Seminars and Colloquia by Series

Analytic complex one-frequency cocycles

Series
CDSNS Colloquium
Time
Friday, January 17, 2014 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Christian H. Sadel – University of British Columbia, Vancouver.
(Joint work with A. Avila and S. Jitomirskaya). An analytic, complex, one-frequency cocycle is given by a pair $(\alpha,A)$ where $A(x)$ is an analytic and 1-periodic function that maps from the torus $\mathbb(R) / \mathbb(Z)$ to the complex $d\times d$ matrices and $\alpha \in [0,1]$ is a frequency. The pair is interpreted as the map $(\alpha,A)\,:\, (x,v) \mapsto (x+\alpha), A(x) v$. Associated to the iterates of this map are (averaged) Lyapunov exponents $L_k(\alpha,A)$ and an Osceledets filtration. We prove joint-continuity in $(\alpha,A)$ of the Lyapunov exponents at irrational frequencies $\alpha$, give a criterion for domination and prove that for a dense open subset of cocycles, the Osceledets filtration comes from a dominated splitting which is an analogue to the Bochi-Viana Theorem.

Low-dimensionality in mathematical signal processing

Series
Job Candidate Talk
Time
Thursday, January 16, 2014 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yaniv Plan – University of Michigan
Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.

Riemann surfaces and non-Archimedean analytic curves

Series
Research Horizons Seminar
Time
Wednesday, January 15, 2014 - 12:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Joe Rabinoff – School of Math
The theory of non-Archimedean analytic spaces closely parallels that of complex analytic spaces, with many theorems holding in both situations. I'll illustrate this principle by giving a survey of the structure theory of analytic curves over non-Archimedean fields, and comparing them to classical Riemann surfaces. I'll draw plenty of pictures and discuss topology, pair-of-pants decompositions, etc.

Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

Series
Job Candidate Talk
Time
Tuesday, January 14, 2014 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Jacob Bedrossian – Courant Institute, NYU –
We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations.That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. Joint work with Nader Masmoudi. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. Recent work with Nader Masmoudi and Clement Mouhot on Landau damping may also be discussed.

A categorification of the cut and flow lattices of graphs

Series
Geometry Topology Seminar
Time
Monday, January 13, 2014 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Zsuzsanna Dancso – University of Toronto –
We discuss a simple construction a finite dimensional algebra("bipartite algebra") to a bipartite oriented graph, and explain how thestudy of the representation theory of these algebras produces acategorification of the cut and flow lattices of graphs. I'll also mentionwhy we suspect that bipartite algebras should arise naturally in severalother contexts. This is joint work with Anthony Licata.

Stable cohomology of toroidal compactifications of the moduli space of abelian varieties

Series
Algebra Seminar
Time
Friday, January 10, 2014 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Orsola Tomassi – Leibniz University Hannover
It is well known that the cohomology of the moduli space A_g of g-dimensional principally polarized abelian varieties stabilizes when the degree is smaller than g. This is a classical result of Borel on the stable cohomology of the symplectic group. By work of Charney and Lee, also the stable cohomology of the minimal compactification of A_g, the Satake compactification, is explicitly known.In this talk, we consider the stable cohomology of toroidal compactifications of A_g, concentrating on the perfect cone compactification and the matroidal partial compactification. We prove stability results for these compactifications and show that all stable cohomology is algebraic. This is joint work with S. Grushevsky and K. Hulek.

Alexander polynomials of curves and Mordell-Weil ranks of Abelian threefolds

Series
Algebra Seminar
Time
Friday, January 10, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Remke Kloosterman – Humboldt University Berlin
Let $C=\{f(z_0,z_1,z_2)=0\}$ be a complex plane curve with ADE singularities. Let $m$ be a divisor of the degree of $f$ and let $H$ be the hyperelliptic curve $y^2=x^m+f(s,t,1)$ defined over $\mathbb{C}(s,t)$. In this talk we explain how one can determine the Mordell-Weil rank of the Jacobian of $H$ effectively. For this we use some results on the Alexander polynomial of $C$. This extends a result by Cogolludo-Augustin and Libgober for the case where $C$ is a curve with ordinary cusps. In the second part we discuss how one can do a similar approach over fields like $\mathbb{Q}(s,t)$ and $\mathbb{F}(s,t)$.

Tree Codes and a Conjecture on Exponential Sums

Series
ACO Colloquium
Time
Thursday, January 9, 2014 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Leonard J. Schulman – Professor, CalTech
Tree codes are the basic underlying combinatorial object in the interactive coding theorem, much as block error-correcting codes are the underlying object in one-way communication. However, even after two decades, effective (poly-time) constructions of tree codes are not known. In this work we propose a new conjecture on some exponential sums. These particular sums have not apparently previously been considered in the analytic number theory literature. Subject to the conjecture we obtain the first effective construction of asymptotically good tree codes. The available numerical evidence is consistent with the conjecture and is sufficient to certify codes for significant-length communications. (Joint work with Cris Moore.)

The local to global principle for rational points

Series
Job Candidate Talk
Time
Thursday, January 9, 2014 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Bianca Viray – Brown University –
Let X be a connected smooth projective variety over Q. If X has a Q point, then X must have local points, i.e. points over the reals and over the p-adic completions Q_p. However, local solubility is often not sufficient. Manin showed that quadratic reciprocity together with higher reciprocity laws can obstruct the existence of a Q point (a global point) even when there exist local points. We will give an overview of this obstruction (in the case of quadratic reciprocity) and then show that for certain surfaces, this reciprocity obstruction can be viewed in a geometric manner. More precisely, we will show that for degree 4 del Pezzo surfaces, Manin's obstruction to the existence of a rational point is equivalent to the surface being fibered into genus 1 curves, each of which fail to be locally solvable.

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