Department of Mathematics,
University of California San Diego
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Math 248 - Analysis Seminar
Yu Deng
USC
Instability of the Couette flow in low regularity spaces
Abstract:
In an exciting paper, J. Bedrossian and N. Masmoudi established the stability of the 2D Couette flow in Gevrey spaces of index greater than 1/2. I will talk about recent joint work with N. Masmoudi, which proves, in the opposite direction, the instability of the Couette flow in Gevrey spaces of index smaller than 1/2. This confirms, to a large extent, that the transient growth predicted heuristically in earlier works does exist and has the expected strength. The proof is based on the framework of the stability result, with a few crucial new observations. I will also discuss related works regarding Landau damping, and possible extensions to infinite time.
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AP&M 7321
AP&M 7321
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Department of Mathematics,
University of California San Diego
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Center for Computational Mathematics Seminar
James Brannick
Penn State University
Algebraic Multigrid: Theory and Practice
Abstract:
This talk focuses on developing a generalized bootstrap algebraic
multigrid algorithm for solving sparse matrix equations. As a
motivation of the proposed generalization, we consider an optimal form
of classical algebraic multigrid interpolation that has as columns
eigenvectors with small eigenvalues of the generalized eigen-problem
involving the system matrix and its symmetrized smoother. We use this
optimal form to design an algorithm for choosing and analyzing the
suitability of the coarse grid. In addition, it provides insights into
the design of the bootstrap algebraic multigrid setup algorithm that we
propose, which uses as a main tool a multilevel eigensolver to compute
approximations to these eigenvectors. A notable feature of the approach
is that it allows for general block smoothers and, as such, is well
suited for systems of partial differential equations. In addition, we
combine the GAMG setup algorithm with a least-angle regression
coarsening scheme that uses local regression to improve the choice of
the coarse variables. These new algorithms and their performance are
illustrated numerically for scalar diffusion problems with highly
varying (discontinuous) diffusion coefficient and for the linear
elasticity system of partial differential equations.
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AP&M 2402
AP&M 2402
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Department of Mathematics,
University of California San Diego
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Food For Thought Seminar
Evangelos ``Vaki'' Nikitopoulos
UCSD
Algebratizing Differential Geometry: Linear Differential Operators
Abstract:
It is frequently the case that certain objects in differential topology/geometry can be described in purely algebraic terms, where the algebraic structures involved are constructed using the smooth structure(s) of the underlying manifold(s). For example, a common equivalent characterization of a smooth vector field on a smooth manifold $M$ is a derivation of the $\mathbb{R}$-algebra of smooth real-valued functions on $M$. I shall discuss this example and describe how it led me to a considerably more involved one: linear differential operators on $M$. This talk should be of interest to anyone who likes differential topology/geometry, algebraic geometry, or algebra.
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AP&M 7321
AP&M 7321
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Department of Mathematics,
University of California San Diego
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Math 243 - Functional Analysis Seminar
David Jekel
UCLA
An Elementary Approach to Free Gibbs Laws Given by Convex Potentials
Abstract:
We present an alternative approach to the theory of free Gibbs
laws with convex potentials developed by Dabrowski, Guionnet, and Shlyakhtenko.
Instead of solving SDE's, we combine PDE techniques with a notion of
asymptotic approximability by trace polynomials for a sequence of
functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose
$\mu_N$ is a probability measure on on $M_N(\mathbb{C})_{sa}^m$ given by
uniformly convex and semi-concave potentials $V_N$, and suppose that the
sequence $DV_N$ is asymptotically approximable by trace polynomials in a
certain sense. Then
the moments of $\mu_N$ converge to a non-commutative law $\lambda$.
Moreover, the free entropies $\chi(\lambda)$, $\underline{\chi}(\lambda)$,
and $\chi^*(\lambda)$ agree and equal the limit of the normalized
classical entropies of $\mu_N$. An upcoming paper will use the same
techniques to obtain transport maps from $\lambda$ to a free semicircular
family as the limit of transport maps for the matrix models $\mu_N$.
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AP&M 6402
AP&M 6402
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Department of Mathematics,
University of California San Diego
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Math 269 - Combinatorics
Graham Hawkes
UC Davis
Characterization of queer supercrystals
Abstract:
We analyze the crystal bases for the quantum queer superalgebra recently introduced by Grantcharov et. al.. Like crystals of type $A$, this crystal can be described by explicit operators on words in the alphabet $\{1, 2, \dots, n\}$. Like crystals of type $A$, each connected component of a queer supercrystal has a unique highest (and lowest) weight. In the type $A$ case, if one is given a certain highest weight, one can reconstruct the connected component containing it using simple axioms introduced by Stembridge. However, given a highest weight, it is much more difficult to reconstruct the queer connected component containing it in this way. Nevertheless, a set of axioms has been conjectured by Assaf and Oguz to do just this. Unfortunately, these axioms are not sufficient, in fact they already fail to uniquely characterize the queer connected component containing highest weight $(4,2,0)$. In this talk, we provide the additional information which is needed to reconstruct the connected queer crystal which contains a given highest weight.
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AP&M 6402
AP&M 6402
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Department of Mathematics,
University of California San Diego
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Math 258 - Differential Geometry
Lei Ni
UCSD
Two curvature notions on K$\ddot{\text{a}}$hler manifolds and some questions
Abstract:
Here I shall introduce the two curvatures and discuss their relations with existing ones and their implications, including comparison theorems, vanishing theorems, projective embeddings and hyperbolicity. If time permits, I shall also discuss some open problems related.
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AP&M 5829
AP&M 5829
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Department of Mathematics,
University of California San Diego
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Math 278C: Optimization and Data Science Seminar
Lingling Xu
Nanjing Normal University
Recent advances in non-convex generalized Nash equilibrium problems
Abstract:
In this talk, we first summarize recent advances in the research on non-convex generalized Nash equilibrium problems(GNEP). That is, the cost functions of some players are non-convex, or the strategy sets are non-convex. Non-convex GNEP has wide applications. Some computational methods and theoretical results are given. We will present a half-space projection method with inertial step for convex GNEP.
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AP&M 5829
AP&M 5829
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Department of Mathematics,
University of California San Diego
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Math 278C: Optimization and Data Science Seminar
Chunfeng Cui
UCSB
Uncertainty quantification with Non-Gaussian Correlated Process Variations: Theory, Algorithms and Applications
Abstract:
Since the invention of generalized polynomial chaos in 2002, uncertainty quantification has impacted many engineering fields. However, almost all existing generalized polynomial chaos methods have a strong assumption: the uncertain parameters are mutually independent or Gaussian correlated. This assumption rarely holds in many realistic applications, and it has been a long-standing challenge for both theorists and practitioners. In this talk, I will presented two algorithms for handling this task. The first one is stochastic collocation algorithm, and the second one is sparse optimization approach. We provide some rigorous proofs for the complexity and error bound of our proposed method. Numerical experiments on synthetic, electronic and photonic integrated circuit examples show the nearly exponential convergence rate and excellent efficiency of our proposed approaches.
This is a joint work with Prof. Zheng Zhang from UC Santa Barbara.
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AP&M 5829
AP&M 5829
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Department of Mathematics,
University of California San Diego
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Math 288 - Probability Seminar
Joshua Frisch
Cal Tech
Proximal actions, Strong amenability, and infinite conjugacy class groups.
Abstract:
A topological dynamical system (i.e. a group acting by homeomorphisms on a compact topological space) is said to be proximal if for any two points p and q we can simultaneously push them together i.e. there is a sequence $g_n$ such that $lim g_n(p)=lim g_n (q)$. In his paper introducing the concept of proximality Glasner noted that whenever $Z$ acts proximally that action will have a fixed point. He termed groups with this fixed point property ``strongly amenable'' and showed that non-amenable groups are not strongly amenable and virtually nilpotent groups are strongly amenable. In this talk I will discuss recent work precisely characterizing which (countable) groups are strongly amenable.
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AP&M 6402
AP&M 6402
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Department of Mathematics,
University of California San Diego
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Math 243 - Functional Analysis Seminar
Pooya Vahidi Ferdowsi
Caltech
Classification of Choquet-Deny Groups
Abstract:
A countable discrete group is said to be Choquet-Deny if it has a trivial Poisson boundary for every non-degenerate probability measure on the group. In other words, a countable discrete group is Choquet-Deny if non-degenerate random walks on the group have trivial behavior at infinity. For example, all abelian groups are Choquet-Deny. It has been long known that all Choquet-Deny groups are amenable. I will present a recent result classifying countable discrete Choquet-Deny groups: a countable discrete group is Choquet-Deny if and only if none of its quotients have the infinite conjugacy class property. As a corollary, a finitely generated group is Choquet-Deny if and only if it is virtually nilpotent. This is a joint work with Joshua Frisch, Yair Hartman, and Omer Tamuz.
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AP&M 6402
AP&M 6402
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Department of Mathematics,
University of California San Diego
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Math 209 - Number Theory
Daniel Le
University of Toronto
Serre weights and affine Grassmannians
Abstract:
A conjecture of Serre (now a theorem of Gross, Edixhoven, and
Coleman-Voloch) classifies pairs of weights where one finds modular forms
congruent modulo a prime p in terms of local behavior at p. We discuss a
generalization of this conjecture in higher rank. A key step in our work
is the study of a certain subscheme of Gaitsgory's $A^1$ affine Grassmannian
which shares properties with some affine Springer fibers. This is joint work
with B. Le Hung, B. Levin, and S. Morra.
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AP&M 7321
AP&M 7321
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Department of Mathematics,
University of California San Diego
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Math 209 - Number Theory
Joe Ferrara
UCSD
A p-adic Stark conjecture in the rank one setting
Abstract:
In the 1970's Stark made precise conjectures about the leading term of the Taylor series at s=0 for Artin L-functions. In the rank one setting when the order vanishing is exactly one, these conjectures relate the derivative of the L-function at s=0 to the logarithm of a unit in an abelian extension of the base field. In this talk, we will define a p-adic L-function and state a p-adic Stark conjecture in the rank one setting when the base field is a quadratic field. We prove our conjecture in the case when the base field is imaginary quadratic and the prime p is split, and discuss numerical evidence in the other cases.
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AP&M 7321
AP&M 7321
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