This Week in Mathematics


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Entries for this week: 5

Monday April 21, 2025

Analysis and PDE Seminar
Global perturbation of isolated equivariant chiral skyrmions from the harmonic maps
- Slim Ibrahim, University of Victoria
Time: 3.05pm Room: LOV 232
More Information
Abstract/Desc: Isolated skyrmion solutions to the 2D Landau-Lifshitz equation with the Dzyaloshinskii-Moriya interaction, Zeeman interaction, and easy-plane anisotropy are considered. In a wide range of parameters illustrating the various interaction strengths, we construct exact solutions and examine their monotonicity, exponential decay, and stability using a careful mathematical analysis. We also estimate the distance between the constructed solutions and the harmonic maps by exploiting the structure of the linearized equation and by proving a resolvent estimate for the linearized operator that is uniform in extra implicit potentials. This is joint work with Ikkei Shimizu (U. Kyoto).

Tuesday April 22, 2025

ACM
Adaptive Reduced-Order Modeling using Statistical Learning
- Xiao Liu, Gatech
Time: 3:05 pm Room: 0231
Abstract/Desc: Projection-based model reduction is among the most widely adopted approaches for constructing parametric Reduced-Order Models (ROMs). Utilizing snapshot data from solving full-order governing equations, the Proper Orthogonal Decomposition (POD) computes the optimal basis modes that span a low-dimensional subspace where the ROM resides. Challenges arise when one would like to investigate how systems behave differently over the parameter space (in design, diagnosis, control, uncertainty quantification and real-time operations). In this case, the optimal basis needs to be efficiently updated so as to adapt ROM that can accurately capture the variation of a system's behavior over its parameter space. In this talk, we focus on a Projected Gaussian Process (pGP) model and formulate the problem of adapting POD basis as a supervised statistical learning problem, for which the goal is to learn a mapping (injective) from the parameter space to the Grassmann Manifold that contains the optimal vector subspaces. To establish such a relationship, a mapping is found between the Euclidean space and the horizontal space of an orthogonal matrix that spans a reference subspace in the Grassmann Manifold. Then, a second mapping from the horizontal space to the Grassmann Manifold is established through the Exponential/Logarithm maps between the manifold and its tangent space. Finally, given a new parameter, the conditional distribution of a vector can be found in the Euclidean space using the Gaussian Process (GP) regression, and such a distribution is then projected to the Grassmann Manifold that enables us to find the optimal subspace, i.e., POD basis, for the new parameter. Compared with existing interpolation method, the proposed statistical learning approach allows us to optimally estimate (or tune) model parameters given data (i.e., the prediction/interpolation becomes problem-specific), and quantify the uncertainty associated with the prediction. Numerical examples are presented to demonstrate the advantages of the proposed pGP for adapting POD basis against parameter changes.

Thursday April 24, 2025

Algebra seminar
How the icosahedral group leads to the Zometool construction toy
- Mark van Hoeij, FSU
Time: 3:05PM Room: 232
Abstract/Desc: Typical building constructions use 90 degree angles, so their symmetries are elements of the symmetry group of a cube. In the talk I will show how A5, the symmetry group of the icosahedron, leads to a fascinating construction toy called zometool. I will use groups, rings and modules (the talk should be easy to follow for any student who has taken GRV I+II) to obtain a theorem that describes exactly when zometool pieces fit together in constructions.

Financial Mathematics Seminar
Point Process Theory and High Frequency Trading
- Farez Siddiqui , FSU
Time: 3:05 Room: LOV 0231
Abstract/Desc: In most of today's electronic liquid markets (stocks, futures, foreign exchange), a continuous-time double auction procedure is carried out using a structure called the limit order book (LOB) in which the use of traditional designated market-makers (the kind that would "quote and clear" at specified levels so that transactions would still occur) is avoided by automatically triggering a trade once any buyer/seller pair agree on submitted price. A typical LOB is meant to be observable by all participants in real-time, making it a valuable source of information when making inferences on short-term price movements. Indeed, the sequence of events that affect the state of the LOB is referred to as "order flow" and will result in tangible transactions to take place, directly affecting price dynamics. As such, well-directed quantitative analysis may lead to reliable signals for short-term price inference, making the LOB a valuable object for analysis. Beyond the obvious financial incentives associated with careful study of the LOB, quantitative analysis of the LOB is a source of rich mathematical interest as well. LOB data is typically collected on extremely high-frequency intervals on the order of milliseconds to even nanoseconds. For many assets the historic data is available for as far back as the initial implementation of the LOB. The complex nature of the LOB coupled with its ample and highly detailed data describing its evolution makes it a rich testing ground for a plethora of statistical phenomena that are seen in other markets. This talk will provide a brief overview of some of the techniques that have been employed in modeling the LOB, as well as begin covering the foundational theory behind the particularly popular approach of modeling the order flow that drives the changes in the LOB as a multivariate point process

Friday April 25, 2025

Financial Math Candidacy Exam
Modeling the Limit Order Book using state-dependent Hawkes processes
- Farez Siddiqui, FSU
Time: 1:00 pm Room: LOV 204A

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