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Entries for this week: 6

Tuesday September 10, 2024

Geometry and Topology Seminar [url]
Existence of quasigeodesic Anosov flows in hyperbolic 3-manifolds
- Sergio Fenley, FSU
Time: 3:05 Room: 301
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Abstract/Desc: A quasigeodesic in a manifold is a curve so that when lifted to the universal cover is uniformly efficient up to a bounded multiplicative and added error in measuring length. A flow is quasigeodesic if all flow lines are quasigeodesics. We prove that an Anosov flow in a closed hyperbolic manifold is quasigeodesic if and only if it is not R-covered. Here R-covered means that the stable 2-dim foliation of the flow, lifts to a foliation in the universal cover whose leaf space is homeomorphic to the real numbers. There are many examples of quasigeodesic Anosov flows in closed hyperbolic 3-manifolds. There are consequences for the continuous extension property of Anosov foliations, and the existence of group invariant Peano curves associated with Anosov flows.

Wednesday September 11, 2024

Biomathematics Seminar
Biofilm Rheology and Uncertainty
- Nick Cogan, FSU
Time: 3:05 Room: 232 Love

Thursday September 12, 2024

Financial Mathematics Seminar
Financial Applications of the Signature Method
- Tyler Gorczycki and Denny Serdarevic, Florida State University
Time: 3:05 Room: LOV 231
Abstract/Desc: We explore the challenges of calibrating a linear combination of signature terms, expressed as polynomials using the Signature Method, to accurately model stochastic processes in financial applications, aiming to improve the calibration of pricing and Implied Volatility models. Originating from rough path theory, the signature method offers a time-invariant transformation that captures the characteristics of multidimensional time series. This method relies on iterated integrals, which form a natural linear basis for functions on data streams, allowing for the extraction of meaningful information in a non-parametric manner, without relying on traditional statistical methods. We extend the Signature Method to financial applications, extracting and tuning key terms to build accurate models. In our simulations, we use Brownian motions as the primary process when computing the signature, providing a foundation for modeling stochastic behavior. Specifically, we assume the asset's price follows a Geometric Brownian Motion and compute the signature of this process. We then extend this approach to calibrate both volatility and price using the Heston Model. For our simulated options contracts, we use Monte Carlo simulations and apply the Black-Scholes model to determine the Implied Volatility of the options. Ultimately, this approach improves our ability to model financial derivatives, leading to more reliable pricing and Implied Volatility models. (This is based on REU 2024 Summer project; Mentor: Dr. Qi Feng.)

Algebra seminar [url]
3-dimensional hypersurfaces and intermediate Jacobians
- Benson Farb, University of Chicago
Time: 3:05 Room: LOV 301
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Abstract/Desc: Clemens-Griffiths discovered in 1971 a surprising connection between two questions (all terms will be defined in the talk) 1. When is a 3-dimensional hypersurface rational (i.e. birational to P^3)? 2. When is a flat torus isometric to the Jacobian of an algebraic curve? I will explain this connection and how I used it recently to give a simple proof of the (already known) irrationality of the general quartic 3-fold. More importantly, we will see some fundamental classical ideas in action: resolution of singularities, the theory of periods, automorphism groups of varieties, and a beautiful example of Felix Klein. This talk will be aimed at beginners (like the speaker) in algebraic geometry.

Friday September 13, 2024

Mathematics Colloquium [url]
Hyperbolic dynamics, partially hyperbolic dynamics and transverse foliations
- Sergio Fenley, FSU
Time: 3:05 Room: Lov 101
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Abstract/Desc: Anosov diffeomorphisms and flows are the primary example of hyperbolic dynamics, which is an extremely important class of dynamical systems. A partially hyperbolic diffeomorphism is one for which besides stable and unstable bundles, there is another "in between bundle" - the center bundle. In dimension 3, these are essentially the only systems that can be robustly transitive and/or stably ergodic. A primary example is the time one map of an Anosov flow. The talk will also cover how transverse foliations can help understand these diffeomorphisms.

Machine Learning and Data Science Seminar
Maximum principles for PDEs and numerical schemes based on gradient ascent
- Arash Fahim, FSU Mathematics
Time: 1:20pm Room: Love 106
Abstract/Desc: We introduce a gradient descent method for solving nonlinear PDEs that satisfy maximum principle. Maximum principle allows to approximate nonlinear PDEs with a sequence of solutions of parametrized simple linear PDEs. Therefore, efficient methods for linear PDEs can be used to approach the solution of the nonlinear ones. We also show how the solution to each of the parametrized linear PDEs provides the direction of maximum increase of the solution with respect to the parameter, and hence, the gradient ascent methodology.

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