Speaker: Shengjing Xu (UPenn) - When: Mon, September 21, 2026 - 3:00pm Where: Kirwan Hall 3206
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Abstract: Moduli spaces of Higgs bundles are holomorphic symplectic varieties arising naturally in hyperkähler geometry, algebraic integrable systems, and the Langlands program. In this talk, I will describe a construction of holomorphic Lagrangians in Higgs moduli spaces using colored Baker–Akhiezer divisor data associated with a Higgs field.
In type A, the construction begins with a Higgs bundle together with a distinguished line subbundle. Iterating the Higgs field generates a cyclic flag, and the failure of this flag to be transverse is recorded by an effective divisor. On the spectral side, this divisor lifts naturally to a generalized Baker–Akhiezer divisor. Fixing the divisor gives rise to holomorphic Lagrangian subvarieties and, in families, to Lagrangian correspondences with Hilbert schemes of points on the cotangent bundle of the underlying curve.
I will then explain how this picture extends to Higgs bundles for a general semisimple group of adjoint type. A choice of Borel reduction replaces the cyclic flag, while the ordinary effective divisor is replaced by a colored divisor recording the vanishing of the simple-root components of the Higgs field. On the spectral side, the analogue of the Baker–Akhiezer divisor is naturally organized into a Weyl-group orbit of colored divisors on the cameral cover. This also suggests a way to find the lagrangian correspondence in G higgs side.
Finally, I will discuss the connection with the Dolbeault and Geometric Langlands. Colored defect data also appear naturally in the Whittaker coefficient functor and in the Drinfeld–Laumon constructions of geometric Langlands. I will explain how these suggest a possible relation between Higgs-side Lagrangian geometry and the classical limit of Langlands.
Speaker: Dave Levermore (U. Maryland) - When: Mon, September 21, 2026 - 3:00pm Where: Kirwan Hall 1313
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Abstract: The kinetic theory of gases was very controversial at the dawn of the twentieth century. Hilbert posed his Sixth Problem to spur the development of mathematics that might help resolve these controversies and advance the understanding of how macroscopic physics emerges from microscopic physics. In particular, he cited the work of Boltzmann, which was built upon the work of Clausius and Maxwell. We will describe the Sixth Problem as Hilbert saw it, through the lens of the work of Clausius, Maxwell, and Boltzmann. It is still an active area of research. Recent results by Deng, Hani and Ma helped earn a 2026 Fields Medal for Yu Deng. We will describe these and other significant recent advances, but show that the problem largely remains open.
Speaker: Dr. Alex Safsten (University of Maryland) - https://sites.google.com/view/alexsafsten?usp=sharing When: Tue, September 22, 2026 - 12:30pm Where: Kirwan Hall 3206
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Abstract: Crawling cells such as keratocytes exhibit a striking transition from nearly circular stationary states to persistent directed motion, while maintaining a coherent shape. Experiments also suggest the possibility of bistability, in which both stationary and rapidly moving states are stable under the same conditions. In this talk, I will discuss a free-boundary model of cell motility that couples the motion of the cell boundary to the transport of myosin, which generates contractile stress inside the cell. I will describe how a symmetric stationary state undergoes a symmetry-breaking bifurcation to traveling-wave solutions, and how the geometry of this bifurcation is tied to their stability. I will then discuss how spectral information can be promoted to nonlinear stability, first in a one-dimensional version of the model and then as part of an ongoing program in two dimensions. Finally, I will turn to the question of bistability. Density-dependent myosin diffusion can reverse the direction of the bifurcation, and a combination of asymptotic analysis and computer-assisted estimates gives a mechanism by which the traveling-wave branch can bend back, allowing stable rest and stable motion to coexist.
Speaker: Alex Safsten (UMD) - When: Tue, September 22, 2026 - 2:00pm Where: Kirwan Hall 3206
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Abstract: Training a classifier typically means minimizing a smooth loss function, such as cross-entropy, while the quantity we ultimately care about is often accuracy. It is tempting to assume that decreasing loss should therefore improve accuracy—but this is not true in general: loss can decrease even as correctly classified points become misclassified. In this talk, I will describe a mathematical notion of stability of training that addresses this mismatch. The key quantity is the classification margin, given by the difference between the network output corresponding to the correct class and its largest competitor. Stability can then be related to how these margins are distributed across the training set. In particular, concentrations of points with similar small or negative margins can allow substantial losses in accuracy even while the loss function decreases. The paper develops quantitative conditions preventing such concentrations and shows that they imply that once sufficiently high accuracy is reached, it remains high throughout subsequent training. These conditions can in turn be related to geometric properties of the training data through a doubling-type “no small isolated data clusters” condition, giving a criterion that depends only on the dataset rather than on the evolving network parameters. I will focus on the basic mathematical ideas behind these results, the simple examples showing why stability is nontrivial, and the connection between the geometry of the data and the behavior of the classifier.
Speaker: Todd Rowland (UMD) - When: Tue, September 22, 2026 - 2:30pm Where: Kirwan Hall 3206
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Abstract: Recent publications have outlined guidelines for AI in higher education, notably the MIT report https://aiandeducation.mit.edu/report/ We will discuss the main ideas from some of these publications, and how it can apply to us.
Speaker: Christian Rosendal (UMD) - When: Tue, September 22, 2026 - 3:30pm Where: Kirwan Hall 1311
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Abstract: We study the dependence of the transfinite Schreier hierarchy on the
choice of fundamental sequences for the countable limit ordinals.
We prove that the bounding number is equal to ω1 exactly
when every infinite compact interval family has infinite
intersection with some member of the hierarchy. We also prove that
the dominating number is equal to ω1 exactly when the fundamental
sequences may be chosen so that every compact family of finite subsets
of N*= {2, 3, . . .} is contained in one Schreier family. The latter equiv-
alence combines Fremlin’s cofinality theorem for compact subsets of the
rationals with an absorption construction for compact families.
Speaker: Gianluca Fabiani (Johns Hopkins University) - When: Tue, September 22, 2026 - 3:30pm Where: Kirwan Hall 3206
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Abstract:Random feature Random feature methods offer a mathematically tractable class of neural-network models: the nonlinear feature map is fixed a priori, typically via random sampling, and learning reduces to a linear least-squares problem. Random projection neural networks (RPNNs) are a prototypical example, where hidden-layer parameters are fixed and only output coefficients are optimized against a finite-dimensional random feature space. This linear structure has enabled efficient approaches to function and operator approximation, as well as numerical solvers for ODEs and PDEs, at costs competitive with classical discretization methods. Yet the mathematical foundations remain incomplete: what approximation properties are guaranteed, when are the resulting systems well-conditioned? This talk develops a theoretical framework addressing these questions. We revisit universal approximation and density results, including connections to reproducing kernel Hilbert spaces, then establish quantitative convergence rates matching classical polynomial approximation rates. We then show that strong approximation properties do not imply numerical stability: constructive examples reveal deterministic ill-conditioning of collocation matrices, exposing a fundamental tension between approximation and stability. Finally, we turn to ODEs, where theory is far less developed, analyzing the numerical linear stability of physics-informed random projection methods in the classical numerical-analysis sense. We close with a set of open problems in approximation, conditioning, and stability, and argue that random feature methods offer a rare setting where such questions can be posed, and answered, with full mathematical rigor, offering a template for tackling analogous questions across machine learning more broadly.
Speaker: Zhengwu Zhang (UNC Chapel Hill) - https://zhengwu.github.io/ When: Thu, September 24, 2026 - 3:30pm Where: Kirwan Hall 1311
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Abstract: Comparing brain connectivity across people presupposes that cortical locations correspond, yet standard registration matches folding patterns, which are only loosely related to the brain's wiring. This talk presents two methods that align cortical surfaces using structural connectivity itself. Representing connectivity as a continuous, atlas-free density over pairs of cortical locations, ENCORE uses a square-root transform and Fisher-Rao geometry to obtain an inverse-consistent, penalty-free registration on the product of two spheres. ConSEAL keeps that geometry but moves the observed streamline endpoints directly, making the alignment grid-robust and diffeomorphic by construction. On Human Connectome Project data, alignment by wiring improves bundle correspondence, reduces nuisance variance, localizes individual variability to association cortex, and improves prediction of cognition. I will end with open statistical problems on registration uncertainty and point processes on manifolds.
Joint work with Martin Cole, Yang Xiang, William Consagra, Anuj Srivastava, and Xing Qiu.