This Week in Mathematics


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Today:

Applied and Computational Math
A generative approach for simulating Wasserstein geometric flows
- Shu Liu, Florida State University
Time: 3:05pm Room: LOV 306
Abstract/Desc: Wasserstein geometric flows (WGFs) constitute a class of time-evolution partial differential equations that play a fundamental role in modeling and simulating physical systems. In this talk, we present a sampling-friendly, optimization-free approach for simulating WGFs by leveraging generative models from deep learning. Specifically, we project the WGF defined on the probability manifold onto a finite-dimensional parameter space induced by the generative model, in a manner that faithfully mimics the dynamics of the original geometric flow. The resulting system of ordinary differential equations, referred to as the parametrized WGF, can then be efficiently solved using classical numerical integration methods. This framework enables direct generation of samples from the time marginals of the WGFs, even in high-dimensional settings. In addition, we establish error analysis results that provide accuracy guarantees for the proposed method. The talk will conclude with a brief discussion of future research directions and potential applications.

Geometry and Topology [url]
Stability theory of Persistence Modules
- Mujtaba Ali, FSU
Time: 3:05 Room: 231
More Information
Abstract/Desc: Topological Data Analysis (TDA) provides a robust framework for extracting meaningful topological features from complex data. A central tool in this framework is persistent homology, which captures how homological features of a dataset appear and disappear across scales. The resulting algebraic structures, known as persistence modules, can be compared using various metrics that quantify their similarity. In this talk, I will introduce the basic ideas of persistent homology and discuss the interleaving distance and bottleneck distance, two fundamental metrics defined on the space of persistence modules and their associated barcodes. I will outline the celebrated result establishing their equivalence in the one-parameter case. Finally, I will discuss how these notions extend to multi-parameter persistence modules, where the situation becomes more intricate, and explain some of the challenges and partial results known about their relationship in higher dimensions.

Entries for this week: 6

Tuesday October 21, 2025

Applied and Computational Math
A generative approach for simulating Wasserstein geometric flows
- Shu Liu, Florida State University
Time: 3:05pm Room: LOV 306
Abstract/Desc: Wasserstein geometric flows (WGFs) constitute a class of time-evolution partial differential equations that play a fundamental role in modeling and simulating physical systems. In this talk, we present a sampling-friendly, optimization-free approach for simulating WGFs by leveraging generative models from deep learning. Specifically, we project the WGF defined on the probability manifold onto a finite-dimensional parameter space induced by the generative model, in a manner that faithfully mimics the dynamics of the original geometric flow. The resulting system of ordinary differential equations, referred to as the parametrized WGF, can then be efficiently solved using classical numerical integration methods. This framework enables direct generation of samples from the time marginals of the WGFs, even in high-dimensional settings. In addition, we establish error analysis results that provide accuracy guarantees for the proposed method. The talk will conclude with a brief discussion of future research directions and potential applications.

Geometry and Topology [url]
Stability theory of Persistence Modules
- Mujtaba Ali, FSU
Time: 3:05 Room: 231
More Information
Abstract/Desc: Topological Data Analysis (TDA) provides a robust framework for extracting meaningful topological features from complex data. A central tool in this framework is persistent homology, which captures how homological features of a dataset appear and disappear across scales. The resulting algebraic structures, known as persistence modules, can be compared using various metrics that quantify their similarity. In this talk, I will introduce the basic ideas of persistent homology and discuss the interleaving distance and bottleneck distance, two fundamental metrics defined on the space of persistence modules and their associated barcodes. I will outline the celebrated result establishing their equivalence in the one-parameter case. Finally, I will discuss how these notions extend to multi-parameter persistence modules, where the situation becomes more intricate, and explain some of the challenges and partial results known about their relationship in higher dimensions.

Thursday October 23, 2025

Financial Math
Deep Learning of Alpha Term Structures from the Order Book
- Petter Kolm, NYU Courant Institute of Mathematical Sciences
Time: 3.05 Room: 105
Abstract/Desc: In recent years, deep learning (DL) models have experienced notable success in predicting high-frequency returns in equities by leveraging extensive order book data and directly extracting features from it. This marks a notable departure from current industry practice, where features are often manually crafted. In this talk, I discuss two of our articles on this topic, addressing several practical open questions in this area, such as determining the most suitable network architecture and input selection for forecasting returns at multiple horizons (e.g. alpha term structures), optimizing the width and depth of neural network components to enhance performance, defining the appropriate historical data window size, and evaluating the benefits of incorporating time as a feature in the models. We evaluate the effectiveness of four DL models in forecasting high-frequency alpha term structures across various settings: a simple LSTM, a multi-head LSTM, an LSTM Seq2Seq without attention, and an LSTM Seq2Seq with attention. We find that surpassing the performance of a simple LSTM in the return forecasting task is surprisingly challenging.

Algebra seminar
Weighted blow-up in nature: wall crossings for Log-Hilbert stacks of points on curves
- Veronica Arena, Cambridge
Time: 3:05pm Room: Zoom
Abstract/Desc: Weighted blow-ups are a birational transformation that naturally appears in moduli spaces. One instance where this happens, is when studying the logarithmic Hilbert scheme of points on a curve $C$ equipped with a log structure. Today we will give a quick introduction to both weighted blow-ups and logarithmic Hilbert schemes of points on curves. Then we will focus our attention on the examples of two and three points on $(P^1|0)$ and will describe the wall crossings between the classical Hilbert scheme of points and the logarithmic ones via weighted blow-ups.

Friday October 24, 2025

Machine Learning and Data Science Seminar
Latest Architectures, Hierarchical Reasoning Model
- Gordon Erlebacher, FSU Scientific Computing
Time: 12pm Room: 499 Dirac Science Library
Abstract/Desc: This is joint with the Scientific Computing AI Seminar. Note the special time and location! Abstract: This talk links standard LLM practice to the Hierarchical Reasoning Model (HRM). Starting from embeddings—vectors that map symbols to geometry—we show how an autoregressive decoder yields next-token predictions under causal masking. We contrast explicit Chain-of-Thought with implicit reasoning emerging in hidden states and attention. HRM introduces nested loops: a fast inner loop refining hypotheses and a slower outer loop coordinating multi-step inference—bringing System-1 speed and System-2 deliberation into one design. We discuss why compact, task-specialised HRMs can rival larger models, the trade-offs (e.g., per-task retraining), and possible extensions.

Mathematics Colloquium [url]
Information Scrambling, Circuit Complexity, and Thermodynamics: From Black Holes to Cryptography
- Eduardo Mucciolo , UCF
Time: 3:05 Room: Lov 101
Abstract/Desc: In this colloquium, I will present some recent ideas and results concerning quantum computing, information scrambling, and how we can use thermodynamic concepts to formulate complexity in classical and quantum computing. I will argue that classical and quantum circuits can scramble information as fast and as thorough as black hole, which is considered Nature’s ultimate scrambler. I will also show how the thermodynamics of mixing applied to reversible circuits can leads to an important application in cryptography.

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Department of Mathematics

This Week in Mathematics