Seminars and Colloquia by Series

Embeddings of lens spaces and rational homology balls in complex projective space II

Series
Geometry Topology Working Seminar
Time
Friday, September 30, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agniva Roy – Georgia Tech

Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. We will discuss a construction of small symplectic caps, using ideas first laid out by Gay in 2002, for rational homology balls bounded by lens spaces. This allows us to explicitly understand embeddings of these rational balls in CP2 that were earlier understood only through almost toric fibrations. This is joint work with John Etnyre, Hyunki Min, and Lisa Piccirillo.

Convexity of quadratic maps and convex hull via aggregation

Series
ACO Student Seminar
Time
Friday, September 30, 2022 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Shengding Sun – Georgia Tech Math –

Quadratic forms and their hidden convexity have been studied for decades, dating back to famous theorems by Toeplitz-Hausdorff, Dines and Brickman. It has very rich connection to optimization via Yakubovich's S-lemma. I will introduce these results, as well as an ongoing work of obtaining convex hull via aggregations, where we introduced the closely related notion of hidden hyperplane convexity.

Overview of GPU-based tools for studying multiscale and complex dynamics

Series
Time
Friday, September 30, 2022 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006 and online
Speaker
Andrey Shilnikov – GSU –
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09

I will review several parallel GPU-based approaches to better understand multistable dynamics of simple neural networks and global bifurcation unfolding of systems with deterministic chaos. 
 

Which magnetic fields support a zero mode?

Series
Math Physics Seminar
Time
Thursday, September 29, 2022 - 16:30 for 1 hour (actually 50 minutes)
Location
Skiles Room 005
Speaker
Michael Loss – School of Mathematics, Georgia Tech –

Please Note: Note the different time!

I present some results concerning the size of magnetic fields that support zero modes for the three dimensional Dirac equation and related problems for spinor equations. Critical quantities measuring this size are the 3/2 norm of the magnetic field B and the 3 norm of the vector potential A.  The point is that the spinor structure enters the analysis in a crucial way. This is joint work with Rupert Frank at LMU Munich.

Perturbation theory for systems with a first integral

Series
Stochastics Seminar
Time
Thursday, September 29, 2022 - 15:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Mark Freidlin – University of Maryland

I will consider the long-time influence of deterministic and stochastic perturbations of dynamical systems and diffusion processes with a first integral . A diffusion process on the Reeb graph of the first integral determines the long-time behavior of the perturbed system. In particular, I will consider stochasticity of long time behavior of deterministic systems close to a system with a conservation law. Which of the invariant  measures of the non-perturbed system will be limiting for a given class of perturbations also will be discussed.

Analyzing developmentally-mediated transitions in patterns of human sleep under homeostatic and circadian variation: A mathematical modeling approach

Series
Research Horizons Seminar
Time
Wednesday, September 28, 2022 - 12:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Christina Athanasouli – Georgia Institute of Technology –

Sleep and wake states are driven by interactions of neuronal populations in many areas of the human brain, such as the brainstem, midbrain, hypothalamus, and basal forebrain. The timing of human sleep is strongly modulated by the 24 h circadian rhythm and the homeostatic sleep drive, the need for sleep that depends on the history of prior awakening. The parameters dictating the evolution of the homeostatic sleep drive may vary with development or interindividual characteristics and have been identified as important parameters for generating the transition from multiple sleeps to a single sleep episode per day. Features of the mean firing rate of the neurons in the suprachiasmatic nucleus (SCN), the central pacemaker in humans, may differ with seasonality. In this talk, I will present our analysis of changes in sleep patterning under variation of homeostatic and circadian parameters using a mathematical model for human sleep-wake regulation. I will also talk about the fundamental tools we employ to understand the dynamics of the model, such as the construction of a circle map that captures the timing of sleep onsets on successive days. Analysis of the structure and bifurcations in the map reveals changes in the average number of sleep episodes per circadian day in a period-adding-like structure caused by the separate or combined effects of circadian and homeostatic variation. Time permitting, I will talk about some of our current work on modeling sleep patterns in early childhood using experimental data.

Hardy spaces for Fourier integral operators

Series
PDE Seminar
Time
Tuesday, September 27, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
Online: https://gatech.zoom.us/j/95574359880?pwd=cGpCa3J1MFRkY0RUeU1xVFJRV0x3dz09
Speaker
Jan Rozendaal – IMPAN –

It is well known that the wave operators cos(t (−∆)) and sin(t (−∆)) are not bounded on Lp(Rn), for n≥2 and 1≤p≤∞, unless p=2 or t=0. In fact, for 1 < p < ∞ these operators are bounded from W2s(p),p  to Lp(Rn) for s(p) := (n−1)/2 | 1/p − 1/2 |, and this exponent cannot be improved. This phenomenon  is symptomatic of the behavior of Fourier integral operators, a class of oscillatory operators which includes wave propagators, on Lp(Rn).

In this talk, I will introduce a class of Hardy spaces HFIOp (Rn), for p ∈ [1,∞],on which Fourier integral operators of order zero are bounded. These spaces also satisfy Sobolev embeddings which allow one to recover the optimal boundedness results for Fourier integral operators on Lp(Rn).

However, beyond merely recovering existing results, the invariance of these spaces under Fourier integral operators allows for iterative constructions that are not possible when working directly on Lp(Rn). In particular, we shall indicate how one can use this invariance to obtain the optimal fixed-time Lp regularity for wave equations with rough coefficients. We shall also mention the connection of these spaces to the phenomenon of local smoothing.

This talk is based on joint work with Andrew Hassell and Pierre Portal (Aus- tralian National University), and Zhijie Fan, Naijia Liu and Liang Song (Sun Yat- Sen University).

The stable cohomology of the level-l subgroup of the mapping class group (Joint Topology Seminar @ UGA)

Series
Geometry Topology Seminar
Time
Monday, September 26, 2022 - 16:30 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 322)
Speaker
Andrew Putman – Notre Dame

After an introduction to how to think about the mapping class groupand its cohomology, I will discuss a recent theorem of mine saying
that passing to the level-l subgroup does not change the rational cohomology in a stable range.

Obstructions to reversing Lagrangian surgery (Joint Topology Seminar @ UGA)

Series
Geometry Topology Seminar
Time
Monday, September 26, 2022 - 15:00 for 1 hour (actually 50 minutes)
Location
University of Georgia (Boyd 322)
Speaker
Orsola Capovilla Searle – UC Davis

Given an immersed, Maslov-0, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-0, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-0, exact Lagrangian filling with genus g ≥ 1 and p double points can be obtained from such a Lagrangian surgery on a filling of genus g − 1 with p+1 double points. To show this, we establish the connection between the existence of an immersed, Maslov-0, exact Lagrangian filling of a Legendrian Λ that has p double points with action 0 and the existence of an embedded, Maslov-0, exact Lagrangian cobordism from p copies of a Hopf link to Λ. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-0, exact Lagrangian cobordisms between Legendrian links. Joint work with Noemie Legout, Maylis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.

Solving decomposable sparse polynomial systems

Series
Algebra Seminar
Time
Monday, September 26, 2022 - 13:30 for 1 hour (actually 50 minutes)
Location
Clough 125 Classroom
Speaker
Thomas Yahl – TAMU –

Polynomial systems can be effectively solved by exploiting structure present in their Galois group. Esterov determined two conditions for which the Galois group of a sparse polynomial system is imprimitive, and showed that the Galois group is the symmetric group otherwise. A system with an imprimitive Galois group can be decomposed into simpler systems, which themselves may be further decomposed. Esterov's conditions give a stopping criterion for decomposing these systems and leads to a recursive algorithm for efficient solving.

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