## Programs & Events

##### The Ceresa Cycle in Arithmetic and Geometry

May 13 - 17, 2024

In the 1980s, Ceresa exhibited one of the first naturally occurring examples of an algebraic cycle, the Ceresa cycle, which is in general homologically trivial but algebraically nontrivial. In the last few years, there has been a renewed interest in the Ceresa cycle, and other cycle classes associated to curves over arithmetically interesting fields, and their interactions with analytic, combinatorial, and arithmetic properties of those curves. We hope to capitalize on this momentum to bring together different communities of arithmetic geometers to fully explore explicit computations around the arithmetic and geometry of cycles when these various approaches are systematically combined.

##### Organizing Committee

- Daniel Corey
- Jordan Ellenberg
- Wanlin Li
- Daniel Litt
- Congling Qiu
- Padmavathi Srinivasan

##### Interacting Particle Systems: Analysis, Control, Learning and Computation

May 6 - 10, 2024

Systems of interacting particles or agents are studied across many scientific disciplines. They are used as effective models in a wide variety of sciences and applications, to represent the dynamics of particles in physics, cells in biology, people in urban mobility studies, but also, more abstractly in the context of mathematics, as sample particles in Monte Carlo simulations or parameters of neural networks in machine learning.

This workshop aims at bringing together researchers in analysis, computation, inference, control and applications, to facilitate cross-fertilization and collaborations.

##### Organizing Committee

- Jose Carrillo
- Katy Craig
- Massimo Fornasier
- Fei Lu
- Mauro Maggioni
- Kavita Ramanan

##### An ICERM Public Lecture: Mathematicians Helping Art Historians and Art Conservators

May 2, 2024

In recent years, mathematical algorithms have helped the art historians and art conservators putting together fragments of world-famous frescos by Andrea Mantegna classify certain paintings as “roll mates,” remove artifacts in preparation for a restoration campaign, and gain insight into the hidden paintings underneath visible ones.

This lecture will review these applications and give a glimpse into the mathematical aspects that make them possible.

##### Nonlocality: Challenges in Modeling and Simulation

Apr 15 - 19, 2024

This workshop focuses on the modeling, analysis, approximation, and applications of nonlocal equations, which have raised new challenges to mathematical modeling, numerical analysis, and their computational implementation. Recent applications include, but are not limited to: heat and mass diffusion, mechanics, pattern formation, image processing, self-organized dynamics, and population dispersal.

Invited speakers and participants will bring expertise from a wide range of related fields, including mathematical and numerical analysis of nonlocal and fractional equations, numerical methods and discretization schemes, multiscale modeling, adaptivity, machine learning, software implementation, peridynamics modeling of material failure and damage, nonlocal and fractional modeling of anomalous heat and mass diffusion, and several engineering and scientific applications in which nonlocal modeling is useful.

##### Organizing Committee

- Marta D'Elia
- Abner Salgado
- Pablo Seleson
- Xiaochuan Tian

##### The Industrialization of SciML

Mar 23 - 24, 2024

Scientific Machine Learning (SciML) is a merger of computational sciences and data-driven machine learning, implemented in software as a set of abstractions to leverage existing domain knowledge and physics models within learning schemes and accelerated computing platforms. Only now are the component technologies becoming mature and integrated, with the purpose of providing robust and reliable ML that encodes domain knowledge and yields interpretable solutions in nearly all sciences at any scaleâ€”from particle physics and protein folding, to epidemiology and economics, to climate and astrophysics.

While recent years have shown immense research progress in accelerating classical physics solvers and discovering new governing laws for complex physical systems, SciML methods and applications fall short of real-world utility. Across all physical and life sciences the technical maturity and validations necessary are lacking, failing, or unknown to SciML research. In the context of... (more)

##### Organizing Committee

- Marta D'Elia
- George Karniadakis
- Alexander Lavin

##### PDEs and Geometry: Numerical Aspects

Mar 11 - 15, 2024

The development and analysis of numerical methods for PDEs whose formulation or interpretation is derived from an underlying geometry is a persistent challenge in numerical analysis. Examples include PDEs posed on complicated manifolds or graphs, PDEs that describe interactions across complex interfaces, and equations derived from intrinsically geometric concepts such as curvature-driven flows or highly nonlinear Monge-Ampere equations arising in optimal transport. In recent years, these PDEs have gained significance in diverse areas such as machine learning, optical design problems, meteorology, medical imaging, and beyond. Hence, the development of numerical methods for this class of PDEs is poised to lead to breakthroughs for a wide range of timely problems. However, designing methods to accurately and efficiently solve these PDEs requires careful consideration of the interactions between discretization methods, the PDE operators, and the underlying geometric properties.

This... (more)

##### Organizing Committee

- Charlie Elliott
- Brittany Hamfeldt
- Michael Neilan
- Maxim Olshanskiy
- Axel Voigt

##### K3 Surfaces

Feb 14 - 17, 2024

The purpose of the workshop is to advance toward the goal of making information relating to families of K3 surfaces and their elliptic fibrations into the LMFDB. We expect to begin with families of Picard number 19, building on the previous week's work on Shimura curves, and entering at least the following information: Picard lattice, frame lattices, transcendental lattice, 2-neighbour relations, elliptic fibrations, singular projective model. For some of these, code exists or is currently under construction; for others, it will need to be developed.

The workshop will consist of a small group of mostly experienced people. We expect that most of the time will be spent writing code in small groups, with at least one meeting a day to make sure that we are all going in the same direction while avoiding duplication of effort.

##### Organizing Committee

- Noam Elkies
- Adam Logan
- John Voight

##### Numerical Analysis of Multiphysics Problems

Feb 12 - 16, 2024

It is practically rare that a natural phenomenon or engineering problem can be accurately described by a single law of physics. The striking diversity of rules of life forces scientists to continuously increase the complexity of models to address the ever-growing requirements for their prediction capabilities. It remains a formidable challenge to derive and analyze numerical methods which are universal enough to handle complex multiphysics problems with the same ease and efficiency as traditional methods do for textbook PDEs.

The workshop will focus on recent trends in the field of numerical methods for multiphysics problems that include the development of monolithic approaches, structure preserving discretizations, geometrically unfitted methods, data-driven techniques, and modern algebraic methods for the resulting linear and nonlinear discrete systems. The topics of interest include models and discretizations for fluid - elastic structure interaction, non-Newtonian fluids, phase... (more)

##### Organizing Committee

- Martina Bukač
- John Evans
- Hyesuk Lee
- Amnon Meir
- Maxim Olshanskiy
- Sara Pollock
- Valeria Simoncini

##### Numerical PDEs: Analysis, Algorithms, and Data Challenges

Jan 29 - May 3, 2024

This Semester Program will bring together both leading experts and junior researchers to discuss the current state-of-the-art and emerging trends in computational PDEs. While there are scores of numerical methodologies designed for a wide variety of PDEs, the program will be designed around three workshops each centered around a specific theme: PDEs and Geometry, Nonlocal PDEs, and Numerical Analysis of Multiphysics problems. This grouping of topics embodies a broad representation of computational mathematics with each set possessing its own skill set of mathematical tools and viewpoints. Nonetheless, all workshops will have the common theme of using rigorous mathematical theory to develop and analyze the convergence and efficiency of numerical methods. The diversity of the workshop topics will bring together historically distinct groups of mathematicians to interact and facilitate new ideas and breakthroughs.

##### Organizing Committee

- Marta D'Elia
- Johnny Guzman
- Brittany Hamfeldt
- Michael Neilan
- Maxim Olshanskiy
- Sara Pollock
- Abner Salgado
- Valeria Simoncini

##### Connecting Higher-Order Statistics and Symmetric Tensors

Jan 8 - 12, 2024

This workshop focuses on connections between higher-order statistics and symmetric tensors, and their applications to machine learning, network science, and other domains. Higher-order statistics refers to the study of correlations between three or more covariates. This is in contrast to the usual mean and covariance, which are based on one and two covariates.

Higher-order statistics are needed to characterize complex data distributions, such as mixture models. Symmetric tensors, meanwhile, are multi-dimensional arrays. They generalize covariance matrices and affinity matrices and can be used to represent higher-order correlations. Tensor decompositions extend matrix factorizations from numerical linear algebra to multilinear algebra. Recently tensor-based approaches have become more practical, due to the availability of bigger datasets and new algorithms.

The workshop brings together applied mathematicians, statisticians, probabilists, machine learning experts, and computational... (more)

##### Organizing Committee

- Joe Kileel
- Tamara Kolda
- Joao Pereira

##### Computational Tools for Single-Cell Omics

Dec 11 - 15, 2023

Single-cell assays provide a tool for investigating cellular heterogeneity and have led to new insights into a variety of biological processes that were not accessible with bulk sequencing technologies. Assays generate observations of many different molecular types and a grand mathematical challenge is to devise meaningful strategies to integrate data gathered across a variety of different sequencing modalities. The first-order approach to do this is to analyze the projected data by clustering. Keeping more refined shape information about the data enables more meaningful and accurate analysis. Geometric methods include (i) Manifold learning: Whereas classical approaches (PCA, metric MDS) assume projection to a low-dimensional Euclidean subspace, manifold learning finds coordinates that lie on a not necessarily flat or contractible manifold. (ii) Topological data analysis: Algebraic topology provides qualitative descriptors of global shape. Integrating these descriptors across feature... (more)

##### Organizing Committee

- Elham Azizi
- Andrew Blumberg
- Lorin Crawford
- Bianca Dumitrascu
- Antonio Moretti
- Itsik Pe'er

##### Extending Inferences to a New Target Population

Nov 17 - 19, 2023

Estimators of various causal or statistical quantities are usually constructed with a particular target population in mind, that is, the population about which the investigators intend to draw inferences (e.g., decide on the implementation of a treatment strategy or use algorithm-derived predictions). Typically, however, the data used for estimation comes from a population that differs from the target population. How to ensure or evaluate whether the estimates generalize to the target population is a question that has received substantial attention in many scientific disciplines, but with the fields not always connecting with one another on overlapping challenges and solutions. This workshop will bring together experts from different disciplines to present state-of-the-science methods to address generalizability and discuss key challenges, and open problems.

##### Organizing Committee

- Issa Dahabreh
- Jon Steingrimsson
- Elizabeth Stuart