Seminars and Colloquia by Series

Bochner-Riesz multipliers associated to convex planar domains with rough boundary

Series
Analysis Seminar
Time
Wednesday, September 30, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Laura Cladek – University of Wisconsin, Madison
We consider generalized Bochner-Riesz multipliers $(1-\rho(\xi))_+^{\lambda}$ where $\rho(\xi)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sj\"{o}lin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. For integers $m\ge 2$, we find domains such that $(1-\rho(\xi))_+^{\lambda}\in M^p(\mathbb{R}^2)$ for all $\lambda>0$ in the range $\frac{m}{m-1}\le p\le 2$, but for which $\inf\{\lambda:\,(1-\rho)_+^{\lambda}\in M_p\}>0$ when $p<\frac{m}{m-1}$. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the ``additive energy" of the boundary of a convex domain. Second, we associate a set of directions to a convex domain and define a sequence of Nikodym-type maximal operators corresponding to this set of directions. We show that domains that have low higher order energy, as well as those which have asymptotically good $L^p$ bounds for the corresponding sequence of Nikodym-type maximal operators, have improved $L^p$ bounds for the associated Bochner-Riesz operators over those proved by Seeger and Ziesler.

Morphogenesis of curved bilayer membranes

Series
Mathematical Biology Seminar
Time
Wednesday, September 30, 2015 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Norbert Stoop – MIT
Morphogenesis of curved bilayer membranes Buckling of curved membranes plays a prominent role in the morphogenesis of multilayered soft tissue, with examples ranging from tissue differentiation, the wrinkling of skin, or villi formation in the gut, to the development of brain convolutions. In addition to their biological relevance, buckling and wrinkling processes are attracting considerable interest as promising techniques for nanoscale surface patterning, microlens array fabrication, and adaptive aerodynamic drag control. Yet, owing to the nonlinearity of the underlying mechanical forces, current theoretical models cannot reliably predict the experimentally observed symmetry-breaking transitions in such systems. Here, we derive a generalized Swift-Hohenberg theory capable of describing the wrinkling morphology and pattern selection in curved elastic bilayer materials. Testing the theory against experiments on spherically shaped surfaces, we find quantitative agreement with analytical predictions separating distinct phases of labyrinthine and hexagonal wrinkling patterns. We highlight the applicability of the theory to arbitrarily shaped surfaces and discuss theoretical implications for the dynamics and evolution of wrinkling patterns.

Whitney differentiability in KAM theory II

Series
Dynamical Systems Working Seminar
Time
Tuesday, September 29, 2015 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rafael de la Llave – Georgia Tech (Math)
We will review the notion of Whitney differentiability and the Whitney embedding theorem. Then, we will also review its applications in KAM theory (continuation of last week's talk).

Tropical Homotopy Continuation

Series
Algebra Seminar
Time
Monday, September 28, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anders Jensen – Aarhus University, Denmark
In numerical algebraic geometry the key idea is to solve systems of polynomial equations via homotopy continuation. By this is meant, that the solutions of a system are tracked as the coefficients change continuously toward the system of interest. We study the tropicalisation of this process. Namely, we combinatorially keep track of the solutions of a tropical polynomial system as its coefficients change. Tropicalising the entire regeneration process of numerical algebraic geometry, we obtain a combinatorial algorithm for finding all tropical solutions. In particular, we obtain the mixed cells of the system in a mixed volume computation. Experiments suggest that the method is not only competitive but also asymptotically performs better than conventional methods for mixed cell enumeration. The method shares many of the properties of a recent tropical method proposed by Malajovich. However, using symbolic perturbations, reverse search and exact arithmetic our method becomes reliable, memory-less and well-suited for parallelisation.

Methods for multiscale inverse problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 28, 2015 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Dr. Christina Frederick – GA Tech
I will discuss inverse problems involving elliptic partial differential equations with highly oscillating coefficients. The multiscale nature of such problems poses a challenge in both the mathematical formulation and the numerical modeling, which is hard even for forward computations. I will discuss uniqueness of the inverse in certain problem classes and give numerical methods for inversion that can be applied to problems in medical imaging and exploration seismology.

Semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in R^n

Series
CDSNS Colloquium
Time
Monday, September 28, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chenchen Mou – Georgia Institute of Technology
In this talk, we will consider semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in R^n. This class of equations includes Bellman equations containing operators of Levy-Ito type. Holder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.

Prospective Student Day

Series
Other Talks
Time
Friday, September 25, 2015 - 14:00 for 3.5 hours
Location
Skiles 006
Speaker
Mohammad Ghomi – School of Mathematics, Georgia Tech
All students interested in graduate studies in the School of Math are invited to attend the "prospective student day." This event will offer the opportunity to hear about our graduate degree options, requirements for admission, as well as meet our Faculty and current graduate students. Prospective students from underrepresented groups in the Mathematical Sciences and students from the Atlanta area are particularly encouraged to attend. If you plan to attend, please send your name, the year you plan to graduate, and the college you are attending to dgs@math.gatech.edu. See the schedule for more details.

Sampling on lattices with free boundary conditions using randomized extensions

Series
ACO Student Seminar
Time
Friday, September 25, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Sarah Cannon – Georgia Institute of Technology
Many statistical physics models are defined on an infinite lattice by taking appropriate limits of the model on finite lattice regions. A key consideration is which boundary to use when taking these limits, since the boundary can have significant influence on properties of the limit. Fixed boundary conditions assume that the boundary cells are given a fixed assignment, and free boundary conditions allow these cells to vary, taking the union of all possible fixed boundaries. It is known that these two boundary conditions can cause significant differences in physical properties, such as whether there is a phase transition, as well as computational properties, including whether local Markov chain algorithms used to sample and approximately count are efficient. We consider configurations with free or partially free boundary conditions and show that by randomly extending the boundary by a few layers, choosing among only a constant number of allowable extensions, we can generalize the arguments used in the fixed boundary setting to infer bounds on the mixing time for free boundaries. We demonstrate this principled approach using randomized extensions for 3-colorings of regions of Z2 and lozenge tilings of regions of the triangle lattice, building on arguments for the fixed boundary cases due to Luby et.al. Our approach yields an efficient algorithm for sampling free boundary 3-colorings of regions with one reflex corner, the first result to efficiently sample free boundary 3-colorings of any nonconvex region. We also consider self-reducibility of free boundary 3-colorings of rectangles, and show our algorithm can be used to approximately count the number of free-boundary 3-colorings of a rectangle.

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