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Series: Other Talks

Club Math Presents The Mathematics of Futurama, by Dr. Michael Lacey.

Series: Other Talks

Series: Other Talks

Let k be a global field or a local field. Class field theory says that every central division algebra over k is cyclic. Let l be a prime not equal to the characteristic of k. If k contains a primitive l-th root of unity, then this leads to the fact that every element in H^2(k, µ_l ) is a symbol. A natural question is a higher dimensional analogue of this result: Let F be a function field in one variable over k which contains a primitive l-th root of unity. Is every element in H^3(F, µ_l ) a symbol? In this talk we answer this question in affirmative for k a p-adic field or a global field of positive characteristic. The main tool is a certain local global principle for elements of H^3(F, µ_l ) in terms of symbols in H^2(F µ_l ). We also show that this local-global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields.

Series: Other Talks

In 1854 Riemann extended Gauss' ideas on curved geometries from two dimensional surfaces to higher dimensions. Since that time mathematicians have tried to understand the structure of geometric spaces based on their curvature properties. It turns out that basic questions remain unanswered in this direction. In this lecture we will give a history of such questions for spaces with positive curvature, and describe the progress that has been made as well as some outstanding conjectures which remain to be settled.

Series: Other Talks

The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions: The University of Alabama at Birmingham; The Georgia Institute of Technology; Emory University; The University of Tennessee Knoxville. The presentations will include topics on geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology. See the Schedule for times and abstracts of talks.

Series: Other Talks

Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the Glauber dynamics for the Ising model on $Z^2$ everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems. A rich interplay exists between the static and dynamic models. At the static phase-transition for Ising, the dynamics is conjectured to undergo a critical slowdown: At high temperature the inverse-gap is $O(1)$, at the critical $\beta_c$ it is polynomial in the side-length and at low temperature it is exponential in it. A long series of papers verified this on $Z^2$ except at $\beta=\beta_c$ where the behavior remained unknown. In this work we establish the first rigorous polynomial upper bound for the critical mixing, thus confirming the critical slowdown for the Ising model in $Z^2$. Namely, we show that on a finite box with arbitrary (e.g. fixed, free, periodic) boundary conditions, the inverse-gap at $\beta=\beta_c$ is polynomial in the side-length. The proof harnesses recent understanding of the scaling limit of critical Fortuin-Kasteleyn representation of the Ising model together with classical tools from the analysis of Markov chains. Based on joint work with Eyal Lubetzky.

Series: Other Talks

Concrete optimization problems, while often nonsmooth, are not
pathologically so. The class of "semi-algebraic" sets and functions -
those arising from polynomial inequalities - nicely exemplifies
nonsmoothness in practice. Semi-algebraic sets (and their
generalizations) are common, easy to recognize, and richly structured,
supporting powerful variational properties. In particular I will discuss
a generic property of such sets - partial smoothness - and its relationship
with a proximal algorithm for nonsmooth composite
minimization, a versatile model for practical optimization.

Series: Other Talks

The purpose of this talk is to highlight some versions of the Krein-Rutman theorem
which have been widely and deeply applied in many fields (e.g., Mathematical Analysis, Geometric Analysis, Physical Sciences, Transport theory and Information Sciences).
These versions are motivated by optimization theory, perturbation theory, bifurcation theory, etc. and give rise to some simple but useful comparison methods, in ordered Banach spaces, such as the Dodds-Fremlin theorem and the De Pagter theorem.

Series: Other Talks

Santosh Vempala and I have been exploring an intriguing new
approach to convex optimization. Intuition about first-order interior
point methods tells us that a main impediment to quickly finding an
inside track to optimal is that a convex body's boundary can get in
one's way in so many directions from so many places. If the surface of
a convex body is made to be perfectly reflecting then from every
interior vantage point it essentially disappears. Wondering about what
this might mean for designing a new type of first-order interior point
method, a preliminary analysis offers a surprising and suggestive
result. Scale a convex body a sufficient amount in the direction of
optimization. Mirror its surface and look directly upwards from
anywhere. Then, in the distance, you will see a point that is as close
as desired to optimal. We wouldn't recommend a direct implementation,
since it doesn't work in practice. However, by trial and error we have
developed a new algorithm for convex optimization, which we are
calling Reflex. Reflex alternates greedy random reflecting steps, that
can get stuck in narrow reflecting corridors, with simply-biased
random reflecting steps that escape. We have early experimental
experience using a first implementation of Reflex, implemented in
Matlab, solving LP's (can be faster than Matlab's linprog), SDP's
(dense with several thousand variables), quadratic cone problems, and
some standard NETLIB problems.

Series: Other Talks

Come celebrate pi day with math club! Pot-luck, so bring food! Math club will be providing the pies, so we ask that everyone else try to bring more substantial food. ;)Bring any games and such you want as well.