The ability to accurately forecast the dynamics of a complex oscillating system is fundamental to the study of many physical and biological systems. This is especially important for understanding and predicting the bodies natural 24 hour oscillations in physiology and behavior known as circadian rhythms. The response of oscillators to external perturbations can be traced out either theoretically or experimentally into phase response curves. In this work, we study the application of model discovery techniques to complex nonlinear oscillator systems using phase response data. Using simulated data we examine how well these techniques perform when the perturbations are generalized from the training data. Finally, we discuss the application of these techniques to the study of circadian rhythms and discuss some preliminary results for using wearable data (apple watch, fitbit) to build personalized models of circadian rhythms. 

At MIC this week, a panel of instructors and support staff will share some of their favorite technologies for teaching remotely.  The Zoom meeting link is https://arizona.zoom.us/j/91785371355.

In this talk, I will discuss the Sparse Identification of Nonlinear Dynamics (SINDy), a model discovery tool developed at the University of Washington. I will give an overview of the method as well as discussing its potential and limitations in dynamical systems and partial differential equations through concrete examples. This will be complemented by a discussion of methods for more rigorous implementation of SINDy when the governing dynamics are entirely unknown as well as using it as a tool to motivate theoretical research for modifications to existing models.

Zoom: https://arizona.zoom.us/j/98593835992    
Password: 021573

Abstract: Diffusion and mixing are two fundamental phenomena that arise in a wide variety of applications. In this talk we quantitatively study the interaction between diffusion and mixing in the context of problems arising in fluid dynamics. The first question we address is how fast the energy can decay in the advection diffusion equation. Even though this is a simple linear equation, the energy decay rate is intrinsically related to the mixing properties of the advecting velocity field, and there are many unresolved open questions. I will present a few recent results involving both upper and lower bounds, and then consider applications to studying the long time dynamics of a few model non-linear equations.

Zoom: https://arizona.zoom.us/j/91826900125  Password:  "Locute"

The pentagram map is a discrete dynamical system on the space of plane polygons. It was proved to be integrable by Ovsienko, Schwartz, and Tabachnikov. Later on, Gekhtman, Shapiro, Vainshtein, and Tabachnikov gave another proof of integrability using a combinatorial model of networks drawn on a cylinder. Recently, a generalization of the pentagram map was defined where the polygons are in some Grassmannian manifold. I will discuss a noncommutative version of the techniques of Gekhtman et. al., and how it can be used to define a Poisson structure for the Grassmannian pentagram map and to show integrability.

The talk will be via Zoom at: https://utoronto.zoom.us/j/99576627828

Passcode: 448487

The Kontsevich-Zorich cocycle is a fundamental object in the study of the chaotic dynamics of the Teichmueller geodesic flow on moduli spaces of hyperbolic surfaces. A dynamical form of the Law of Large
Numbers shows that the exponential growth rate of the norm of the cocycle along typical orbits converges to its space average. In this talk, we show that these growth rates also obey a Central Limit Theorem. This provides one of the first instances in which such a result holds for deterministic cocycles over flows and generalizes a long history of results in the study of random walks on groups. The main ingredient is a spectral gap result which is established via the theory of anisotropic Banach spaces. No background in Teichmueller theory or spectral theory will be assumed.

For a real four manifold M, the S-duality conjecture of Vafa-Witten (1994) predicts that the S-transformation sends the gauge group SU(r)-invariants counting instantons on M to the Langlands dual gauge group SU(r)/Z_r-invariants counting SU(r)/Z_r-instantons on M; and both of the invariants satisfy modularity properties. This is a generalization of electro-magnetic duality in physics. On mathematics side the SU(r)-Vafa-Witten invariants have been constructed by Tanaka-Thomas using the moduli space of semistable Higgs bundle or sheaves on a smooth complex projective surface underlying M. In this talk I will present the idea of using moduli space of twisted sheaves and twisted Higgs sheaves on a projective surface to define the Langlands dual gauge group SU(r)/Z_r-Vafa-Witten invariants, and provide the proposal to prove the S-duality conjecture of Vafa-Witten for algebraic surfaces. A particular case of K3 surface is proved under this proposal.

In the mid-1800s, Kummer observed some striking congruences between certain values of the Riemann zeta function, which have important consequences in algebraic number theory, in particular for unique factorization in certain rings. In spite of its potential, this topic lay mostly dormant for nearly a century until it was revived by Iwasawa in the mid-1950s. Since then, advances in arithmetic geometry and number theory (in particular, for modular forms, certain analytic functions that play a central role in number theory) have enabled substantial extension to congruences in the context of other arithmetically significant data, and this has remained an active area of research. In this talk, I will survey old and new tools for studying such congruences. I will conclude by introducing some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art.

This will be a Zoom Meeting and the link information will be shared to all@math the week of the talk.  If you are not on this list and would like to attend, please email headoffice@math.arizona.edu

Ming Jiang

Andrew Gillette

Jeffrey Hittinger

This is a series of NSF Noyce Seminars for the AZ Noyce Mathematics Teaching (MaTh) Scholars Project in Secondary Mathematics Education (PI: Cynthia Anhalt; Co-PI: Marta Civil; Co-PI: Rebecca McGraw; Co-PI: Jennifer Wolfe)

Please email Jen Wolfe (jwolfe628@math.arizona.edu) if you are interested in attending.

Title: Maximum sampled likelihood estimation for informative  subsample

Abstract: Subsampling is an effective approach to extract useful information from massive data sets when computing resources are limited. Existing investigations focus on developing better sampling procedures and deriving probabilities with higher estimation efficiency. After a subsample is taken from the full data, most available methods use an inverse probability weighted target function to define the estimator. This type of weighted estimator reduces the contributions of more informative data points, and thus it does not fully utilize information in the selected subsample. This paper focuses on parameter estimation with selected subsample, and proposes to use the maximum sampled likelihood estimator (MSLE) based on the sampled data. We established the asymptotic normality of the MSLE, and prove that its variance covariance matrix reaches the lower bound of asymptotically unbiased estimators. Specifically, the MSLE has a higher estimation efficiency than the weighted estimator. We further discuss the asymptotic results with the L-optimal subsampling probabilities, and illustrate the estimation procedure with generalized linear models. Numerical experiments are provided to evaluate the practical performance of the proposed method.

Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in these areas are driven by neural networks. There are also growing applications in finance, engineering, robotics, and medicine. Despite their immense success in practice, there is limited mathematical understanding of neural networks. Our work shows how neural networks can be studied via stochastic analysis, and develops approaches for addressing some of the technical challenges which arise. We analyze both multi-layer and one-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. In the case of single layer neural networks, we rigorously prove that the empirical distribution of the neural network parameters converges to the solution of a nonlinear partial differential equation. In addition, we rigorously prove a central limit theorem, which describes the neural network's fluctuations around its mean- field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. For multilayer neural networks we rigorously derive the limiting behavior of the neural networks output. We also prove convergence to the global minimum under appropriate conditions. We demonstrate the theoretical results in the study of the evolution of parameters in the well known MNIST and CIFAR10 data sets.

Advisor: Andrew Gillette, Department of Mathematics, University of Arizona and Computational Scientist, Lawrence Livermore National Laboratory

Place:  Zoom:  https://arizona.zoom.us/j/92695578455    Password: TBA

Abstract: TBA

Zoom: TBA

Peter Behroozi

Title: Assessment of Case Influence in Support Vector Machine

Abstract: Support vector machine (SVM) is a very popular technique for classification. A key property of SVM is that its discriminant function depends only on a subset of data points called support vectors. This comes from the representation of the discriminant function as a linear combination of kernel functions associated with individual cases. Despite the direct relation between each case and the corresponding coefficient in the representation, the influence of cases and outliers on the classification rule has not been examined formally. Borrowing ideas from regression diagnostics, we define case influence measures for SVM and study how the classification rule changes as each case is perturbed. To measure case sensitivity, we introduce a weight parameter for each case and reduce the weight from one to zero to link the full data solution to the leave-one-out solution. We develop an efficient algorithm to generate case-weight adjusted solution paths for SVM. The solution paths and the resulting case influence graphs facilitate evaluation of the influence measures and allow us to examine the relation between the coefficients of individual cases in SVM and their influences comprehensively. We present numerical results to illustrate the benefit of this approach.

Abstract: TBA

Zoom: TBA

Daniel Bienstock

This is a series of NSF Noyce Seminars for the AZ Noyce Mathematics Teaching (MaTh) Scholars Project in Secondary Mathematics Education (PI: Cynthia Anhalt; Co-PI: Marta Civil; Co-PI: Rebecca McGraw; Co-PI: Jennifer Wolfe)

Please email Jen Wolfe (jwolfe628@math.arizona.edu) if you are interested in attending.