Friday, January 17, 2014 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Choongbum Lee – MIT
Sidorenko's conjecture states that the number of homomorphisms from a bipartite graph $H$ to a graph $G$ is at least the expected number of homomorphisms from $H$ to the binomial random graph with the same expected edge density as $G$. In this talk, I will present two approaches to the conjecture. First, I will introduce the notion of tree-arrangeability, where a bipartite graph $H$ with bipartition $A \cup B$ is tree-arrangeable if neighborhoods of vertices in $A$ have a certain tree-like structure, and show that Sidorenko's conjecture holds for all tree-arrangeable bipartite graphs. In particular, this implies that Sidorenko's conjecture holds if there are two vertices $a_1, a_2$ in $A$ such that each vertex $a \in A$ satisfies $N(a) \subseteq N(a_1)$ or $N(a) \subseteq N(a_2)$. Second, I will prove that if $T$ is a tree and $H$ is a bipartite graph satisfying Sidorenko's conjecture, then the Cartesian product of $T$ and $H$ also satisfies Sidorenko's conjecture. This result implies that, for all $d \ge 2$, the $d$-dimensional grid with arbitrary side lengths satisfies Sidorenko's conjecture. Joint work w/ Jeong Han Kim (KIAS) and Joonkyung Lee (Oxford).
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.
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.
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.
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.
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.
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.
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 $\mathbb{F}(s,t)$.