Georgia Institute of Technology College of Sciences

A Jones surface for a knot in the three-sphere is an essential surface whose boundary slopes, Euler characteristic, and number of sheets correspond to quantities defined from the asymptotics of the degrees of colored Jones polynomial. The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran predicts that there are Jones surfaces for every knot. A link diagram D is said to be a Murasugi sum of two links D' and D'' if a state graph of D has a cut vertex, which separates the graph into two state graphs of D' and D'', respectively. We may obtain a state surface in the complement of the link K represented by D by gluing the state surface for D and the state surface for D' along the disk filling the circle represented by the cut vertex in the state graph. The resulting surface is called the Murasugi sum of the two state surfaces. We consider near-adequate links which are certain Murasugi sums of near-alternating link diagrams with an adequate link diagram along their all-A state graphs with an additional graphical constraint. For a near-adequate knot, the Murasugi sum of the corresponding state surface is a Jones surface by the work of Ozawa. We discuss how this proves the Strong Slope Conjecture for this class of knots.

New and proposed interplanetary missions increasingly require the design of trajectories within challenging multi-body environments that stress or exceed the capabilities of the two-body design methodologies typically used over the last several decades. These current methods encounter difficulties because they often require appreciable user interaction, result in trajectories that require significant amounts of propellant, or miss potential mission-enabling options. The use of dynamical systems methods applied to three-body and multi-body models provides a pathway to obtain a fuller theoretical understanding of the problem that can then result in significant improvements to trajectory design in each of these areas. In particular, the computation of periodic Lagrange point and resonant orbits along with their associated invariant manifolds and heteroclinic connections are crucial to finding the dynamical channels that provide new or more optimal solutions. These methods are particularly effective for mission types that include multi-body tours, Earth-Moon transfers, approaches to moons, and trajectories to asteroids. The inclusion of multi-body effects early in the analysis for these applications is key to providing a more complete set of solutions that includes improved trajectories that may otherwise be missed when using two-body methods. This seminar will focus on two representative trajectory design applications that are especially challenging. The first is the design of tours using flybys of planets or moons with a particular emphasis on the Galilean moons and Europa. In this case, the exploration of the design space using the invariant manifolds of resonant and Lyapunov orbits provides information such as the resonance transitions that are required as part of the tour. The second application includes endgame scenarios, which typically involve an approach to a moon with an objective of either capturing into orbit around the moon or landing on the surface. Often, the invariant manifolds of particular orbits may be used in this case to provide a wide set of approach options for both capture and landing analyses. New methods will also be discussed that provide a foundation for rigorously analyzing the transit of trajectories through the libration point regions that is necessary for the approach and capture phase for bodies such as Europa and the Moon. These methods provide a fundamentally new method to search for the invariant manifolds of orbits and hyperbolic invariant sets associated with libration points while giving additional insight into the dynamics of the flow in these regions.

It is well-known, that any univariate polynomial matrix A over the complex numbers that takes only positive semidefinite values on the real line, can be factored as A=B^*B for a polynomial square matrix B. For real A, in general, one cannot choose B to be also a real square matrix. However, if A is of size nxn, then a factorization A=B^tB exists, where B is a real rectangular matrix of size (n+1)xn. We will see, how these correspond to the factorizations of the Smith normal form of A, an invariant not usually associated with symmetric matrices in their role as quadratic forms. A consequence is, that the factorizations canusually be easily counted, which in turn has an interesting application to minimal length sums of squares of linear forms on varieties of minimal degree.

In this talk, we will have an overview of: the Gaming Industry, specifically on the Video Slot Machine segment; the top manufactures in the world; the game design studio Gimmie Games, who we are, what we do; what is the process of making a video slot game; what is the basic structure of the math model of a slot game; current strong math models in the market; what is the roll of a game designer in the game development process; the skill set needed to be a successful Game Designer. Only basic probability knowledge is required for this talk.

We will discuss a way of explicitly constructing ribbon knots using one-two handle canceling pairs. We will also mention how this is related to some recent work of Yasui, namely that there are infinitely many knots in (S^3, std) with negative maximal Thurston-Bennequin invariant for which Legendrian surgery yields a reducible manifold.

Motivated by the investigation on the dependence on ``epsilon" in the Dvoretzky's theorem, I will show some refinements of the classical concentration of measure for convex functions. Applications to convexity will be presented if time permits. The talk will be based on joint works with Peter Pivovarov and Petros Valettas.

In this talk, I will discuss some examples of sparse signal detection problems in the context of binary outcomes. These will be motivated by examples from next generation sequencing association studies, understanding heterogeneities in large scale networks, and exploring opinion distributions over networks. Moreover, these examples will serve as templates to explore interesting phase transitions present in such studies. In particular, these phase transitions will be aimed at revealing a difference between studies with possibly dependent binary outcomes and Gaussian outcomes. The theoretical developments will be further complemented with numerical results.

Braid and knot theory in 3-dimensional Euclidean space are related by classical theorems of Alexander and Markov. We will talk about closed braids in higher dimensions, and generalizations of Alexander's theorem.

Excitable media are characterized by a local tendency towards synchronization, which can lead to waves of excitement through the system. Two classical discrete, deterministic models of excitable media are the cyclic cellular automaton and Greenberg-Hastings models, which have been extensively studied on lattices, Z^d. One is typically interested in whether or not sites are excited (change states) infinitely often (fluctuation vs fixation), and if so, whether the density of domain walls between disagreeing sites tends to 0 (clustering). We introduce a new comparison process for the 3-color variants of these models, which allows us to study the asymptotic rate at which a site gets excited. In particular, for a class of infinite trees we can determine whether the rate is 0 or positive. Using this comparison process, we also analyze a new model for pulse-coupled oscillators in one dimension, introduced recently by Lyu, called the firefly cellular automaton (FCA). Based on joint works with Lyu and Gravner.

A conversation with Fr. Mike May, S.J., a UC Berkeley math PhD (1988) and chair of the SLU Department of Mathematics and Computer Science (2000-2002, 2004-2011).

In enumerative combinatorics, it is quite common to have in hand a number of known initial terms of a combinatorial sequence whose behavior you'd like to study. In this talk we'll describe two techniques that can be used to shed some light on the nature of a sequence using only some known initial terms. While these methods are, on the face of it, experimental, they often lead to rigorous proofs. As we talk about these two techniques -- automated conjecturing of generating functions, and the method of differential approximation -- we'll exhibit their usefulness through a variety of combinatorial topics, including matchings, permutation classes, and inversion sequences.

In a high harmonic generation (HHG) experiment, an intense laser pulse is sent through an atomic gas, and some of that light is converted to very high harmonics through the interaction with the gas. The spectrum of the emitted light has a particular, nearly universal shape. In this seminar, I will describe my efforts to derive a classical reduced Hamiltonian model to capture this phenomenon. Beginning with a parent Hamiltonian that yields the equations of motion for a large collection of atoms interacting self-consistently with the full electromagnetic field (Lorentz force law + Maxwell's equations), I will follow a sequence of reductions that lead to a reduced Hamiltonian which is computationally tractable yet should still retain the essential physics. I will conclude by pointing out some of the still-unresolved issues with the model, and if there's time I will discuss the results of some preliminary numerical simulations.

The Georgia Scientific Computing Symposium (GSCS) is a forum for professors, postdocs, graduate students and other researchers in Georgia to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The symposium has been held every year since 2009 and is open to the entire research community. The format of the day-long symposium is a set of invited presentations, poster sessions and a poster blitz, and plenty of time to network with other attendees. More information at http://euler.math.uga.edu/cms/GSCS-2017

Mozghan Entekhabi (Wichita State University) Radial Limits of Bounded Nonparametric Prescribed Mean Curvature Surfaces ; Miyuki Koiso (Kyushu University) Stability and bifurcation for surfaces with constant mean curvature ; Vladimir Oliker (Emory University) Freeform lenses, Jacobian equations, and supporting quadric method(SQM) ; Sungho Park (Hankuk University of Foreign Studies) Circle-foliated minimal and CMC surfaces in S^3 ; Yuanzhen Shao (Purdue University) Degenerate and singular elliptic operators on manifolds with singularities ; Ray Treinen (Texas State University) Surprising non-uniqueness for the 2D floating ball ; See http://www.math.uab.edu/sgs/ for abstracts and further details.