Colloquium Abstracts Fall 2025

Abstracts will be posted here for the colloquium talks when they are available.

Bo Deng

University of Nebraska
October 2, 2025

Error-free Training for Artificial Neural Network

If we define intelligence as not making the same mistake twice, then a system achieves this artificial intelligence if and only if it can learn from its mistakes every time. For a feedforward neural network under supervised training, this means that it can be trained error-free for every data set. This problem is known as the discrete classification problem in mathematics. Its solution was obtained more than thirty years ago by what is now known as the Universal Approximation Theorem. In this talk, I will present a numerical algorithm to fulfill the UAT. I will illustrate the algorithm by both abstract and practical classification problems.

Eduardo Cardoso de Abreu

State University of Campinas, Brazil
October 23, 2025

On the Lagrangian-Eulerian Approach: Mathematical and Computational Aspects with Examples

Mathematical modeling and numerical analysis challenges for the study of hyperbolic partial differential equations (PDEs) is in the realm of basic and applied sciences, for instance, ranging from fluid mechanics and modeling of vehicular traffic flows to fluid dynamics in porous media flows. In this Colloquium, we will discuss on a new approach [1,2,3] for studying some hyperbolic conservation laws of first order PDEs with examples for some multi-dimensional problems subject to irregular vector fields either in fluid mechanics linked to vortex sheet [4] or in the context of flow is porous media with spatially discontinuous coefficients [5]. In the context of multidimensional hyperbolic systems of conservation laws, the resulting Lagrangian-Eulerian method [6] satisfies a weak positivity principle in view of results of P. Lax and X.-D. Liu [Computational Fluid Dynamics Journal, 5(2) (1996) 133-156 and [Journal of Computational Physics, 187 (2003) 428-440]. We also found [2] an interesting connection between the notion of no-flow curves [1,2,3] (viewed as a vector field with locally bounded variation) and the results of A. Bressan in the context of (local) existence and continuous de pendence for discontinuous O.D.E.’s as introduced by A. Bressan (1988) [Proc. Amer. Math. Soc. 104, 772-778]. The method is based on the concept of multidimensional no-flow curves/surfaces/manifolds [1,2,3,6]. Roughly speaking, one reduces the hyperbolic PDE into a family of ODEs along the forward untangled space-time no-flow Lagrangian trajectories. As a by-product of the no-flow framework, there is no need to compute the eigenvalues (exact or approximate values), and in fact there is no need to construct the Jacobian matrix of the hyperbolic flux functions, and thus giving rise to an effective (weak) CFL-stability condition useful in the computing practice. The no-flow framework might be also applied to nonlinear balance laws [3].  We present numerical computations for nontrivial (local and nonlocal) hyperbolic problems, as such compressible Euler flows with positivity of the density, the Orszag-Tang problem, which is well-known to satisfy the notable involution-constrained partial differential equation div B = 0, a nonstrictly hyperbolic three-phase flow system in porous media with a resonance point, and the classical 3 by 3 shallow-water system (with and without discontinuous bottom topography). We will also provide numerical 1-D and Multi-D examples to verify the theory and exemplify the capabilities of the proposed approach. E. Abreu thanks the grant support of CNPq (307641/2023-6) and FAPESP (2025/07662-6).

References:

[1] A fast, robust, and simple Lagrangian-Eulerian solver for balance laws and applications (2019). Computers and Mathematics with Applications, 77(9) (2019) 2310-2336. https://doi.org/10.1016/j.camwa.2018.12.019 joint work  E. A and J. Pérez.

[2] A Lagrangian-Eulerian Method on Regular Triangular Grids for Hyperbolic Problems: Error Estimates for the Scalar Case and a Positive Principle for Multidimensional Systems,  Journal of Dynamics and Differential Equations,  Published: 26 June 2023, Volume 37, pages 749–814, (2025)  https://doi.org/10.1007/s10884-023-10283-1  joint work E.A., J. Agudelo, J. Pérez and W. Lambert.

[3] An Enhanced Lagrangian-Eulerian Method for a Class of Balance Laws: Numerical Analysis via a Weak Asymptotic Method With Applications, Numerical Methods for Partial Differential Equations, Volume 41, Issue 1 (January 2025; e23163) https://doi.org/10.1002/num.23163  joint work E.A., E. Pandini and W. Lambert.

[4] On a 1D model with nonlocal interactions and mass concentrations: an analytical-numerical approach. Nonlinearity, 35(4) (2022) 1734.  https://doi.org/10.1088/1361-6544/ac5097  joint work E.A., L.C.F. Ferreira, J. Galeano and J. Pérez.

[5] On the conservation properties in multiple scale coupling and simulation for Darcy flow with hyperbolic-transport in complex flows. Multiscale Model. Simul. 18(4) (2020) 1375-1408.  https://doi.org/10.1137/20M1320250  joint work E.A., C. Diaz, J. Galvis and J. Pérez.

[6] A Class of Positive Semi-discrete Lagrangian–Eulerian Schemes for Multidimensional Systems of Hyperbolic Conservation Laws, Journal of Scientific Computing (2022) https://link.springer.com/article/10.1007/s10915-021-01712-8; joint work E.A., J. François, W. Lambert and J. Pérez.

Carl Lian

Washington University, St. Louis
October 30, 2025

How (Not) to Count Curves

Given a geometric space X, how many curves of a certain shape lie in X? For instance, Euclid knew that there is a single line in the plane passing through two points. In the 1990s, Kontsevich, using ideas inspired from string theory, generalized Euclid’s count to determine the number of rational plane curves of degree d through 3d-1 points. This was the birth of Gromov-Witten (GW) theory, which offers a robust framework to think about counting curves, and which has seen an explosion of progress in the last three decades. However, the GW machine comes with a serious caveat: it often spits out a count that includes unwanted “excess” contributions that are less geometrically meaningful, and that are hard to control. I will outline a broad program to confront this problem in a particular class of examples, which forms a bridge between classical and modern enumerative geometry. The story is largely complete in the case of curves on projective spaces, but there are many interesting directions which lie beyond.

Junsheng Zhang

Courant Institute of Mathematical Sciences, New York University
November 13, 2025

Kähler-Ricci Shrinkers and Polarized Fano Fibrations

We prove that every Kähler-Ricci shrinker, (i.e. complete shrinking gradient Kähler-Ricci soliton) naturally admits a polarized Fano fibration structure. The proof relies on Kähler reductions and boundedness result in birational geometry. Moreover, we propose several conjectures for Kähler-Ricci shrinkers, unifying the well-developed theories of Kähler-Einstein metrics and Calabi-Yau cones. This is joint work with Song Sun