Title: Arithmetic criteria of spectral dimension for quasiperiodic Schrodinger operators.

Date: 10/31/2016

Time: 4:02 PM - 4:52 PM

Place: C517 Wells Hall

Shiwen Zhang (msu), joint work with Svetlana Jitomirskaya (uci)
Abstract: We introduce a notion of β-almost periodicity and prove quantitative lower spectral/
quantum dynamical bounds for general bounded β-almost periodic potentials.
Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic
quasiperiodic Schrodinger operators in the positive Lyapunov exponent regime
and arithmetic criteria for families with zero Lyapunov exponents, with applications
to Sturmian potentials and the critical almost Mathieu operator.

Title: How to define the Torelli group of a surface with boundary-2

Date: 10/31/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

The first talk was devoted mostly to the motivation. The second talk will be devoted to the extension problem in Torelli topology and some speculations about the possible answer to the question posed in the talk title.

Speaker: Panupong Vichitkunakorn, University of Illinois at Urbana-Champaign

Title: Conserved quantites of Q-systems from dimer integrable systems

Date: 11/01/2016

Time: 1:00 PM - 1:50 PM

Place: C304 Wells Hall

In 2013, Goncharov and Kenyon constructed integrable systems from a class of quivers on a torus, parametrized by integral convex polygons. Associating a Y-pattern to the quiver, the phase space coordinates of the dynamical systems are y-variables together with two extra variables. A Y-seed mutation at a vertex having two incoming and two outgoing arrows gives a change of coordinates. We study this dimer integrable system on Cluster variables, extend it to some quivers outside the class, and construct conserved quantities of Q-systems.

Title: Riemann Roch Theorem for Compact Riemann Surfaces

Date: 11/01/2016

Time: 3:00 PM - 3:50 PM

Place: C304 Wells Hall

We will introduce Riemann Surface and define related objects like meromorphic function, holomorphic mapping and meromorphic one form. We will also define divisors on a Riemann Surface, and state the Riemann Roch theorem for compact Riemann Surface.

Algebraic K-theory brings together classical invariants of rings with homotopy groups of topological spaces. In general algebraic K-theory groups are difficult to compute, but in recent years methods in equivariant stable homotopy theory have led to many important K-theory computations. I will introduce this approach to K-theory computations, and discuss some of my recent joint work with Angeltveit on the algebraic K-theory of the group ring Z[C_2].

Speaker: Thomas Sideris, University of California, Santa Barbara

Title: Affine motion of 3d incompressible ideal fluids surrounded by vacuum

Date: 11/03/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

We shall discuss the existence and long-time behavior of affine (spatially linear) solutions to the initial
free boundary value problem for incompressible fluids surrounded by vacuum in 3d. The general problem is known to be
locally well-posed. For affine motion, the equations of motion reduce to a globally solvable system of ordinary differential
equations corresponding to geodesic flow in SL(3,R) viewed as a submanifold embedded in R^9 with the Euclidean metric.
The domain occupied by the fluid at each time is an ellipsoid of constant volume whose diameter grows linearly in time,
provided the pressure remains nonnegative. We shall examine the motion in several representative cases, including
swirling flow geometry where elementary phase plane analysis can be used.

Title: Threading Fáry and Milnor with Buffon's Needle

Date: 11/04/2016

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

There is a fundamental invariant of a curve in Euclidean 3-space called the total curvature which, roughly speaking, measures how 'twisted' the curve is. A beautiful result of Fáry and Milnor from the early 1950's relates this to a branch of mathematics called knot theory, and shows that if the curvature of a curve is small enough, then the curve is actually 'unknotted'; that is, it can be untied without cutting it. Their proofs were rather technical, and in this talk I'll relate their problem to an experimental method for computing the value of pi known as the Buffon Needle Problem. Using this interpretation I will give an elegant and self-contained proof of their result, due to Ari Turner. The talk should be accessible to anyone who has taken multivariable calculus.

Speaker: Tom Sideris, University of California at Santa Barbara; Thomas C. Sideris, University of California, Santa Barbara

Title: Affine motion of 3d compressible ideal fluids surrounded by vacuum

Date: 11/04/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

We shall discuss the existence and long-time behavior of affine (spatially linear) solutions to the initial
free boundary value problem for compressible fluids surrounded by vacuum in 3d. The general problem is known to be
locally well-posed. For affine motion, the equations of motion reduce to a globally solvable Hamiltonian system of ordinary
differential equations on GL_+(3,R). The domain occupied by the fluid at each time is an ellipsoid whose diameter grows
linearly in time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid domain is determined by a semi-definite
quadratic form of rank r = 1,2, or 3, corresponding to a collapse of the ellipsoid along 2,1, or 0 of its principal axes. When
the adiabatic index \gamma determining the pressure law lies in the interval ( 4/3 , 2 ), the rank of this form is 3 and there
is a scattering operator, i.e. a bijection between states at minus and plus infinity. However, larger values of \gamma can
lead to collapsed asymptotic states of rank 1 or 2.

Speaker: Tom Sideris, University of California at Santa Barbara; Thomas C. Sideris, University of California, Santa Barbara

Title: Affine motion of 3d compressible ideal fluids surrounded by vacuum

Date: 11/04/2016

Time: 4:10 PM - 5:00 PM

Place: 1502 Engineering Building

We shall discuss the existence and long-time behavior of affine (spatially linear) solutions to the initial free boundary value problem for compressible fluids surrounded by vacuum in 3d. The general problem is known to be locally well-posed. For affine motion, the equations of motion reduce to a globally solvable Hamiltonian system of ordinary differential equations on GL_+(3,R). The domain occupied by the fluid at each time is an ellipsoid whose diameter grows linearly in time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid domain is determined by a semi-definite
quadratic form of rank r = 1,2, or 3, corresponding to a collapse of the ellipsoid along 2,1, or 0 of its principal axes. When
the adiabatic index \gamma determining the pressure law lies in the interval ( 4/3 , 2 ), the rank of this form is 3 and there is a scattering operator, i.e. a bijection between states at minus and plus infinity. However, larger values of \gamma can lead to collapsed asymptotic states of rank 1 or 2.