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Series: Math Physics Seminar

This is part of the 2017 Quolloquium series.

Starting from the classical Berezin- and Li-Yau-bounds onthe eigenvalues of the Laplace operator with Dirichlet boundaryconditions I give a survey on various improvements of theseinequalities by remainder terms. Beside the Melas inequalitywe deal with modifications thereof for operators with and withoutmagnetic field and give bounds with (almost) classical remainders.Finally we extend these results to the Heisenberg sub-Laplacianand the Stark operator in domains.

Friday, November 17, 2017 - 10:00 ,
Location: Skiles 114 ,
Timothy Duff ,
GA Tech ,
Organizer: Timothy Duff

Motivated by the general problem of polynomial system solving, we state and sketch a proof Kushnirenko's theorem. This is the simplest in a series of results which relate the number of solutions of a "generic" square polynomial system to an invariant of some associated convex bodies. For systems with certain structure (here, sparse coefficients), these refinements may provide less pessimistic estimates than the exponential bounds given by Bezout's theorem.

Series: Math Physics Seminar

This is part of the 2017 Quolloquium series.

We use the weighted isoperimetric inequality of J. Ratzkin for a wedge domain in higher dimensions to prove new isoperimetric inequalities for weighted $L_p$-norms of the fundamental eigenfunction of a bounded domain in a convex cone-generalizing earlier work of Chiti, Kohler-Jobin, and Payne-Rayner. We also introduce relative torsional rigidity for such domains and prove a new Saint-Venant-type isoperimetric inequality for convex cones. Finally, we prove new inequalities relating the fundamental eigenvalue to the relative torsional rigidity of such a wedge domain thereby generalizing our earlier work to this higher dimensional setting, and show how to obtain such inequalities using the Payne interpretation in Weinstein fractional space. (Joint work with A. Hasnaoui)

Series: Joint ACO and ARC Seminar

Is matching in NC, i.e., is there a deterministic fast parallel
algorithm for it? This has been an outstanding open question in TCS for
over three decades, ever since the discovery of Random NC matching
algorithms. Within this question, the case of planar graphs has remained
an enigma: On the one hand, counting the number of perfect matchings is
far harder than finding one (the former is #P-complete and the latter
is in P), and on the other, for planar graphs, counting has long been
known to be in NC whereas finding one has resisted a solution!The
case of bipartite planar graphs was solved by Miller and Naor in 1989
via a flow-based algorithm. In 2000, Mahajan and Varadarajan gave an
elegant way of using counting matchings to finding one, hence giving a
different NC algorithm.However, non-bipartite
planar graphs still didn't yield: the stumbling block being odd tight
cuts. Interestingly enough, these are also a key to the solution: a
balanced odd tight cut leads to a straight-forward divide and conquer NC
algorithm. The remaining task is to find such a cut in NC. This
requires several algorithmic ideas, such as finding a point in the
interior of the minimum weight face of the perfect matching polytope and
uncrossing odd tight cuts.Joint work with Nima Anari.

Series: Analysis Seminar

t-Haar multipliers are examples of Haar multipliers were the symbol depends both on the frequency variable (dyadic intervals) and on the space variable, akin to pseudo differential operators. They were introduced more than 20 years ago, the corresponding multiplier when $t=1$ appeared first in connection to the resolvent of the dyadic paraproduct, the cases $t=\pm 1/2$ is intimately connected to direct and reverse inequalities for the dyadic square function in $L^2$, the case $t=1/p$ naturally appears in the study of weighted inequalities in $L^p$. Much has happened in the theory of weighted inequalities in the last two decades, highlights are the resolution of the $A_2$ conjecture (now theorem) by Hyt\"onen in 2012 and the resolution of the two weight problem for the Hilbert transform by Lacey, Sawyer, Shen and Uriarte Tuero in 2014. Among the competing methods used to prove these results were Bellman functions, corona decompositions, and domination by sparse operators. The later method has gained a lot of traction and is being widely used in contexts beyond what it was originally conceived for in work of Lerner, several of these new applications have originated here at Gatech. In this talk I would like to tell you what I know about t-Haar multipliers (some work goes back to my PhD thesis and joint work with Nets Katz and with my former students Daewon Chung, Jean Moraes, and Oleksandra Beznosova), and what we ought to know in terms of sparse domination.

Wednesday, November 15, 2017 - 13:55 ,
Location: Skiles 006 ,
Surena Hozoori ,
Georgia Tech ,
Organizer: Jennifer Hom

It is known that a class of codimension one foliations, namely "taut foliations", have subtle relation with the topology of a 3-manifold. In early 80s, David Gabai introduced the theory of "sutured manifolds" to study these objects and more than 20 years later, Andres Juhasz developed a Floer type theory, namely "Sutured Floer Homology", that turned out to be very useful in answering the question of when a 3-manifold with boundary supports a taut foliation.

Series: Research Horizons Seminar

A motivating problem in number theory and algebraic geometry is to find
all integer-valued solutions of a polynomial equation. For example,
Fermat's Last Theorem asks for all integer solutions to x^n + y^n = z^n,
for n >= 3. This kind of problem is easy
to state, but notoriously difficult to solve. I'll explain a p-adic
method for attacking Diophantine equations, namely, p-adic integration
and the Chabauty--Coleman method. Then I'll talk about some recent
joint work on the topic.

Series: Algebra Seminar

Given a Galois cover of curves X to Y with Galois group G which is totally ramified at a point x and unramified elsewhere,
restriction to the punctured formal neighborhood of x induces a Galois extension of Laurent series rings k((u))/k((t)). If
we fix a base curve Y , we can ask when a Galois extension of Laurent series rings comes from a global cover of Y in this
way. Harbater proved that over a separably closed field, this local-to-global principle holds for any base curve if G is a
p-group, and gave a condition for the uniqueness of such an extension. Using a generalization of Artin-Schreier theory to
non-abelian p-groups, we characterize the curves Y for which this lifting property holds and when it is unique, but over
a more general ground field.

Series: Geometry Topology Seminar

We discuss the growth of homonoly in finite coverings, and show that the growth of the torsion part of the first homology of finite coverings of 3-manifolds is bounded from above by the hyperbolic volume of the manifold. The proof is based on the theory of L^2 torsion.

Series: AMS Club Seminar

Join us for a discussion of making professional mathematics diagrams and
illustrations with free vector graphics editing software Inkscape.
We'll discuss and tinker with Bezier curves, TexTex, and vectorization
of scanned images.