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Series: PDE Seminar

We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.

Series: Analysis Seminar

Abstract: I will state a version of Voiculescu's noncommutative
Weyl-von Neumann theorem for operators on l^p that I obtained.
This allows certain classical results concerning unitary
equivalence of operators on l^2 to be generalized to operators on
l^p if we relax unitary equivalence to similarity. For example,
the unilateral shift on l^p, 1

Series: PDE Seminar

(Due to a flight cancellation, this talk will be moved to Thursday (Apr 26) 3pm at Skiles 257). We prove an abstract theorem giving a $t^\epsilon$ bound for any $\epsilon> 0$ on the growth of the Sobolev norms in some abstract linear Schrödinger equations. The abstract theorem is applied to nonresonant Harmonic oscillators in R^d. The proof is obtained by conjugating the system to some normal form in which the perturbation is a smoothing operator. Finally, time permitting, we will show how to construct a perturbation of the harmonic oscillator which provokes growth of Sobolev norms.

Series: Algebra Seminar

The talk reports on joint work with Wayne Raskind and concerns the conjectural definition of a new type of regulator map into a quotient of an algebraic torus by a discrete subgroup, that should fit in "refined" Beilinson type conjectures, exteding special cases considered by Gross and Mazur-Tate.The construction applies to a smooth complete variety over a p-adic field K which has totally degenerate reduction, a technical term roughly saying that cycles acount for the entire etale cohomology of each component of the special fiber. The regulator is constructed out of the l-adic regulators for all primes l simulateously. I will explain the construction, the special case of the Tate elliptic curve where the regulator on cycles is the identity map, and the case of K_2 of Mumford curves, where the regulator turns out to be a map constructed by Pal. Time permitting I will also say something about the relation with syntomic regulators.

Series: Geometry Topology Seminar

Novikov homology was introduced by Novikov in the
early 1980s motivated by problems in hydrodynamics. The Novikov
inequalities in the Novikov homology theory
give lower bounds for the number of critical points of a Morse closed 1-form on a compact
differentiable manifold M. In the first part of my talk I shall
survey the Novikov homology theory in finite dimensional setting
and its further developments in infinite dimensional setting with applications in the theory of symplectic fixed points and Lagrangian intersection/embedding problems. In the second part of my
talk
I shall report on my recent joint work with Jean-Francois Barraud and Agnes Gadbled on construction of the Novikov fundamental group
associated to a cohomology
class of a closed 1-form on M and its application to obtaining
new lower bounds for the number of
critical points of a Morse 1-form.

Friday, April 20, 2018 - 15:05 ,
Location: Skiles 271 ,
Prof. Rafael de la Llave ,
GT Math ,
Organizer: Jiaqi Yang

A well known paper of H. Federer on Flat chains contains a remarkable example attributed to F. Almgren. We intend to give a geometric exposition of the example and explain its relevance in the global theory of geodesic flows and some global problems such as homogenization in quasi-periodic media. This is part of an expository paper with X. Su.

Friday, April 20, 2018 - 15:05 ,
Location: Skiles 271 ,
Prof. Rafael de la Llave ,
GT Math ,
Organizer: Jiaqi Yang
A well known paper of H. Federer on Flat chains contains a remarkable example attributed to F. Almgren. We intend to give a geometric exposition of the example and explain its relevance in the global theory of geodesic flows and some global problems such as homogenization in quasi-periodic media. This is part of an expository paper with X. Su.

Series: ACO Student Seminar

Consider the problem of selling items to a unit-demand buyer. Most work
on maximizing seller revenue considers either a setting that is single
dimensional, such as where the items are identical, or
multi-dimensional, where the items are heterogeneous. With respect to
revenue-optimal mechanisms, these settings sit at extreme ends of a
spectrum: from simple and fully characterized (single-dimensional) to
complex and nebulous (multi-dimensional).
In this paper, we identify a setting that sits in between these
extremes. We consider a seller who has three services {A,B,C} for sale
to a single buyer with a value v and an interest G from {A,B,C}, and
there is a known partial ordering over the services. For example,
suppose the seller is selling {internet}, {internet, phone}, and
{internet, cable tv}. A buyer with interest {internet} would be
satisfied by receiving phone or cable tv in addition, but a customer
whose interest is {internet, phone} cannot be satisfied by any other
option. Thus this corresponds to a partial-ordering where {internet}
> {internet, phone} and {internet} > {internet, cable tv}, but
{internet, phone} and {internet, cable tv} are not comparable.
We show formally that partially-ordered items lie in a space of their
own, in between identical and heterogeneous items: there exist
distributions over (value, interest) pairs for three partially-ordered
items such that the menu complexity of the optimal mechanism is
unbounded, yet for all distributions there exists an optimal mechanism
of finite menu complexity. So this setting is vastly more complex than
identical items (where the menu complexity is one), or even
“totally-ordered” items as in the FedEx Problem [FGKK16] (where the menu
complexity is at most seven, for three items), yet drastically more
structured than heterogeneous items (where the menu complexity can be
uncountable [DDT15]). We achieve this result by proving a
characterization of the class of best duals and by giving a primal
recovery algorithm which obtains the optimal mechanism. In addition, we
(1) extend our lower-bound to the Multi-Unit Pricing setting, (2) give a
tighter and deterministic characterization of the optimal mechanism
when the buyer’s distribution satisfies the declining marginal revenue
condition, and (3) prove a master theorem that allows us to reason about
duals instead of distributions.
Joint work with Nikhil Devanur, Raghuvansh Saxena, Ariel Schvartzman, and Matt Weinberg.

Friday, April 20, 2018 - 10:00 ,
Location: Skiles 006 ,
Jose Acevedo ,
Georgia Tech ,
Organizer: Kisun Lee

In this talk we show how to obtain some (sometimes sharp) inequalities between subgraph densities which are valid asymptotically on any sequence of finite simple graphs with an increasing number of vertices. In order to do this we codify a simple graph with its edge monomial and establish a nice graphical notation that will allow us to play around with these densities.

Series: Stochastics Seminar

We shall prove that a certain stochastic ordering defined in terms of
convex symmetric sets is inherited by sums of independent symmetric
random vectors. Joint work with W. Bednorz.