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4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics Chunyang Liao - (UCLA) Radius of Information for Two Intersected Centered Hyperellipsoids and Implications in Optimal Recovery from Inaccurate Data For objects belonging to a known model set and observed through a prescribed linear process, we aim at determining methods to recover linear quantities of these objects that are optimal from a worst-case perspective. Working in a Hilbert setting, we show that, if the model set is the intersection of two hyperellipsoids centered at the origin, then there is an optimal recovery method which is linear. It is specifically given by a constrained regularization procedure whose parameters, short of being explicit, can be precomputed by solving a semidefinite program. This general framework can be swiftly applied to several scenarios: the two-space problem, the problem of recovery from l2-inaccurate data, and the problem of recovery from a mixture of accurate and l2-inaccurate data. With more effort, it can also be applied to the problem of recovery from l1-inaccurate data. For the latter, we reach the conclusion of existence of an optimal recovery method which is linear, again given by constrained regularization, under a computationally verifiable sufficient condition. Experimentally, this condition seems to hold whenever the level of l1-inaccuracy is small enough. We also point out that, independently of the inaccuracy level, the minimal worst-case error of a linear recovery method can be found by semidefinite programming. |
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4:00pm to 5:50pm - RH 340N - Logic Set Theory Julian Eshkol - (UC Irvine) The combinatorics of Large Cardinals At and above the level of measurability, large cardinal notions are typically characterized by the existence of certain elementary embeddings of the universe into an inner model. We may contrast this with smaller large cardinal notions, whose characterizations tend to be strictly combinatorial. In this series of talks, we survey results from Magidor's thesis, in which he shows that the large notion of supercompactness can also be viewed combinatorially, and in this light supercompactness is seen to be a natural strengthening of ineffability. We will also survey modern results which show how these strong combinatorial principles can be forced to hold at small successor cardinals. |
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2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Ji Zeng - (UCSD) Variation of no-three-in-line problem The famous no-three-in-line problem by Dudeney more than a century ago asks whether one can select $2n$ points from the grid $[n]^2$ such that no three are collinear. We present two results related to this problem. First, we give a non-trivial upper bound for the maximum size of a set in $[n]^4$ such that no four are coplanar. Second, we characterize the behavior of the maximum size of a subset such that no three are collinear in a random set of $\mathbb{F}_q^2$, that is, the plane over the finite field of order $q$. We discuss their proofs and related open problems. |
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9:00am to 10:00am - Zoom - Inverse Problems Martin Burger - (Deutsches Elektronen-Synchrotron DESY and Universität Hamburg) Image Reconstruction – The Dialectic of Modelling and Learning |
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1:00pm - 306 Rowland Hall - Harmonic Analysis Francis White - (UCI) Bounds for Eigenfunctions of Semiclassical Pseudodifferential Operators with Double Characteristics In mathematical physics, non self-adjoint operators often arise in connection with processes that do not conserve energy. From the mathematical point of view, such operators are of interest because they arise as the quantizations of complex-valued symbols, and the associated classical dynamics must be extended into the complex domain. In this talk, I will discuss the special class of non self-adjoint pseudodifferential operators with double characteristics, and I will present some new results on L^p-bounds for eigenfunctions of such operators in the semiclassical limit. The main tools used are the Fourier-Bros-Iagolnitzer (FBI) transform and microlocal analysis in exponentially weighted spaces of holomorphic functions. |
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3:00pm to 4:00pm - RH 306 - Number Theory Sudhir Ghorpade - (IITB) Shellability and Homology of q-matroids with applications to Rank Metric Codes The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. It is a classical result that matroid complexes, that is, simplicial complexes formed by the class of independent subsets in a matroid, are shellable. This has some bearing on the study of linear block codes, especially in regard to their Betti numbers and generalized weight enumerator polynomials. We now know that q-matroids have close connections with rank metric codes in a manner similar to the connection between matroids and codes. A recent result establishes shellability of q-matroid complexes and also determines the homology of these complexes in many cases. The determination of homology has now been completed for arbitrary q-matroid complexes. We will outline these developments whlie making an attempt to keep the prerequisites at a minimum. The contents of this talk are based on a joint work with Rakhi Pratihar and Tovohery Randrianarisoa, and also with Rakhi Pratihar, Tovohery Randrianarisoa, Hugues Verdure and Glen Wilson. |
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1:00pm - DBH 1200 - Graduate Seminar Chris Davis - (UC Irvine) Planning for Spring/Fall |