Navigating the Frontiers of Mathematics: A Glimpse into UCLA's Math Seminar
Mathematics, a realm of abstract thought and rigorous logic, constantly evolves through the dedicated efforts of researchers and scholars. UCLA's math seminar serves as a vibrant hub where these individuals converge to share their latest discoveries, explore novel ideas, and engage in stimulating discussions. This article offers a glimpse into the diverse topics presented at a UCLA math seminar, showcasing the breadth and depth of contemporary mathematical research.
Unveiling Hidden Structures: Recovering Planted Matchings in High Dimensions
One fascinating area of exploration involves the problem of recovering an unknown "planted matching." Imagine a set of randomly scattered points in a multi-dimensional space, $\mathbb{R}^d$. Now, picture these points being subtly shifted or perturbed. The challenge lies in reconstructing the original correspondence - the "matching" - between the initial and displaced points.
Lucas R. presented recent progress on establishing results which hold under general distributional assumption for both the the initial positions and noise. Some recent works have established results for the error rates for this problem under Gaussian assumptions for both initial positions and noise at different scaling regimes of sample size and dimension. More precisely, he showed a general recipe to establish lower bounds via showing the existence of large matchings in random geometric graphs, which leads to simplified and generalized proofs of previous results. Time allowing he also made some remarks regarding sufficient conditions for perfect recovery in high-dimensions for models where the noise is not isotropic.
The Nonlinear Universe: Computational Methods for Complex Systems
Beyond the realm of abstract geometry lies the study of complex systems, ubiquitous in both nature and human-designed environments. These systems, often governed by nonlinear equations, present formidable challenges to traditional analytical methods. The overarching goal of research in this area is to leverage advanced computational methods with fundamental theoretical analysis to model the nonlinear behavior of systems that are not otherwise amenable to integrable systems techniques.
Examples include: Studies of superfluidity and superconductivity in ultra-cold atomic physics (e.g., Bose-Einstein condensation), extreme and rare events (e.g., tsunamis and rogue waves), and collapse phenomena in optics (e.g., light propagation through a medium without diffraction). Researchers have developed computational methods for bifurcation analysis that explain the structure of the parameter space of these systems and continuation methods (pseudo-arclength and Deflated Continuation Method (DCM)) for efficient tracking of solution branches and connecting them to physical observations. In a talk, a wide pallete of results that were obtained by using the developed computational methods were presented. Specifically, inconspicuous solutions of the Nonlinear Schrödinger (NLS) equation were discovered by developing DCM specifically for NLS to uncover previously unknown behavior and weakly nonlinear unstable solutions that are potential targets for experimental verification. Furthermore, a novel Kuznetsov-Ma breather (time-periodic) solution to the discrete and non-integrable NLS equation relevant to predicting periodic extreme and rare events in optical systems was discovered by employing pseudo-arclength continuation. The combination of perturbation methods with pseudo-arclength continuation enabled the elucidation of collapsing waveforms associated with the 1D focusing NLS and Korteweg-de Vries equations.
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Dynamics and Sturmian Hamiltonians: Exploring Continuity Properties
Motivated by the spectral analysis of Sturmian Hamiltonians, research delves into their underlying dynamical systems. This class of subshifts admits a natural extension to a larger family parameterized by the unit interval. A talk focused on studying continuity properties of this parametrization with respect to the Hausdorff distance. Starting from the well known fact that these discontinuities arise at rational parameters only, a pass to a non-Euclidean metric in order to characterize limits at rational parameters was shown.
Curvature and Complex Spaces: Characterizing Cohomology Projective Spaces
In the realm of differential geometry, a celebrated result by Sui-Yau states that manifolds with positive bisectional curvature are biholomorphic to complex projective space. A talk introduced new curvature conditions that provide characterizations of cohomology complex projective spaces. For example, the curvature tensor of a Kaehler manifold induces an operator on symmetric holomorphic 2-tensors, called Calabi operator. This operator is the identity for complex projective space with the Fubini Study metric. It was shown that a compact n-dimensional Kaehler manifold with n/2-positive Calabi curvature operator has the rational cohomology of complex projective space. The complex quadric shows that this result is sharp if n is even. This talk is based on joint work with K. Broder, J. Nienhaus, P. Petersen, J.
Generalizations of Fermat's Last Theorem: Delving into Arithmetic Geometry
Fermat's Last Theorem, a famous problem that stumped mathematicians for centuries, has spurred countless advancements in number theory. The seminar explored generalizations of this theorem, delving into the field of arithmetic geometry.
Numerous generalizations of FLT were discussed -- for instance, for fixed integers $a,b,c \geq 2$ satisfying $1/a + 1/b + 1/c < 1$, Darmon and Granville proved the single generalized Fermat equation $x^a + y^b = z^c$ has only finitely many coprime integer solutions. Conjecturally something stronger is true: for $a,b,c \geq 3$ there are no non-trivial solutions. More generally, the subfield "arithmetic geometry", and in particular the geometric intuitions that underlie the conjectural framework of modern number theory were discussed.
The Limits of Estimation: Impossibility in High-Dimensional Eigenvector Alignment
Large covariance estimation plays a central role in high-dimensional statistics and underpins much of modern multivariate data analysis. A common modeling strategy introduces pairwise correlations among variables through a small number of latent factors, yielding a spiked covariance structure in which a few large eigenvalues separate from a bulk spectrum. The associated eigenvectors form a population quantity of interest, denoted B, while their empirical counterparts, H, are typically estimated via the leading eigenvectors of the sample covariance matrix. The matrix BTH, encoding the alignment between sample and population eigenvectors, offers a fine-grained measure of estimation accuracy.
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An analysis considered the high-dimensional, low-sample-size (HDLSS) regime, where the sample size is fixed and the dimension grows. Drawing on the HDLSS theory developed in recent PCA literature, the focus here is not on asymptotic limits of BTH, but rather on the feasibility of estimating it from the observed data. An impossibility theorem was established, showing that under mild conditions, no consistent estimator of BTH exists when the number of spiked eigenvalues exceeds one. This result highlights fundamental limitations of spectral methods in extreme dimensional settings and contributes to the broader understanding of PCA behavior in the HDLSS regime, which is increasingly relevant due to practical constraints in experimental design and the nonstationarity of real-world time series data.
The speaker, Hubeyb Gurdogan, is a Hedrick Assistant Adjunct Professor in the Department of Mathematics at UCLA. He earned his Ph.D. in Financial Engineering from the Department of Mathematics at Florida State University in December 2021, under the supervision of Alec Kercheval. Prior to his doctoral studies, he received M.S. degrees in Mathematics from Syracuse University and Bilkent University. Following his Ph.D., Dr. Gurdogan held a research appointment at UC Berkeley’s Center for Risk Management Research (CDAR), where he led a collaborative project with the Swiss Re Institute. In this role, he served as research lead on a project focused on modeling supply chain risk propagation, with particular emphasis on pricing Non-Damage Business Interruption (NDBI) insurance products. Dr.
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