. As we find Gaussian Mixture Model the best in describing the joint distribution of income and score (in Year 5 Math Test conducted in 2016), we apply Gaussian Mixture Model further to define an alternative score for each school that account for the income distribution of of students in each school, and compare this score with the existing grades of the elementary schools. By looking at the schools in Durham specifically, we notice that there is a significant difference in our alternative score and the official grades of the schools.

• Monday, April 19, 2021, 3:15pm, Zoom link, Geometry/topology Seminar
  Limits of manifolds with a Kato bound on the Ricci curvature
  Ilaria Mondello (Université de Paris Est Créteil, Laboratoire d’Analyse et Mathématiques Appliquées)

   

  Starting from Gromov pre-compactness theorem, a vast theory about the structure of limits of manifolds with a lower bound on the Ricci curvature has been developed thanks to the work of J. Cheeger, T.H. Colding, M. Anderson, G. Tian, A. Naber, W. Jiang. Nevertheless, in some situations, for instance in the study of geometric flows, there is no lower bound on the Ricci curvature. It is then important to understand what happens when having a weaker condition. In this talk, we present new results about limits of manifolds with a Kato bound on the negative part of the Ricci tensor. Such a bound is weaker than the previous L^p bounds considered in the literature (P. Petesern, G. Wei, G. Tian, Z. Zhang, C. Rose, L. Chen, C. Ketterer…). In the non-collapsing case, we recover part of the regularity theory that was known in the setting of Ricci lower bounds: in particular, we obtain that all tangent cones are metric cones, a stratification result and volume convergence to the Hausdorff measure. After presenting the setting and main theorem, we will focus on proving that tangent cones are metric cones, and in particular on the study of the appropriate monotone quantities that leads to this result.

• Tuesday, April 20, 2021, 3:15pm, Virtual, Applied Math And Analysis Seminar
  Deep learning for solving inverse imaging problems
  Carola-Bibiane Schönlieb (University of Cambridge, Applied Mathematics and Theoretical Physics)

   

  Inverse problems are about the reconstruction of an unknown physical quantity from indirect measurements. In imaging, they appear in a variety of places, from medical imaging, for instance MRI or CT, to remote sensing, for instance Radar, to material sciences and molecular biology, for instance electron microscopy. Here, imaging is a tool for looking inside specimen, resolving structures beyond the scale visible to the naked eye, and to quantify them. It is a mean for diagnosis, prediction and discovery. Most inverse problems of interest are ill-posed and require appropriate mathematical treatment for recovering meaningful solutions. Classically, such approaches are derived almost conclusively in a knowledge driven manner, constituting handcrafted mathematical models. Examples include variational regularization methods with Tikhonov regularization, the total variation and several sparsity-promoting regularizers such as the L1 norm of Wavelet coefficients of the solution. While such handcrafted approaches deliver mathematically rigorous and computationally robust solutions to inverse problems, they are also limited by our ability to model solution properties accurately and to realise these approaches in a computationally efficient manner. Recently, a new paradigm has been introduced to the regularization of inverse problems, which derives solution to inverse problems in a data driven way. Here, the inversion approach is not mathematically modelled in the classical sense, but modelled by highly over-parametrised models, typically deep neural networks, that are adapted to the inverse problems at hand by appropriately selected (and usually plenty of) training data. Current approaches that follow this new paradigm distinguish themselves through solution accuracies paired with computational efficieny that were previously unconceivable. In this talk I will provide a glimpse into such deep learning approaches and some of their mathematical properties. I will finish with open problems and future research perspectives.

• Thursday, April 22, 2021, 3:15pm, Virtual, Probability Seminar
  Mean-field spin glasses: beyond Parisi's formula?
  J.C. Mourrat (NYU)

   

  Spin glasses are models of statistical mechanics encoding disordered interactions between many simple units. One of the fundamental quantities of interest is the free energy of the model, in the limit when the number of units tends to infinity. For a restricted class of models, this limit was predicted by Parisi, and later rigorously proved by Guerra and Talagrand. I will first show how to rephrase this result using an infinite-dimensional Hamilton-Jacobi equation. I will then present partial results suggesting that this new point of view may allow to understand limit free energies for a larger class of models, focusing in particular on the case in which the units are organized over two layers, and only interact across layers.

• Tuesday, April 27, 2021, 3:15pm, Virtual, Applied Math And Analysis Seminar
  Low-rank techniques for PDE solving and PDE learning
  Alex Townsend (Cornell University, Mathematics)

   

  Matrices and tensors in computational mathematics are so often well-approximated by low-rank objects. In the first part of the talk, we will use the ADI method, a classic partial differential equation (PDE) solver, to understand the prevalence of compressible matrices and tensors, and resolve a long-standing problem of finding an optimal complexity spectrally-accurate Poisson solver. In the second part of the talk, we will use low-rank techniques for PDE learning where one is given input-output training data from an unknown uniformly elliptic PDE and would like to recover the PDE operator. By exploiting the hierarchical low-rank structure of Green’s functions and randomized linear algebra, we will describe a rigorous scheme for PDE learning with a provable “learning rate.”

• Thursday, April 29, 2021, 3:15pm, Virtual, Probability Seminar
  Integrability of SLE and Liouville Conformal Field Theory through Conformal Welding
  Xin Sun (U. Penn)

   

  here are two frameworks to study quantum surfaces in Liouville quantum gravity: random planar geometry and Liouville conformal field theory (LCFT). A key feature of the first framework is the coupling with Schramm-Loewner evolution (SLE). A key feature of the second framework is the application of conformal field theoretical techniques. We report an ongoing program that unifies these two frameworks using conformal welding of quantum surfaces, which gives strong links between the integrability of SLE and LCFT. We obtain exact formulae for both SLE and LCFT which are hard to obtain from methods within their own framework. Based on a joint work with Ang and Holden, a joint work with Ang and Remy; and a series of ongoing projects with Ang.

• Friday, May 7, 2021, 12:00pm, Virtual, Mathematical Biology Seminar
  An introduction and invitation to Quantitative Systems Pharmacology
  Blerta Shtylla (Pfizer Worldwide Research, Development, and Medical, Clinical Pharmacology, Early Clinical Development)

   

  Email ciocanel@math.duke.edu to request the Zoom link and password for the talk (or subscribe to announcements at https://lists.duke.edu/sympa/info/mathbio-seminar).

• Friday, May 14, 2021, 12:00pm, Virtual, Mathematical Biology Seminar
  A dynamical model of the visual cortex
  Lai-Sang Young (New York University, Courant Institute of Mathematical Sciences)

   

  Email ciocanel@math.duke.edu to request the Zoom link and password for the talk (or subscribe to announcements at https://lists.duke.edu/sympa/info/mathbio-seminar).