Seminars and Colloquia Schedule

Legendrian contact homology and products of Legendrian knots

Series
Geometry Topology Seminar
Time
Monday, April 15, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Peter Lambert-Cole – LSU
Legendrian contact homology is an invariant in contact geometry that assigns to each Legendrian submanifold a dg-algebra. While well-defined, it depends upon counts of holomorphic curves that can be hard to calculate in practice. In this talk, we introduce a class of Legendrian tori constructed as the product of collections of Legendrian knots. For this class, we discuss how to explicitly compute the dg-algebra invariant of the tori in terms of diagram projections of the constituent Legendrian knots.

Central-Upwind Schemes for Shallow Water Models

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 15, 2013 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alexander Kurganov – Tulane University
I will first give a brief review on simple and robust central-upwind schemes for hyperbolic conservation laws. I will then discuss their application to the Saint-Venant system of shallow water equations. This can be done in a straightforward manner, but then the resulting scheme may suffer from the lack of balance between the fluxes and (possibly singular) geometric source term, which may lead to a so-called numerical storm, and from appearance of negative values of the water height, which may destroy the entire computed solution. To circumvent these difficulties, we have developed a special technique, which guarantees that the designed second-order central-upwind scheme is both well-balanced and positivity preserving. Finally, I will show how the scheme can be extended to the two-layer shallow water equations and to the Savage-Hutter type model of submarine landslides and generated tsunami waves, which, in addition to the geometric source term, contain nonconservative interlayer exchange terms. It is well-known that such terms, which arise in many different multiphase models, are extremely sensitive to a particular choice their numerical discretization. To circumvent this difficulty, we rewrite the studied systems in a different way so that the nonconservative terms are multiplied by a quantity, which is, in all practically meaningful cases, very small. We then apply the central-upwind scheme to the rewritten system and demonstrate robustness and superb performance of the proposed method on a number numerical examples.

Stark-Heegner/Darmon points on elliptic curves over totally real fields

Series
Algebra Seminar
Time
Monday, April 15, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Amod Agashe – Florida State University –
The classical theory of complex multiplication predicts the existence of certain points called Heegner points defined over quadratic imaginary fields on elliptic curves (the curves themselves are defined over the rational numbers). Henri Darmon observed that under certain conditions, the Birch and Swinnerton-Dyer conjecture predicts the existence of points of infinite order defined over real quadratic fields on elliptic curves, and under such conditions, came up with a conjectural construction of such points, which he called Stark-Heegner points. Later, he and others (especially Greenberg and Gartner) extended this construction to many other number fields, and the points constructed have often been called Darmon points. We will outline a general construction of Stark-Heegner/Darmon points defined over quadratic extensions of totally real fields (subject to some mild restrictions) that combines past constructions; this is joint work with Mak Trifkovic.

The rank of elliptic curves

Series
School of Mathematics Colloquium
Time
Tuesday, April 16, 2013 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Benedict Gross – Harvard University
The problem of finding rational solutions to cubic equations is central in number theory, and goes back to Fermat. I will discuss why these equations are particularly interesting, and the modern theory of elliptic curves that has developed over the past century, including the Mordell-Weil theorem and the conjecture of Birch and Swinnerton-Dyer. I will end with a description of some recent results of Manjul Bhargava on the average rank.

Asymptotic behavior of globally modified non-autonomous 3D Navier-Stokes equations with memory effects and stochastic perturbations

Series
PDE Seminar
Time
Tuesday, April 16, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chen, Zhang – Shandong University
In this talk, globally modified non-autonomous 3D Navier-Stokes equations with memory and perturbations of additive noise will be discussed. Through providing theorem on the global well-posedness of the weak and strong solutions for the specific Navier-Stokes equations, random dynamical system (continuous cocycle) is established, which is associated with the above stochastic differential equations. Moreover, theoretical results show that the established random dynamical system possesses a unique compact random attractor in the space of C_H, which is periodic under certain conditions and upper semicontinuous with respect to noise intensity parameter.

Athens-Atlanta number theory seminar 1 - The arithmetic of hyperelliptic curves

Series
Algebra Seminar
Time
Tuesday, April 16, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Dick Gross – Harvard University
Hyperelliptic curves over Q have equations of the form y^2 = F(x), where F(x) is a polynomial with rational coefficients which has simple roots over the complex numbers. When the degree of F(x) is at least 5, the genus of the hyperelliptic curve is at least 2 and Faltings has proved that there are only finitely many rational solutions. In this talk, I will describe methods which Manjul Bhargava and I have developed to quantify this result, on average.

The Ruelle zeta for C^\infty Anosov flows

Series
CDSNS Colloquium
Time
Tuesday, April 16, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mark Pollicott – Univ. of Warwick
In joint work with P. Guilietti and C. Liverani, we show that the Ruelle zeta function for C^\infty Anosov flows has a meromorphic extension to the entire complex plane. This generalises results of Selberg (for geodesic flows in constant curvature) and Ruelle. I

Athens-Atlanta number theory seminar 2 - Arithmetic statistics over function fields

Series
Algebra Seminar
Time
Tuesday, April 16, 2013 - 17:15 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jordan Ellenberg – University of Wisconsin
What is the probability that a random integer is squarefree? Prime? How many number fields of degree d are there with discriminant at most X? What does the class group of a random quadratic field look like? These questions, and many more like them, are part of the very active subject of arithmetic statistics. Many aspects of the subject are well-understood, but many more remain the subject of conjectures, by Cohen-Lenstra, Malle, Bhargava, Batyrev-Manin, and others. In this talk, I explain what arithmetic statistics looks like when we start from the field Fq(x) of rational functions over a finite field instead of the field Q of rational numbers. The analogy between function fields and number fields has been a rich source of insights throughout the modern history of number theory. In this setting, the analogy reveals a surprising relationship between conjectures in number theory and conjectures in topology about stable cohomology of moduli spaces, especially spaces related to Artin's braid group. I will discuss some recent work in this area, in which new theorems about the topology of moduli spaces lead to proofs of arithmetic conjectures over function fields, and to new, topologically motivated questions about counting arithmetic objects.

Generation and Synchronization of Oscillations in Synthetic Gene Networks

Series
Mathematical Biology Seminar
Time
Wednesday, April 17, 2013 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles Bldg, Room 006
Speaker
Lev Tsimring – UC San Diego, BIOCircuits Inst.
In this talk, I will describe our recent experimental and theoretical work on small synthetic gene networks exhibiting oscillatory behavior. Most living organisms use internal genetic "clocks" to govern fundamental cellular behavior. While the gene networks that produce oscillatory expression signals are typically quite complicated, certain recurring network motifs are often found at the core of these biological clocks. One common motif which may lead to oscillations is delayed auto-repression. We constructed a synthetic two-gene oscillator based on this design principle, and observed robust and tunable oscillations in bacteria. Computational modeling and theoretical analysis show that the key mechanism of oscillations is a small delay in the negative feedback loop. In a strongly nonlinear regime, this time delay can lead to long-period oscillations that can be characterized by "degrade and fire'' dynamics. We also achieved synchronization of synthetic gene oscillators across cell population as well as multiple populations using variants of the same design in which oscillators are synchronized by chemical signals diffusing through cell membranes and throughout the populations.

A Brief Tour of Lattice Cryptography

Series
Research Horizons Seminar
Time
Wednesday, April 17, 2013 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chris Peikert – Georgia Tech, Colloge of Computing
I will give an overview of how lattices in R^n are providing a powerful new mathematical foundation for cryptography. Lattices yield simple, fast, and highly parallel schemes that, unlike many of today's popular cryptosystems (like RSA and elliptic curves), even appear to remain secure against quantum computers. What's more, lattices were recently used to solve a cryptographic "holy grail" problem known as fully homomorphic encryption. No background in lattices, cryptography, or quantum computers will be necessary for this talk -- but you will need to know how to add and multiply matrices.

Admissible Risks and Convex Order

Series
Mathematical Finance/Financial Engineering Seminar
Time
Wednesday, April 17, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruodu Wang – University of Waterloo

Hosts: Christian Houdre and Liang Peng

We introduce the admissible risk class as the set of possible aggregate risks when the marginal distributions of individual risks are given but the dependence structure among them is unspecified. The convex ordering upper bound on this class is known to be attained by the comonotonic scenario, but a sharp lower bound is a mystery for dimension larger than 2. In this talk we give a general convex ordering lower bound over this class. In the case of identical marginal distributions, we give a sufficient condition for this lower bound to be sharp. The results are used to identify extreme scenarios and calculate bounds on convex risk measures and other quantities of interest, such as expected utilities, stop-loss premiums, prices of European options and TVaR. Numerical illustrations are provided for different settings and commonly-used distributions of risks.

Quasirandomness of permutations

Series
Graph Theory Seminar
Time
Thursday, April 18, 2013 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Daniel Kral – University of Warwick
A systematic study of large combinatorial objects has recently led to discovering many connections between discrete mathematics and analysis. In this talk, we apply analytic methods to permutations. In particular, we associate every sequence of permutations with a measure on a unit square and show the following: if the density of every 4-element subpermutation in a permutation p is 1/4!+o(1), then the density of every k-element subpermutation is 1/k!+o(1). This answers a question of Graham whether quasirandomness of a permutation is captured by densities of its 4-element subpermutations. The result is based on a joint work with Oleg Pikhurko.

Universality for beta ensembles

Series
Stochastics Seminar
Time
Thursday, April 18, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skyles 006
Speaker
Paul Bourgade – Harvard University
Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis concerns large but finite dimensional matrices. Spectacular progress was done in the past two decades to prove universality of random matrices presenting an orthogonal, unitary or symplectic invariance. These models correspond to log-gases with respective inverse temperature 1, 2 or 4. I will report on a joint work with L. Erdos and H.-T. Yau, which yields universality for log-gases at arbitrary temperature at the microscopic scale. A main step consists in the optimal localization of the particles, and the involved techniques include a multiscale analysis and a local logarithmic Sobolev inequality.

Conormals and contact homology IX

Series
Geometry Topology Working Seminar
Time
Friday, April 19, 2013 - 12:05 for 1.5 hours (actually 80 minutes)
Location
Skiles 006
Speaker
John Etnyre – Georgia Tech
In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.

Role of chemotaxis in enhancement of biological reactions

Series
PDE Seminar
Time
Friday, April 19, 2013 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Prof. Alexandaer Kiselev – Univrsity of Wisconsin,-Madison –
We discuss a system of two equations involving two diffusing densities, one of which is chemotactic on the other, and absorbing reaction. The problem is motivated by modeling of coral life cycle and in particular breeding process, but the set up is relevant to many other situations in biology and ecology. The models built on diffusion and advection alone seem to dramatically under predict the success rate in coral reproduction. We show that presence of chemotaxis can significantly increase reproduction rates. On mathematical level, the first step in understanding the problem involves derivation of sharp estimates on rate of convergence to bound state for Fokker-Planck equation with logarithmic potential in two dimensions.

Stochastic Representation of Solutions to Degenerate Elliptic Boundary Value and Obstacle Problems with Dirichlet Boundary Conditions

Series
Mathematical Finance/Financial Engineering Seminar
Time
Friday, April 19, 2013 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Ruoting Gong – Rutgers University

Hosts: Christian Houdre and Liang Peng

We prove stochastic representation formulae for solutions to elliptic boundary value and obstacle problems associated with a degenerate Markov diffusion process on the half-plane. The degeneracy in the diffusion coefficient is proportional to the \alpha-power of the distance to the boundary of the half-plane, where 0 < \alpha < 1 . This generalizes the well-known Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process. The generator of this degenerate diffusion process with killing, is a second-order, degenerate-elliptic partial differential operator where the degeneracy in the operator symbol is proportional to the 2\alpha-power of the distance to the boundary of the half-plane. Our stochastic representation formulae provides the unique solution to the degenerate partial differential equation with partial Dirichlet condition, when we seek solutions which are suitably smooth up to the boundary portion \Gamma_0 contained in the boundary of the half-plane. In the case when the full Dirichlet condition is given, our stochastic representation formulae provides the solutions which are not guaranteed to be any more than continuous up to the boundary portion \Gamma_0 .

Bounds on the eigenvalues of Laplace-Beltrami operators and Witten Laplacians on Riemannian manifolds

Series
Math Physics Seminar
Time
Friday, April 19, 2013 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ahmad El Soufi – Université François Rabelais, Tours, France –

El Soufi will be visiting Harrell for the week leading up to this seminar

We shall survey some of the classical and recent results giving upper bounds of the eigenvalues of the Laplace-Beltrami operator on a compact Riemannian manifold (Yang-Yau, Korevaar, Grigor'yan-Netrusov-Yau, etc.). Then we discuss extensions of these results to the eigenvalues of Witten Laplacians associated to weighted volume measures and investigate bounds of these eigenvalues in terms of suitable norms of the weights.