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Series: Geometry Topology Seminar

It is generally a difficult problem to compute the Betti numbers of a
given finite-index subgroup of an infinite group, even if the Betti
numbers of the ambient group are known. In this talk, I will describe a
procedure for obtaining new lower
bounds on the first Betti numbers of certain finite-index subgroups of
the braid group. The focus will be on the level 4 braid group, which is
the kernel of the mod 4 reduction of the integral Burau representation.
This is joint work with Dan Margalit.

Monday, November 20, 2017 - 14:00 ,
Location: Skiles 005 ,
Yat Tin Chow ,
Mathematics, UCLA ,
ytchow@math.ucla.edu ,
Organizer: Prasad Tetali

In this talk, we will introduce a family of stochastic processes on the
Wasserstein space, together with their infinitesimal generators. One of
these processes is modeled after Brownian motion and plays a central
role in our work. Its infinitesimal generator defines a partial
Laplacian on the space of Borel probability measures, taken as a
partial trace of a Hessian. We study the eigenfunction of this partial
Laplacian and develop a theory of Fourier analysis. We also consider
the heat flow generated by this partial Laplacian on the Wasserstein
space, and discuss smoothing effect of this flow for a particular class
of initial conditions. Integration by parts formula, Ito formula and an
analogous Feynman-Kac formula will be discussed.
We note the use of the infinitesimal generators in the theory of Mean
Field Games, and we expect they will play an important role in future
studies of viscosity solutions of PDEs in the Wasserstein space.

Series: CDSNS Colloquium

Several modern footbridges around the world have experienced large lateral vibrations during crowd loading events. The onset of large-amplitude bridge wobbling has generally been attributed to crowd synchrony; although, its role in the initiation of wobbling has been challenged. In this talk, we will discuss (i) the contribution of a single pedestrian into overall, possibly unsynchronized, crowd dynamics, and (ii) detailed, yet analytically tractable, models of crowd phase-locking. The pedestrian models can be used as "crash test dummies" when numerically probing a specific bridge design. This is particularly important because the U.S. code for designing pedestrian bridges does not contain explicit guidelines that account for the collective pedestrian behavior. This talk is based on two recent papers: Belykh et al., Science Advances, 3, e1701512 (2017) and Belykh et al., Chaos, 26, 116314 (2016).

Series: AMS Club Seminar

Sage is widely considered to be the defacto open-source alternative to
Mathematica that is freely available for download to users on most
standard platforms at sagemath.org.
New users to Sage are also able to use its capabilities from any
webbrowser and
other useful Linux-only software by registering for a free account on
the Sage Math Cloud platform (SMC). In addition to providing users with
excellent documentation, Sage allows its users to develop spohisticated
mathematics applications using Python and
other excellent open-source developer tools that are well tested under
both Unix / Linux and Windows environments. In this two-week workshop we
provide a user-friendly introduction to Sage for beginners starting
from first principles in Python, though some
coding experience in other languages will of course be helpful to
participants. The main project we will be focusing on over the course of
the workshop is an extension of the open-source library provided by the
Tilings Gap Distributions and Pair Correlation
Project developed by the workshop guide at the University of Washington
this and last year. This application will allow participants in the
workshop to hone their coding skills in Sage by working on an extension
of a real-world computational mathematics application
in statistics and geometry. Prospective participants can gain a
heads-up on the workshop by visiting the syllabus webpage freely
available for modification online at https://github.com/maxieds/WXMLTilingsHOWTO/wiki.
The workshop guide will also offer continued free technical support on
Sage, Python programming, and Linux to participants in the workshop
after the two-week session is complete.
Future AMS workshop sessions focusing on
other Sage programming topics may be run later based on feedback from
this proto-session. Faculty and postdocs are welcome to attend. See you
all there on Friday!

Friday, November 17, 2017 - 15:00 ,
Location: Skiles 154 ,
Bhanu Kumar ,
GT Math ,
Organizer:

This lecture will discuss
the stability of perturbations of integrable Hamiltonian systems. A
brief discussion of history, integrability, and the Poincaré
nonintegrability theorem will be followed by the proof of the theorem of
Kolmogorov on persistence of
invariant tori. Time permitting, the problem of small divisors may be
briefly discussed. This lecture wIll follow the slides from the
Satellite Dynamics and Space Missions 2017 summer school held earlier
this semester in Viterbo, Italy.

Series: Combinatorics Seminar

A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graph G_{n,1/2} typically admits a covering of a constant fraction of its edges by edge-disjoint, nearly maximum cliques. We show that this is not the case. The disproof is based on some (partial) understanding of a more basic question: for k ≪ \sqrt{n} and A_1, ..., A_t chosen uniformly and independently from the k-subsets of {1…n}, what can one say about P(|A_i ∩ A_j|≤1 ∀ i≠j)?
Our main concern is trying to understand how closely the answers to this and a related question about matchings follow heuristics gotten by pretending that certain (dependent) choices are made independently. Joint work with Jeff Kahn.

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: 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: 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)