Seminars and Colloquia by Series

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.

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

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 –
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.

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