### An introduction to Nonlinear Algebra

- Series
- Research Horizons Seminar
- Time
- Wednesday, November 9, 2022 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Papri Dey – Georgia Institute of Technology – pdey33@gatech.edu

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- Series
- Research Horizons Seminar
- Time
- Wednesday, November 9, 2022 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Papri Dey – Georgia Institute of Technology – pdey33@gatech.edu

Nonlinear algebra is a newly evolving field which borrows ideas from the various core areas of mathematics.

In this talk, the theoretical and computational aspects of nonlinear algebra emerging from algebraic geometry, tropical geometry, tensor algebra, and semidefinite programming will be briefly discussed and demonstrated with examples.

This talk is mainly based on the book "Invitation to Nonlinear Algebra" by Mateusz Michalek and Bernd Sturmfels.

- Series
- Research Horizons Seminar
- Time
- Wednesday, November 2, 2022 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Gong Chen – Georgia Institute of Technology – gc@math.gatech.edu

Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”. Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations. After an informal introduction to dispersive equations, I will illustrate how to understand the long-time behavior solutions to dispersive waves via various results I obtained over the years.

- Series
- Research Horizons Seminar
- Time
- Wednesday, October 12, 2022 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Michael Damron – Georgia Tech – mdamron6@math.gatech.edu

Random and irregular growth is all around us. We see it in the form of cancer growth, bacterial infection, fluid flow through porous rock, and propagating flame fronts. Simple models for these processes originated in the '50s with percolation theory and have since given rise to many new models and interesting mathematics. I will introduce a few models (percolation, invasion percolation, first-passage percolation, diffusion-limited aggregation, ...), along with some of their basic properties.

- Series
- Research Horizons Seminar
- Time
- Wednesday, October 5, 2022 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Tom Kelly – Georgia Tech – tom.kelly@gatech.edu

An order-n Latin square is an n by n array of n symbols such that each row and column contains each symbol exactly once. Latin squares were famously studied by Euler in the 1700s, and at present they are still a central object of study in modern extremal and probabilistic combinatorics. In this talk, I will give some history about Latin squares, share some simple-to-state yet notoriously difficult open problems, and present some of my own research on Latin squares.

- 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 – cathanasouli3@gatech.edu

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.

- Series
- Research Horizons Seminar
- Time
- Wednesday, November 10, 2021 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Jonathan Simone – Georgia Institute of Technology – jsimone7@gatech.edu

Given a knot $K$ in the 3-sphere, one can ask: what kinds of surfaces in the 3-sphere are bounded by $K$? One can also ask: what kinds of surfaces in the 4-ball (which is bounded by the 3-sphere) are bounded by $K$? In this talk we will discuss how to construct surfaces in both the 3-sphere and in the 4-ball bounded by a given knot $K$, how to obstruct the existence of such surfaces, and explore what is known and unknown about surfaces bounded by so-called torus knots.

- Series
- Research Horizons Seminar
- Time
- Wednesday, November 3, 2021 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Diego Cifuentes – Georgia Tech – diego.cifuentes@isye.gatech.edu

Semidefinite programming (SDP) is a very well behaved class of convex optimization problems. We will introduce this class of problems, illustrate how it allows to approximate many practical nonconvex optimization problems, and discuss the role of low rank structure.

- Series
- Research Horizons Seminar
- Time
- Wednesday, October 27, 2021 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006 / https://bluejeans.com/396232086/4264
- Speaker
- Hannah Turner – Georgia Tech – hannah.turner@math.gatech.edu

**Please Note:** Talk will be presented live as well as streamed. Questions will be fielded by the organizer.

We'll discuss various operations which can be applied to a knot to "simplify" or "unknot" it. Study of these "unknotting operations" began in the 1800s and continues to be an active area of research in low-dimensional topology. Many of these operations have applications more broadly in topology including to 3- and 4-manifolds and even to DNA topology. I will define some of these operations and highlight a few open problems.

- Series
- Research Horizons Seminar
- Time
- Wednesday, October 20, 2021 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Alexander Ruys De Perez – Georgia Tech – amrp3@gatech.edu

Neural codes are inspired by John O'Keefe's discovery of the place cell, a neuron in the mammalian brain which fires if and only if its owner is in a particular region of physical space. Mathematically, a neural code $C$ on n neurons is a collection of subsets of $\{1,...,n\}$, with the subsets called codewords. The implication is that $C$ encodes how the members of some collection $\{U_i\}_{i=1}^n$ of subsets of $\mathbb{R}^d$ intersect one another.

The principal question driving the study of neural codes is that of convexity. Given just the codewords of $C$, can we determine if there is a collection of open convex subsets $ \{U_i\}_{i=1}^n$ of some $\mathbb{R}^d$ for which $C$ is the code? A convex code is a code for which there is such a realization of open convex sets. While the question of determining which codes are convex remains open, there has been significant progress as many large families of codes can now be ruled as convex or nonconvex. In this talk, I will give an overview of some of the results from this work. In particular, I will focus on a phenomenon called a local obstruction, which if found in a code forbids convexity.

- Series
- Research Horizons Seminar
- Time
- Wednesday, October 13, 2021 - 12:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Hannah Choi – Georgia Tech – hannahch@gatech.edu

**Please Note:** The seminar will also be streamed live at https://bluejeans.com/787128769/7101 . Questions will be fielded by the organizer.

The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. One of many unique features of the mammalian brain network is its spatial embedding and hierarchical organization. I will discuss effects of these structural characteristics on network dynamics as well as their computational implications with a focus on the flexibility between modular and global computations and predictive coding.

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