Abstract: Neural network training is a challenging nonconvex optimization to analyze its convergence. While many studies focus on proving the linear convergence of the empirical risk, the analysis of the population risk is limited. In this talk, I will present my paper, Curse of Dimensionality in Neural Network Optimization, which addresses this problem regarding the target function's regularity. I will start with a brief overview of the neural network optimization literature and introduce Barron spaces, which are crucial for explaining the training convergence of shallow neural networks in the mean-field regime. This is a joint work with Professor Haizhao Yang (UMD).
Abstract: I will discuss joint work with Juan Felipe Ariza Mejia, Ionuţ Chifan, and Denis Osin on McDuff factors. Recall that a II1 factor is called McDuff if it absorbs the hyperfinite II1 factor under tensor product. McDuff factors are generally considered soft, partially due to the presence of non-trivial central sequences. However, we construct McDuff group factors that exhibit rigidity. More precisely, we construct a non-amenable group G such that for any group H satisfying L(H) ∼= L(G), there exists an amenable group A with H ∼= G ⊕ A.
Abstract: Fluorescence microscopy and single-molecule fluorescent methods have played a crucial role in shedding light on various subcellular mechanisms and providing insights into different subcellular structures and their functions. However, these techniques still face multiple challenges in data analysis, including high photon budget requirements, rigorous noise treatment, model selection, and others. In this seminar, I will discuss my research on leveraging tools from Bayesian framework to address questions in single-molecule localization microscopy, particle tracking and spectral imaging.
Abstract: We begin with a brief overview of the rapidly developing research area of active matter, a.k.a. active materials. These materials are intrinsically out of equilibrium resulting in novel physical properties whose modeling requires the development of new mathematical tools. We present a free boundary PDE model a cytoskeleton of a moving cell. The key features of our model are the Keller-Segel cross-diffusion term and nonlocal boundary conditions. We first present a recent result on the nonlinear stability of stationary and traveling wave solutions in this model. We discuss novel mathematical features of this free boundary model with a focus on non-self-adjointness, which plays a key role in the spectral stability analysis. We next consider the model above with nonlinear diffusion and prove this nonlinearity results in the change of the bifurcation from subcritical to subcritical. leading that to two drastically different scenarios of the onset of the cell motion. Finally we derive an explicit formula that governs the change of the bifurcation type in terms of measurable physical parameters and therefore can be used for both qualitative and quantitative biological predictions.
This work resulted in two published papers with A. Safsten and V. Rybalko ( Phys. Rev . E, 2022, Transactions of AMS, 2023) as well as recent papers with A. Safsten and L. Truskinovsky (ARMA, 2025, accepted subject to revision) and with O. Krupchytski and T. Laux (Nonlinear Science, 2025, accepted subject to revision).
Abstract: The talk presents the first rigorous error analysis of an unfitted finite element method for linear parabolic problems posed on time-dependent domains that may undergo topological changes. The domain evolution is assumed to be smooth away from a critical time, at which the topology may change. To accommodate such transitions in the error analysis, we introduce several structural assumptions on the evolution of the domain in the vicinity of the critical time. These assumptions guarantee a specific control over the variation of a solution norm in time, even across singularities, and form the foundation for the numerical analysis. We demonstrate the applicability of our assumptions with examples of level-set domains undergoing topological transitions and discuss cases where analysis fails. The theoretical error estimate is supported by the results of a numerical experiment. Questions that remain open will be outlined.
Abstract: The Beilinson-Bloch conjecture is a generalization of the Birch and Swinnerton-Dyer conjecture, which relates the ranks of Chow groups of smooth projective varieties over global fields to the order of vanishing of L-functions. We construct classes of non-isotrivial hypersurfaces over global function fields for which the conjecture can be verified. These include some quartic K3 surfaces, whose groups of zero-cycles of degree zero we prove to be finite. We also prove that the Chow motive of a smooth cubic threefold over any field has a certain summand coming from its intermediate Jacobian, which is an associated abelian fivefold. We thus deduce the Birch and Swinnerton-Dyer conjecture for the intermediate Jacobians of cubic threefolds constructed in the previous step. Finally, combining this case of BSD with results of Roulleau and Geisser, we prove some new cases of the Tate conjecture over finite fields.
Abstract: Mean curvature flow, the gradient flow of the area functional, is the most natural geometric heat flow for embedded hypersurfaces. Being non linear, the flow develops singularities, at which it stops being smooth. One fundamental, often delicate, question for such non linear flows is that of backwards uniqueness. In this talk I will discuss recent backwards uniqueness results, obtained jointly with Josh Daniels-Holgate, which can address some singularities. I will also compare these results to (commonly more robust) forward uniqueness results, and also to the situation in  other equations.
Abstract: We describe a model for a network time series whose evolution is governed by an underlying stochastic process, known as the latent position process, in which network evolution can be represented in Euclidean space by a curve, called the Euclidean mirror. We define the notion of a first-order changepoint for a time series of networks, and construct a family of latent position process networks with first-order changepoints. We show how a spectral estimate of the associated Euclidean mirror can localize these changepoints and provide simulated and real data examples of such localization.
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