## Fall 2023

**Emily Gunawan (UMass Lowell): Triangulations and maximal almost rigid modules over gentle algebras**

September 13, 11 a.m. - noon, Room: Southwick Hall 303C

Abstract: A type A path algebra is an algebra whose basis is the set of all paths in an orientation of a type A Dynkin diagram. We introduce a new class of modules over a type A path algebra and call them maximal almost rigid (MAR). They are counted by the Catalan numbers and are naturally modeled by triangulations of a polygon. The endomorphism algebras of the MAR modules are classical tilted algebras of type A. Furthermore, their oriented flip graph is the oriented exchange graph of a smaller type A cluster algebra which is known to define a Tamari or Cambrian poset. The type A path algebras are special cases of gentle algebras, a family of finite-dimensional algebras whose indecomposable modules are classified by certain walks called strings and bands. We generalize the notion of MAR to this setting. First, we use the surface models developed by Opper, Plamondon, and Schroll and by Baur and Coelho Simões to show that the MAR modules correspond bijectivity to triangulations of a marked surface. We then show that the endomorphism algebra of a MAR module is the endomorphism algebra of a tilting module over a bigger gentle algebra. Finally, we define an oriented flip graph of the MAR modules and conjecture that it is acyclic. This talk is based on joint projects with Emily Barnard, Raquel Coelho Simões, Emily Meehan, and Ralf Schiffler.

Emily Gunawan's Website

**Adar Kahana (Brown University): Hybrid Iterative Method based on Deep Operator Networks for Solving Differential Equations**

September 27, 11 a.m. - noon, Room: Southwick Hall 313

Abstract: Iterative solver of linear systems is a key component for numerical solutions of differential equations, playing an important role in numerical analysis and scientific computing. While there have been intensive studies on classical methods such as Jacobi, Gauss-Seidel, conjugate gradient, multigrid methods and their more advanced variants, there is still a pressing need to develop faster, more robust, and more reliable solvers. Based on recent advances in scientific deep learning for operator regression, we propose a hybrid numerical solver for solving differential equations. Through a series of numerical experiments, we show that this hybrid solver is capable of providing fast, accurate solutions for a wide class of differential equations, some of which otherwise diverge when using other numerical solvers.

Adar Kahana's LinkedIn Page

**Abinand Gopal (Yale University): Direct solvers for variable-coefficient scattering problems**

October 18, 11 a.m. - noon, Room: Southwick Hall 313

Time-harmonic scattering in variable media can often be modeled by linear elliptic partial differential equations. Such equations pose several numerical challenges. For example, they can be intrinsically ill-conditioned, necessitate the imposition of radiation conditions, and produce pollution errors when discretized with standard finite difference or finite element methods.

To avoid these issues, it is often convenient to first reformulate the differential equation as an integral equation. The tradeoffs are that an integral operator with a singular kernel must be discretized and that the resulting linear system that must be inverted is dense. Sometimes, the latter issue can be handled using a fast matrix-vector product algorithm (e.g., the fast Fourier transform or the fast multipole method) paired with an iterative solver (e.g., GMRES). However, this approach can be prohibitively slow when there is a large amount of backscattering and when multiple incident fields are of interest. In these cases, it is better to use direct solvers.

In this talk, I describe some recent projects on developing direct solvers in this regime with applications to acoustic scattering and metasurface design. Various parts of this work are joint with Gunnar Martinsson (UT Austin), Bowei Wu (UMass Lowell), Wenjin Xue (Yale), Hanwen Zhang (Yale), Vladimir Rokhlin (Yale), and Owen Miller (Yale).

Abinand Gopal's Yale University Bio

**Cory Palmer (University of Montana): A generalization of derangements**

October 25, 11 a.m. - noon, Virtual Meeting

A derangement is a permutation with no fixed points (i.e.no element that appears in its original position). The problem of counting derangements was introduced by de Montmort in 1708 and solved by him in 1713. For large n, the number of derangements of an n-element set is approximately n!/e. The number of derangements of an n-element set can be realized as the number of perfect matchings in a complete bipartite graph K_{n,n} with a perfect matching removed. A related problem is the number of perfect matchings in the complete graph K_{2n} with a perfect matching removed. For large n, this value is approximately (2n-1)!! / sqrt{e}. In this talk we discuss these two parameters and a common generalization.

Cory Palmer's Website

**Andreas Seeger (University of Wisconsin-Madison): Old and new problems about spherical maximal functions**

November 1, 11 a.m. - noon, Room: Southwick Hall 313

We survey old and new problems and results on spherical means, regarding pointwise convergence, p-improving properties of local spherical maximal operators and consequences for sparse domination. We are especially interested in how the outcomes depend on suitable notions of fractal dimension of the dilation sets.

Andreas Seeger's Website

**Joint talk at UMass Lowell Center for Health Statistics Seminar Series: Jong Soo Lee (UMass Lowell): Some statistical analyses of head impacts and concussion in high school sports**

November 15, Noon - 1 p.m., Room: Coburn 275 (South Campus).

In this talk, we present the statistical results and challenges from the projects undertaken at Department of Kinesiology and Rehabilitation Science, University of Hawaii. The projects encompass various aspects of high school and college athlete health, but the focus will be on head impacts and concussion in high school football. The main project consists of analyzing the head impacts of football players between the athletes who adhered to a training intervention - the helmetless tackling - and the athletes who did not. Through thoughtful planning and careful analyses, we demonstrate that the helmetless tackling show reduction in head impacts that affect the concussion in athletes. We also discuss the statistical analyses conducted from secondary aims and peripheral projects.

Jong Soo Lee's UMass Lowell Bio

**Fruzsina Agocs (Flatiron Institute): Chirps and waves: adaptive high-order methods for oscillatory ODEs and PDEs**

November 29, 11 a.m. - Noon, Room: Southwick Hall 313

Oscillatory problems have long posed a challenge to numerical computation, whether they take the form of ordinary or partial differential equations (ODEs or PDEs), yet they are ubiquitous in applications ranging from engineering to astrophysics. Oscillations force algorithms to use more discretization nodes (thus more computational effort) as the frequency grows. In PDEs, there are additional difficulties associated with waves interacting with the geometry, e.g. corners, periodic boundaries, cavities.

In this talk, I will present two classes of fast, high-order accurate numerical methods to solve oscillatory problems. For ODEs, I introduce two algorithms that exploit asymptotic expansions when the solution oscillates, but behave as ``standard'' solvers otherwise, thus achieving O(1) (frequency-independent) runtime. I will show how they eliminate computational bottlenecks in early-universe astrophysics. For PDEs, I will focus on two-dimensional acoustic and electromagnetic scattering from a nonperiodic source by a periodic boundary. I show how these may be solved using boundary integral equation (BIE) methods, and by integrating over a family of quasiperiodic solutions. I will use this to discuss how dispersive trapped waves explain wave-guiding phenomena and an acoustic effect at Mayan pyramids.

Fruzsina Agocs' Simons Institute profile