TBA by Alden Waters
- Series
- Analysis Seminar
- Time
- Wednesday, January 8, 2014 - 15:04 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Alden Water – Univesity of Paris
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Tropical Scheme Theory
- Series
- Algebra Seminar
- Time
- Wednesday, January 8, 2014 - 11:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Noah Giansiricusa – UC Berkeley
I'll discuss joint work with J.H. Giansiracusa (Swansea) in which we study scheme theory over the tropical semiring T, using the notion of semiring schemes provided by Toen-Vaquie, Durov, or Lorscheid. We define tropical hypersurfaces in this setting and a tropicalization functor that sends closed subschemes of a toric variety over a field with non-archimedean valuation to closed subschemes of the corresponding toric variety over T. Upon passing to the set of T-valued points this yields Payne's extended tropicalization functor. We prove that the Hilbert polynomial of any projective subscheme is preserved by our tropicalization functor, so the scheme-theoretic foundations developed here reveal a hidden flatness in the degeneration sending a variety to its tropical skeleton.
Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian
- Series
- CDSNS Colloquium
- Time
- Tuesday, January 7, 2014 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Xifeng Su – Beijing Normal University
We consider the semi-linear elliptic PDE driven by the fractional Laplacian:
\begin{equation*}\left\{%\begin{array}{ll} (-\Delta)^s u=f(x,u) & \hbox{in $\Omega$,} \\ u=0 & \hbox{in $\mathbb{R}^n\backslash\Omega$.} \\\end{array}%
\right.\end{equation*}An $L^{\infty}$ regularity result is given, using De Giorgi-Stampacchia iteration method.By
the Mountain Pass Theorem and some other nonlinear analysis methods,
the existence and multiplicity of non-trivial solutions for the above
equation are established. The validity of the Palais-Smale condition
without Ambrosetti-Rabinowitz condition for non-local elliptic equations
is proved. Two non-trivial solutions are given under some weak
hypotheses. Non-local elliptic equations with concave-convex
nonlinearities are also studied, and existence of at least six solutions
are obtained.
Moreover, a global result of
Ambrosetti-Brezis-Cerami type is given, which shows that the effect of
the parameter $\lambda$ in the nonlinear term changes considerably the
nonexistence, existence and multiplicity of solutions.
Intertwinings, wave equations and growth models
- Series
- Job Candidate Talk
- Time
- Tuesday, January 7, 2014 - 11:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Mykhaylo Shkolnikov – Berkeley Univ – mshkolni@gmail.com
We will discuss a general theory of intertwined diffusion processes of any dimension. Intertwined processes arise in many different contexts in probability theory, most notably in the study of random matrices, random polymers and path decompositions of Brownian motion. Recently, they turned out to be also closely related to hyperbolic partial differential equations, symmetric polynomials and the corresponding random growth models. The talk will be devoted to these recent developments which also shed new light on some beautiful old examples of intertwinings. Based on joint works with Vadim Gorin and Soumik Pal.
Analyzing Phylogenetic Treespace
- Series
- Mathematical Biology Seminar
- Time
- Monday, January 6, 2014 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Katherine St. John – Lehman College, CUNY
Evolutionary histories, or phylogenies, form an integral part of much work in biology. In addition to the intrinsic interest in the interrelationships between species, phylogenies are used for drug design, multiple sequence alignment, and even as evidence in a recent criminal trial. A simple representation for a phylogeny is a rooted, binary tree, where the leaves represent the species, and internal nodes represent their hypothetical ancestors. This talk will focus on some of the elegant mathematical and computational questions that arise from assembling, summarizing, visualizing, and searching the space of phylogenetic trees, as well as delve into the computational issues of modeling non-treelike evolution.
Some metric properties of Houghton's groups
- Series
- Geometry Topology Seminar
- Time
- Monday, January 6, 2014 - 14:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Sean Cleary – CUNY
Houghton's groups are a family of subgroups of infinite permutation groups known for their cohomological properties. Here, I describe some aspects of their geometry and metric properties including families of self-quasi-isomtries. This is joint work with Jose Burillo, Armando Martino and Claas Roever.
Incoherence and Synchronization in the Hamiltonian Mean Field Model
- Series
- CDSNS Colloquium
- Time
- Monday, January 6, 2014 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- James Meiss* – Department of Applied Mathematics, University of Colorado, Boulder
Synchronization of coupled oscillators, such as grandfather clocks or
metronomes, has been much studied using the approximation of strong
damping in which case the dynamics of each reduces to a phase on a limit
cycle. This gives rise to the famous Kuramoto model. In contrast, when
the oscillators are Hamiltonian both the amplitude and phase of each
oscillator are dynamically important. A model in which all-to-all
coupling is assumed, called the Hamiltonian Mean Field (HMF) model, was
introduced by Ruffo and his colleagues. As for the Kuramoto model, there
is a coupling strength threshold above which an incoherent state loses
stability and the oscillators synchronize.
We study the case when the moments of inertia and coupling strengths of
the oscillators are heterogeneous. We show that finite size fluctuations
can greatly modify the synchronization threshold by inducing
correlations between the momentum and parameters of the rotors. For
unimodal parameter distributions, we find an analytical expression for
the modified critical coupling strength in terms of statistical
properties of the parameter distributions and confirm our results with
numerical simulations. We find numerically that these effects disappear
for strongly bimodal parameter distributions.
*This work is in collaboration with Juan G. Restrepo.
Random matrix theory and the informational limit of eigen-analysis
- Series
- Stochastics Seminar
- Time
- Thursday, December 12, 2013 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Raj Rao Nadakuditi – University of Michigan
Motivated by the ubiquity of signal-plus-noise type models in high-dimensional statistical signal processing and machine learning, we consider the eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Applications in mind are as diverse as radar, sonar, wireless communications, spectral clustering, bio-informatics and Gaussian mixture cluster analysis in machine learning. We provide an application-independent approach that brings into sharp focus a fundamental informational limit of high-dimensional eigen-analysis. Building on this success, we highlight the random matrix origin of this informational limit, the connection with "free" harmonic analysis and discuss how to exploit these insights to improve low-rank signal matrix denoising relative to the truncated SVD.
Global regularity for water waves in two dimensions
- Series
- Job Candidate Talk
- Time
- Thursday, December 12, 2013 - 11:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Fabio Pusateri – Princeton University
We will start by describing some general features of quasilinear
dispersive and wave equations. In particular we will discuss a few
important aspects related to the question of global regularity for such
equations.
We will then consider the water waves system for the evolution of a
perfect fluid with a free boundary. In 2 spatial dimensions, under the
influence of gravity, we prove the existence of global irrotational
solutions for suitably small and regular initial data. We also prove
that the asymptotic behavior of solutions as time goes to infinity is
different from linear, unlike the 3 dimensional case.
Localization and delocalization in the Anderson model on random regular graphs
- Series
- Job Candidate Talk
- Time
- Tuesday, December 10, 2013 - 11:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Leander Geisinger – Princeton University
The Anderson model on a discrete graph is given by the graph Laplacian perturbed by a random potential. I study spectral properties of this random Schroedinger operator on a random regular graph of fixed degree in the limit where the number of vertices tends to infinity.The choice of model is motivated by its relation to two important and well-studied models of random operators: On the one hand there are similarities to random matrices, for instance to Wigner matrices, whose spectra are known to obey universal laws. On the other hand a random Schroedinger operator on a random regular graph is expected to approximate the Anderson model on the homogeneous tree, a model where both localization (characterized by pure point spectrum) and delocalization (characterized by absolutely continuous spectrum) was established.I will show that the Anderson model on a random regular graph also exhibits distinct spectral regimes of localization and of delocalization. One regime is characterized by exponential decay of eigenvectors. In this regime I analyze the local eigenvalue statistics and prove that the point process generated by the eigenvalues of the random operator converges in distribution to a Poisson process.In contrast to that I will also show that the model exhibits a spectral regime of delocalization where eigenvectors are not exponentially localized.