This Week in Mathematics

Department of Mathematics

This Week in Mathematics


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Current Week [Feb 22, 2026 - Feb 28, 2026]

Today:

Biomath lab meetings
Solving the 2D Poisson equation using Finite Element Methods
    - Jonathan Engle, FSU
Time: 2:00 Room: LOV102
Abstract/Desc: In this talk, we will discuss the construction of the 2D Finite Element solver for Poisson's equation. This FEM solver will use triangular mesh generation, reference-to-physical mappings, triangle quadrature methods, and global assembly to solve the 2D Poisson equation. To validate our results, we perform a convergence study in the L^2 and H^1 seminorms. Additionally, we will compare these results and methods with a classical Integral equations solver.

Biomathematics Seminar
Optimal Control and its Application to Finding Optimal Treatment For HIV-Related and Triple-Negative Breast Cancers
    - Zachary Deskin, FSU Mathematics
Time: 3:05 Room: Love 232
Abstract/Desc: Optimal control is a useful mathematical tool in applied mathematics. It optimizes variables that can be controlled within a dynamical system. In this talk, we will briefly discuss what optimal control is and how it can be solved. We will look at two applications of optimal control: HIV-related cancer and triple-negative breast cancer. The biology of each type of cancer will be discussed and how optimal control can be used to find their respective optimal treatments.

Applied and Computational Mathematics [url]
Classifying material topology in real space using matrix homotopy
    - Alexander Cerjan, Center for Integrated Nanotechnologies, Sandia National Laboratories
Time: 3:05 Room: 231
Abstract/Desc: The classification of topological phases of matter has traditionally relied on assumptions about the underlying material, requiring that the material be an insulator with a well-defined band structure. However, many experimentally relevant systems violate one or more of these assumptions, raising fundamental questions about how topology should be defined and diagnosed in realistic settings. In this talk, I will present an overview of these outstanding challenges and describe how a real-space, operator-based approach called the spectral localizer framework provides a platform-agnostic unifying theory capable of addressing these challenges. In particular, the spectral localizer enables the formulation of local, energy-resolved topological markers that remain well-defined in systems lacking a global spectral gap, translational symmetry, or sharp interfaces. This perspective allows one to meaningfully classify gapless heterostructures, such as photonic systems embedded in air, and can be applied directly to continuum models without first finding a low-energy approximation. Beyond stable topological phases, the spectral localizer also offers new insights into recently identified classes of topology, including fragile topological phases that induce Wannier obstructions resulting in novel forms of quantum materials. I will discuss applications of these ideas to nonlinear polariton systems to achieve reconfigurable topological interfaces and to electronic platforms such as two-dimensional electron gases in semiconductor heterostructures to show the emergence of Hofstadter’s butterfly for intermediate scales of the periodic potential’s strength, demonstrating the versatility of the spectral localizer as a general tool for topological classification in modern materials physics.

Entries for this week: 8

Monday February 23, 2026

Stochastic Computing
Optimal mean-time path planning for unmanned underwater vehicles under uncertain ocean dynamics
- Jonathan Valyou, Florida State University
Time: 3:05pm Room: LOV232
Abstract/Desc: Unmanned underwater vehicles (UUV) integrate ocean forecasts with path planning algorithms to identify energy- or time-minimizing paths that allow for task completion. A crucial account in determining the optimal path is the local ocean current velocity field. However, ocean forecast models for this field may disagree introducing uncertainty. In this talk, we will detail a deterministic Hamilton-Jacobi approach to path planning and propose a novel system of Hamilton-Jacobi (HJ) partial differential equations that incorporates the forecast uncertainty and yields the optimal mean reachability travel time along with the corresponding controls to ascertain the associated optimal path. We will propose an extension of the Lax-Friedrich’s Fast Sweeping method for solving a single HJ PDE to a system of HJ PDEs. Numerical examples will be provided for verification and to demonstrate the impact of uncertainty on the path planning result.

Tuesday February 24, 2026

Disc-presheaves and smooth structures
- Alexander Kupers, University of Toronto
Time: 3:05PM Room: Zoom
More Information
Abstract/Desc: The configuration spaces of points with framings in a manifold, together with natural point-forgetting and -splitting maps, assemble to an object known as a Disc-presheaf. To what extent is this Disc pre-sheaf sensitive to the smooth structure? I will explain joint work with Ben Knudsen, Manuel Krannich, and Fadi Mezher on this problem.

Wednesday February 25, 2026

Biomath lab meetings
Solving the 2D Poisson equation using Finite Element Methods
- Jonathan Engle, FSU
Time: 2:00 Room: LOV102
Abstract/Desc: In this talk, we will discuss the construction of the 2D Finite Element solver for Poisson's equation. This FEM solver will use triangular mesh generation, reference-to-physical mappings, triangle quadrature methods, and global assembly to solve the 2D Poisson equation. To validate our results, we perform a convergence study in the L^2 and H^1 seminorms. Additionally, we will compare these results and methods with a classical Integral equations solver.

Biomathematics Seminar
Optimal Control and its Application to Finding Optimal Treatment For HIV-Related and Triple-Negative Breast Cancers
- Zachary Deskin, FSU Mathematics
Time: 3:05 Room: Love 232
Abstract/Desc: Optimal control is a useful mathematical tool in applied mathematics. It optimizes variables that can be controlled within a dynamical system. In this talk, we will briefly discuss what optimal control is and how it can be solved. We will look at two applications of optimal control: HIV-related cancer and triple-negative breast cancer. The biology of each type of cancer will be discussed and how optimal control can be used to find their respective optimal treatments.

Applied and Computational Mathematics [url]
Classifying material topology in real space using matrix homotopy
- Alexander Cerjan, Center for Integrated Nanotechnologies, Sandia National Laboratories
Time: 3:05 Room: 231
Abstract/Desc: The classification of topological phases of matter has traditionally relied on assumptions about the underlying material, requiring that the material be an insulator with a well-defined band structure. However, many experimentally relevant systems violate one or more of these assumptions, raising fundamental questions about how topology should be defined and diagnosed in realistic settings. In this talk, I will present an overview of these outstanding challenges and describe how a real-space, operator-based approach called the spectral localizer framework provides a platform-agnostic unifying theory capable of addressing these challenges. In particular, the spectral localizer enables the formulation of local, energy-resolved topological markers that remain well-defined in systems lacking a global spectral gap, translational symmetry, or sharp interfaces. This perspective allows one to meaningfully classify gapless heterostructures, such as photonic systems embedded in air, and can be applied directly to continuum models without first finding a low-energy approximation. Beyond stable topological phases, the spectral localizer also offers new insights into recently identified classes of topology, including fragile topological phases that induce Wannier obstructions resulting in novel forms of quantum materials. I will discuss applications of these ideas to nonlinear polariton systems to achieve reconfigurable topological interfaces and to electronic platforms such as two-dimensional electron gases in semiconductor heterostructures to show the emergence of Hofstadter’s butterfly for intermediate scales of the periodic potential’s strength, demonstrating the versatility of the spectral localizer as a general tool for topological classification in modern materials physics.

Thursday February 26, 2026

Algebra seminar
Log-concavity of Polynomials Arising from Equivariant Cohomology
- Yupeng Li, Michigan State
Time: 3:05pm Room: Zoom
Abstract/Desc: A remarkable result of Brändén and Huh tells us that volume polynomials of projective varieties are Lorentzian polynomials. The dual notion of covolume polynomials was introduced by Aluffi by considering the cohomology classes of subvarieties of a product of projective spaces. In this talk, we shall address the equivariant cohomology classes of torus-equivariant subvarieties of the space of matrices. For a large class of torus actions, we shall show that the polynomials representing these classes (up to suitably changing signs) are covolume polynomials in the sense of Aluffi. If time permits, we shall present a description of the cohomology rings of smooth complex varieties in terms of a generic Macaulay inverse system over the integers. This is based on joint work with Yairon Cid-Ruiz and Jacob Matherne.

Financial Math
Identifying Patterns in Financial Markets: Extending the Statistical Jump Model for Regime Identification
- Petter Kolm, NYU Courant Institute of Mathematical Sciences
Time: 3.05 Room: LOV 231
Abstract/Desc: Regime-driven models are popular for addressing temporal patterns in both financial market performance and underlying stylized factors, wherein a regime describes periods with relatively homogeneous behavior. Recently, statistical jump models have been proposed to learn regimes with high persistence, based on clustering temporal features while explicitly penalizing jumps across regimes. In this article, we extend the jump model by generalizing the discrete hidden state variable into a probability vector over all regimes. This allows us to estimate the probability of being in each regime, providing valuable information for downstream tasks such as regime-aware portfolio models and risk management. Our model’s smooth transition from one regime to another enhances robustness over the original discrete model. We provide a probabilistic interpretation of our continuous model and demonstrate its advantages through simulations and real-world data experiments. The interpretation motivates a novel penalty term, called mode loss, which pushes the probability estimates to the vertices of the probability simplex thereby improving the model’s ability to identify regimes. We demonstrate through a series of empirical and real world tests that the approach outperforms traditional regime-switching models. This outperformance is pronounced when the regimes are imbalanced and historical data is limited, both common in financial markets.

Friday February 27, 2026

Data Science and Machine Learning Seminar [url]
Mathematical Exploration and Discovery at Scale
- Javier Gómez-Serrano , Brown
Time: 1:20 Room: Zoom
Abstract/Desc: Machine learning is transforming mathematical discovery, enabling advances on longstanding open problems. In this talk, I will discuss AlphaEvolve, a general-purpose evolutionary coding agent that uses large language models to autonomously discover old and new mathematical constructions and potentially go beyond them. AlphaEvolve tackles a wide variety of problems across analysis, geometry, combinatorics, and number theory. In some instances, AlphaEvolve is also able to generalize results for a finite number of input values into a formula valid for all input values. Furthermore, we are able to combine this methodology with Deep Think and AlphaProof in a broader framework where the additional proof-assistants and reasoning systems provide automated proof generation and further mathematical insights. This illustrates how general-purpose AI systems can systematically successfully explore broad mathematical landscapes at an unprecedented speed, leading us to do mathematics at scale.

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February