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Series: Research Horizons Seminar

Quantum topology is a collection of ideas and techniques for studying
knots and manifolds using ideas coming from quantum mechanics and
quantum field theory. We present a gentle introduction to this topic via
Kauffman bracket skein algebras of surfaces,
an algebraic object that relates "quantum information" about knots
embedded in the surface to the representation theory of the fundamental
group of the surface. In general, skein algebras are difficult to
compute. We associate to every triangulation of the
surface a simple algebra called a "quantum torus" into which the skein
algebra embeds. In joint work with Thang Le, we make use of this
embedding to give a simple proof of a difficult theorem.

Series: Research Horizons Seminar

If Google Scholar gives you everything you want, what could Georgia Tech Library possibly do for you? Come learn how to better leverage the tools you know and discover some resources you may not. Get to know your tireless Math Librarian and figure out how to navigate the changes coming with Library Next. This is also an opportunity to have a voice in the Library’s future, so bring ideas for discussion. Refreshments will be served.

Series: Research Horizons Seminar

A matrix completion problem starts with a partially specified matrix, where some entries are known and some are not. The goal is to find the unknown entries (“complete the matrix”) in such a way that the full matrix satisfies certain properties. We will mostly be interested in completing a partially specified symmetric matrix to a full positive semidefinite matrix. I will give some motivating examples and then explain connections to nonnegative polynomials and sums of squares.

Series: Research Horizons Seminar

Refreshments will be provided before the seminar.

Collective behavior can be seen in many animal species, such as flocking birds, herding mammals, and swarming bacteria. In the continuum limit, these phenomena can be modeled by nonlocal PDEs. In this talk, after discussing some PDE models for collective dynamics, I will focus on the analysis of the Keller-Segel equation, which models bacterial chemotaxis. Mathematically, this equation exhibits an intriguing "critical mass phenomenon": namely, solutions exist globally in time for all initial data whose mass is below some certain constant, whereas finite-time blow-up always happen if the initial mass is above this constant. I will introduce some useful analysis tools that lead to this result, and discuss some active areas of current research.

Series: Research Horizons Seminar

Refreshments will be provided before the seminar.

It's important to have a personal academic webpage—one that is up-to-date, informative, and easy to navigate. This workshop will be a hands-on guide to making an academic webpage and hosting it on the School of Math website. Webpage templates will be provided. Please bring a laptop if you have one, as well as a photograph of yourself for your new website. Come and get the help you need to create a great webpage!

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

For every surface (sphere, torus, etc.) there is an associated graph called the curve graph. The vertices are curves in the surface and two vertices are connected by an edge if the curves are disjoint. The curve graph turns out to be very important in the study of surfaces. Even though it is well-studied, it is quite mysterious. Here are two sample problems: If you draw two curves on a surface, how far apart are they as edges of the curve graph? If I hand you a surface, can you draw two curves that have distance bigger than three? We'll start from the beginning and discuss these problems and some related computational problems on surfaces.

Series: Research Horizons Seminar

If Google Scholar gives you everything you want, what could Georgia Tech Library possibly do for you? Come learn how to better leverage the tools you know and discover some resources you may not. Get to know your tireless Math Librarian and figure out how to navigate the changes coming with Library Next. This is also an opportunity to have a voice in the Library’s future, so bring ideas for discussion.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

We present a multiscale approach for identifying objects submerged in ocean beds by solving inverse problems in high frequency seafloor acoustics. The setting is based on Sound Navigation And Ranging (SONAR) imaging used in scientific, commercial, and military applications. The forward model incorporates simulations, by solving Helmholtz equations, on a wide range of spatial scales in the seafloor geometry. This allows for detailed recovery of seafloor parameters including the material type. Simulated backscattered data is generated using microlocal analysis techniques. In order to lower the computational cost of large-scale simulations, we take advantage of a library of representative acoustic responses from various seafloor parametrizations.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

A knot is a smooth embedding of S^1 into S^3 or R^3. There is a natural way to "add" two knots, called the connected sum. Under this operation, the set of knots forms a monoid. We will quotient by an equivalence relation called concordance to obtain a group, and discuss what is known about the structure of this group.

Series: Research Horizons Seminar

Food and Drinks will be provided before the seminar.

A fundamental result in Harmonic Analysis states that many functions defined over the interval [-\pi,\pi] can be decomposed into a Fourier series, that is, decomposed as sums of sines and cosines with integer frequencies. This allows one to describe very complicated functions in a simple way, and therefore provides with a strong tool to study the properties of different families of functions.However, the above decomposition does not hold -- or holds but is not efficient enough-- if the functions are no longer defined over an interval,( e.g. if a function is defined over a union of two disjoint intervals). We will discuss the question of whether similar decompositions are possible also in such cases, with the frequencies of the sines and cosines possibly being no longer integers.