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4:00pm to 5:00pm - Zoom - https://uci.zoom.us/j/8706895753 - Special Colloquium Angxiu Ni - (Tsinghua University) Backpropagation and adjoint differentiation of chaos Computing the derivative of long-time-averaged observables with respect to system parameters is a central problem for many numerical applications. Conventionally, there are three straight-forward formulas for this derivative: the pathwise perturbation formula (including the backpropagation method used by the machine learning community), the divergence formula, and the kernel differentiation formula. We shall explain why none works for the general case, which is typically chaotic (also known as the gradient explosion phenomenon), high-dimensional, and small-noise. We present the fast response formula, which is a 'Monte-Carlo' type formula for the parameter-derivative of hyperbolic chaos. It is the average of some function of u-many vectors over an orbit, where u is the unstable dimension, and those vectors can be computed recursively. The fast response overcomes all three difficulties under hyperbolicity assumptions. Then we discuss how to further incorporate the kernel differentiation trick to overcome non-hyperbolicity. |
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4:00pm to 5:30pm - RH 340N - Logic Set Theory Michael Hehmann - (UC Irvine) Algorithmic Randomness Part 2 We give an introductory survey of the theory of algorithmic randomness. The primary question we wish to answer is: what does it mean for a set of natural numbers, or equivalently an infinite binary sequence, to be random? We will focus on three intuitive paradigms of randomness: (i) a random sequence should be hard to describe, (ii) a random sequence should have no rare properties, and (iii) a random sequence should be unpredictable, in the sense that we should not be able to make large amounts of money by betting on the next bit of the sequence. Using ideas from computability theory, we will make each of these three intuitive notions of randomness precise and show that the three define the same class of sets.
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2:00pm to 3:00pm - RH 440R - Dynamical Systems Victor Kleptsyn - (CNRS, University of Rennes 1, France) Non-stationary Anderson Localization Using recent results on dynamics of non-stationary random matrix products, we establish spectral and dynamical localization for 1D Schrodinger operators with potentials given by independent but not identically distributed random variables. |
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4:00pm to 5:00pm - ISEB 1200 - Differential Geometry Bo Guan - (Ohio State University) The roles of concavity, symmetry and sub-solutions in geometric PDEs In this talk we discuss the roles of concavity, symmetry and subsolutions in the study of fully nonlinear PDEs, especially those on real or complex manifolds with connection to geometric problems. We shall report some of our results along the line, which give the optimal conditions for the existence of classical solutions, either of the Dirichlet problem, or of equations on closed manifolds. If time permits, we shall also discuss the possibility to weaken or extend these conditions, and a class of equations involving differential forms of higher rank, more specifically real (p, p) forms for p > 1 on complex manifolds. Part of the talk is based on joint work with my student Mathew George. |
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2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability Paul Duncan - (Hebrew University of Jerusalem) Homological Percolation on a Torus Many well-studied properties of random graphs have interesting generalizations to higher dimensions in random simplicial complexes. We will discuss a version of percolation in which, instead of edges, we add two (or higher)-dimensional cubes to a subcomplex of a large torus at random. In this setting, we see a phase transition that marks the appearance of giant "sheets," analogous to the appearance of giant components in graph models. This phenomenon is most naturally described in the language of algebraic topology, but this talk will not assume any topological background. Based on joint work with Ben Schweinhart and Matt Kahle. |
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1:00pm to 2:00pm - RH 340R - Algebra Reginald Anderson - (Claremont McKenna) Presentations of Derived Categories The bounded derived category of coherent sheaves gives an isomorphism invariant for smooth projective varieties which are Fano or general type. King conjectured in 1997 that any smooth complete toric variety has a strong, full exceptional collection of line bundles. King's conjecture was proven false, but determining the exact criteria for a smooth complete (weak-)Fano toric variety to have a strong full exceptional collection of line bundles remains open, as does finding a method to generate this SFEC of line bundles for varieties which admit such a collection. In this talk I will give a cellular resolution of the diagonal for smooth projective toric varieties which yields a SFEC of line bundles on unimodular toric surfaces, as well as for a smooth non-unimodular example in dimension 2. This is joint work with Gabriel Kerr. |
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3:00pm to 4:00pm - RH 306 - Number Theory Junxian Li - (UC Davis) Two dimensional delta symbol and applications to quadratic forms The delta symbol developed by Duke-Friedlander-Iwaniec and |
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1:00pm - DBH 1200 - Graduate Seminar Kenneth Ascher - (UC Irvine) Moduli spaces in algebraic geometry |