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4:00pm to 5:00pm - RH 340N - Applied and Computational Mathematics Liza Rebrova - (Princeton University) Randomized iterative sketch-and-project methods as efficient large-scale linear solvers Randomized Kaczmarz methods -- popular special case of the sketch-and-project optimization framework -- solve linear systems through iterative projections onto randomly selected equations, resulting in exponential expected convergence via cheap, local updates. While known to be effective in highly overdetermined problems or under the restricted data access, identifying generic scenarios where these methods are advantageous compared to classical solvers remained open. I will talk about better ways to quantify the convergence of the sketch-and-project methods and present our recent work showing that -- if properly designed -- these methods outperform Krylov-based solvers complexity-wise for both square and rectangular systems. Since the proposed solvers quickly capture the large outlying singular values of the linear system they are particularly advantageous for approximately low-rank systems common in machine learning (e.g., kernel matrices, signal-plus-noise models). Our approach combines novel spectral analysis of randomly sketched projection matrices with classical numerical analysis techniques, such as including momentum, adaptive regularization, and memoization. |
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4:00pm to 5:00pm - RH 340P - Geometry and Topology Dan Minahan - (UChicago) The second rational homology of the Torelli group The Torelli subgroup of the mapping class group of a surface is the subgroup acting trivially on the first homology of the surface. We will discuss recent joint work with Andrew Putman where we compute the second rational homology of the Torelli group for closed, orientable surfaces of genus at least 6. Along the way we compute the first twisted homology of the Torelli group with coefficients in the abelianization of the maximal abelian cover of the closed surface. We also prove some new purely representation theoretic results about certain infinitely presented representations of the symplectic group |
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3:00pm to 3:50pm - RH 306 - Analysis Serap Öztop Kaptanoglu - (Istanbul University, Istanbul, Turkey) Geometric Aspects of Renorming of Orlicz Spaces
Let Φ be an Orlicz function and LΦ(X,Σ,μ) be the corresponding Or- licz space on a nonatomic, σ-finite, complete measure space (X, Σ, μ). Our main interest is to rellate analytic properties of LΦ(X,Σ,μ) to various geo- metric conditions centered around convexity as the norm of the Orlicz space varies. We give criteria for the strict convexity of Orlicz spaces equipped with s-norms which lead to results about smoothness and Gateaux differen- tiability of s-norms. We also provide a new perspective for studying extremal problems on Orlicz spaces. (Joint work with Esra Ba ̧sar, Hu ̈seyin Uysal, and S ̧eyma Ya ̧sar.)
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4:00pm to 5:00pm - Rowland Hall 340P - Differential Geometry Shih-Kai Chiu - (UCI) Calabi-Yau manifolds with maximal volume growth Calabi–Yau manifolds with maximal volume growth arise naturally as smoothings or resolutions of certain log terminal singularities and play a central role in understanding the formation of singularities in degenerating families of compact Calabi–Yau manifolds, particularly through bubbling phenomena. In this talk, I will survey recent progress on the existence and classification of such non-compact Calabi–Yau manifolds. |
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3:00pm to 4:00pm - 510R Rowland Hall - Combinatorics and Probability Jorge Garza Vargas - (Caltech) A new approach to strong convergence: nearly optimal expanders with little randomness In joint work with Chi-Fang Chen, Joel Tropp, and Ramon van Handel, we developed a new method for establishing strong convergence. In this talk I will explain what strong convergence is, and, as an application of our results I will discuss a simple way of generating nearly optimal expanders using very little randomness. |
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2:15pm to 3:05pm - ISEB 1200 - Geometry and Topology Steve Trettel - (University of San Francisco) Seeing Space(time) For two thousand years geometry was synonymous with the perfectly flat expanse imagined by Euclid. But nineteenth‑century investigations into the parallel postulate lifted a veil from our eyes, revealing the richer realms charted by Gauss and Riemann. In this talk we’ll take an “insider’s tour” of those curved landscapes. We begin by thinking carefully about what it means to see, and use this to step inside new geometries, by tracing light rays along their geodesics. Modern computing affords us the ability to make this thought experiment a reality, with interactive ray-traced demos and experiments. Using these, we will explore curved spaces important to modern mathematics, and physics, including the curved spacetime we live in.
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4:00pm to 5:00pm - ISEB 1010 - Distinguished Lectures Bin Yu - (UC Berkeley) Veridical Data Science and Alignment in Medical AI Alignment and trust are crucial for the successful integration of AI in healthcare including digital twin projects, a field involving diverse stakeholders such as medical personnel, patients, administrators, public health officials, and taxpayers, all of whom influence how these concepts are defined. This talk presents a series of collaborative medical case studies where AI algorithms progressively become, from transparency to more opaque thus with increasing difficulty of alignment assessment. These range from tree-based methods for trauma diagnosis, to LLM-based emergency department co-pilot, and mechanistic circuits for structured data extraction from pathology reports. They are guided by Veridical Data Science (VDS) principles—Predictability, Computability, and Stability (PCS)—for the goal of building trust and interpretability, enabling doctors to assess alignment. The talk concludes with a discussion on applying VDS to medical foundation models and next steps for evaluating AI algorithm alignment in healthcare. |