This Week in Mathematics

Department of Mathematics

This Week in Mathematics


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

Entries for this week: 4

Wednesday February 25, 2026

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
TBA
- Yupeng Li, Michigan State
Time: 3:05pm Room: Zoom
Abstract/Desc: TBA

Financial Math
TBA
- Petter Kolm, NYU Courant Institute of Mathematical Sciences
Time: 3.05 Room: LOV 231

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