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4:00pm to 5:00pm - RH306 - Applied and Computational Mathematics George Stepaniants - (Caltech) Learning Memory and Material Dependent Constitutive Laws The simulation of multiscale viscoelastic materials poses a significant challenge in computational materials science, requiring expensive numerical solvers that can resolve dynamics of material deformations at the microscopic scale. The theory of homogenization offers an alternative approach to modeling, by locally averaging the strains and stresses of multiscale materials. This procedure eliminates the smaller scale dynamics but introduces a history dependence between strain and stress that proves very challenging to characterize analytically. In the one-dimensional setting, we give the first full characterization of the memory-dependent constitutive laws that arise in multiscale viscoelastic materials. Using this theory, we develop a neural differential equation architecture, that simultaneously across a wide range of material microstructures, accurately predicts their homogenized constitutive laws, thus enabling us to simulate their deformations under forcing. We use the approximation theory of neural operators to provide guarantees on the generalization of our approach to unseen material samples.
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1:00pm to 2:00pm - RH 340N - Dynamical Systems Alexandro Luna - (UCI) On the Regularity of the Dimension of Cookie-Cutter-Like Sets Abstract: We provide an example of a one-parameter family of Cookie-Cutter-Like sets that are generated by a one-parameter family of sequences of analytic maps (varying analytically in the parameter), but for which, the Hausdorff dimension is not even differentiable as a function of the parameter. This motivates an interesting conjecture concerning the regularity of the dimension of the spectrum of a Sturmian Hamiltonian operator as a function of the coupling constant. This is a joint work with Victor Kleptsyn. |
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3:00pm - RH 440R - Logic Set Theory Julian Eshkol - (UCI) Bounding the Ultrafilter Number at Successors For an infinite cardinal κ, the ultrafilter number u_κ is the smallest size of a family that generates a uniform ultrafilter on κ. We say that u_κ is bounded if u_κ<2^κ. The case κ=ω is well understood, with the consistency of a bounded ultrafilter number at $\omega$ first noted in the early 1970s by Kunen. The method used in Kunen’s construction does not generalize to the case κ=ω_1, and Kunen asked whether u_{ω_1}<2^{ω_1} is even consistent; this remains wide open. The consistency of u_κ<2^κ for general uncountable κ has been studied extensively in the last decade. In this series of talks, we will survey some recent progress towards bounding the ultrafilter number at successor values of κ, with emphasis on a key theorem of Raghavan and Shelah. We will also discuss some crucial limitations to applying this theorem for a bounded ultrafilter number at any of the ω_n’s. |
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4:00pm to 5:00pm - 306 Rowland Hall - Differential Geometry Jingze Zhu - (UCI) Ancient solutions to the mean curvature flow in higher dimensions In this talk, we discuss recent developments in ancient solutions to the mean curvature flow in |
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3:00pm to 4:00pm - RH 340P - Combinatorics and Probability Collen Robichaux - (UCLA) Positivity of Schubert coefficients Schubert coefficients are nonnegative integers that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation. A related open problem seeks an algorithm to determine when a Schubert coefficient is positive. In this talk we explore this problem and present an algorithmic solution. This is joint work with Igor Pak. |
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9:00am to 9:50am - Zoom - Inverse Problems Giovanni Alberti - (University of Genoa) Non-zero constraints in PDEs and applications to hybrid inverse problems |
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1:00pm - RH 340N - Algebra William Ren - (UC Irvine) Learning seminar - representation theory |
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3:00pm to 4:00pm - RH 306 - Number Theory Yifeng Huang - (USC) Quot schemes of points on torus knot singularities (Joint with Ruofan Jiang and Alexei Oblomkov) The Hilbert scheme of points on planar singularities is an object with rich connections (q,t-Catalan numbers, HOMFLY polynomials, Oblomkov–Rasmussen–Shende conjecture). The Quot scheme of points is a higher rank generalization of the Hilbert scheme of points. As our main result, we prove that for the "torus knot singularity" $x^a = y^b$ with $\gcd(a,b)=1$, the Quot scheme admits a cell decomposition: every Birula-Białynicki stratum is “as nice as possible” despite poor global geometry. The proof uses two key properties of the rectangular‑grid poset: an Ext‑vanishing for certain quiver representations and a structural result on the poset flag variety. Time permitting, I will discuss a conjectured Rogers–Ramanujan type identity, whose sum side is a summation on (nested) $a \times b$ Dyck paths and product side has modulus $a+b$. |