Georgia Institute of Technology College of Sciences

Algebraic matroids record the algebraic dependencies among elements in a field extension, similar to the linear dependencies of vectors in a vector space. Realizing a given matroid by elements in a field extension can depend on the characteristic of the field. I will talk about the possible characteristic sets of algebraic matroids. An essential tool is the one-dimensional group construction of an algebraic matroid, which turns the realization problem for algebraic matroids into a linear problem over the endomorphism ring of a one-dimensional algebraic group.

Understanding the behavior of large physical systems is a problem of fundamental importance in mathematical physics. Analysis of systems of many interacting particles is key for understanding various phenomena from physical sciences (e.g. gases in non-equilibrium, galactic dynamics) to social sciences (e.g. modeling social networks). Similarly, the description of systems of weakly nonlinear interacting waves, referred to as wave turbulence theory, finds a wide range of applications from solid state physics and water waves to plasma theory and oceanography. However, with the size of the system of interest being extremely large, deterministic prediction of its behavior is practically impossible, and one resorts to an averaging description. In this talk, we will discuss about kinetic theory, which is a mesoscopic framework to study the qualitative properties of large systems. As we will see, the main idea behind kinetic theory is that, in order to identify averaging quantities of large systems, one studies their asymptotic behavior as the size of the system tends to infinity, with the hope that the limiting effective equation will reveal properties observed in a system of large, but finite size. We will focus on the Boltzmann equation, which is the effective equation for systems of interacting particles, and its higher order extensions, as well as the kinetic wave equation which describes systems of many nonlinearly interacting waves.

The Torelli group of a surface is a natural yet mysterious subgroup of the mapping class group.  We will discuss a few recent results about finiteness properties of the Torelli group, as well as a result about the cohomological dimension of the Johnson filtration.  

Smooth ergodic theory provides a framework for studying systems exhibiting dynamical chaos, features of which include sensitive dependence with respect to initial conditions, correlation decay (even for deterministic systems!) and complicated fractal-like attractor geometry. This talk will be an overview of some of these ideas as they apply to evolutionary PDE, with an emphasis on dissipative semilinear parabolic problems, and a discussion of some of my work in this direction, joint with: Lai-Sang Young and Sam Punshon-Smith. 

Mobile sampling concerns finding surfaces upon which any function with Fourier transform supported in a symmetric convex set must have some large values.   We shall describe a sharp sufficient for mobile sampling in terms of the surface density introduced by Unnikrishnan and Vetterli.  Joint work with Mishko Mitkovski and Manasa Vempati.

A central question in the field of optimal transport studies optimization problems involving two measures on a common metric space, a source and a target. The goal is to find a mapping from the source to the target, in a way that minimizes distances. A remarkable fact discovered by Caffarelli is that, in some specific cases of interest, the optimal transport maps on a Euclidean metric space are Lipschitz. Lipschitz regularity is a desirable property because it allows for the transfer of analytic properties between measures. This perspective has proven to be widely influential, with applications extending beyond the field of optimal transport.

In this talk, we will further explore the Lipschitz properties of transport maps. Our main observation is that, when one seeks Lipschitz mappings, the optimality conditions mentioned above do not play a major role. Instead of minimizing distances, we will consider a general construction of transport maps based on interpolation of measures, and introduce a set of techniques to analyze the Lipschitz constant of this construction. In particular, we will go beyond the Euclidean setting and consider Riemannian manifolds as well as infinite-dimensional spaces.

Some applications, such as functional inequalities, normal approximations, and generative diffusion models will also be discussed.

Gutzwiller semi-classical quantization, Boven-Sinai-Ruelle dynamical zeta functions for chaotic dynamical systems, statistical mechanics partition functions, and path integrals of quantum field theory are often presented in ways that make them appear as disjoint, unrelated theories. However, recent advances in describing fluid turbulence by its dynamical, deterministic Navier-Stokes underpinning, without any statistical assumptions, have led to a common field-theoretic description of both (low-dimension) chaotic dynamical systems, and (infinite-dimension) spatiotemporally turbulent flows. 

I have described the remarkable experimental progress connecting turbulence to deterministic dynamics in the Sept 24, 2023 colloquium (the recoding is available on the website below). In this seminar I will use a lattice discretized field theory in 1 and 1+1 dimension to explain how temporal `chaos', `spatiotemporal chaos' and `quantum chaos' are profitably cast into the same field-theoretic framework.

https://ChaosBook.org/overheads/spatiotemporal/

The talk will also be on Zoom:   GaTech.zoom.us/j/95338851370

Chip-firing asks a simple question: Given a group of people and an initial integer distribution of dollars among the people including people in debt, can we redistribute the money so that no one ends up in debt? This simple question with its origins in combinatorics can be reformulated using concepts from graph theory, linear algebra, graph orientation algorithms, and even divisors in Riemann surfaces. This presentation will go over a summary of Part 1 of Divisors and Sandpiles by Scott Corry and David Perkinson. Moreover, we will cover three various approaches to solve this problem: a linear algebra approach with the Laplacian, an algorithmic approach with Dhar's algorithm, and an algebraic geometry approach with a graph-theoretic version of the Riemann-Roch theorem by Baker and Norine. If we have time, we will investigate additional topics from Part 2 and Part 3. As true to the title, there will be a non-alcoholic drinking game involved with this presentation and participation will be completely voluntary. Limited refreshments (leftover Coca-Cola I found in the grad student lounge) and plastic cups will be served.

After finishing the proof of equivalence of the Chebyshev constant of a set and its logarithmic capacity, we will start to discuss classical and recent results on estimates and asymptotics of Chebyshev numbers.

We consider a service system where incoming tasks are instantaneously assigned to one out of many heterogeneous server pools. All the tasks sharing a server pool are executed in parallel and the execution times do not depend on the class of the server pool or the number of tasks currently contending for service. However, associated with each server pool is a utility function that does depend on the class of the server pool and the number of tasks currently sharing it. These features are characteristic of streaming and online gaming services, where the duration of tasks is mainly determined by the application but congestion can have a strong impact on the quality-of-service (e.g., video resolution and smoothness). We derive an upper bound for the mean overall utility in steady-state and introduce two load balancing policies that achieve this upper bound in a large-scale regime. Also, the transient and stationary behavior of these asymptotically optimal load balancing policies is characterized in the same large-scale regime.

Reaction networks are commonly used to model a variety of physical systems ranging from the microscopic world like cell biology and chemistry, to the macroscopic world like epidemiology and evolution biology. A biologically relevant property that reaction networks can have is absolute concentration robustness (ACR), which refers to when a steady-state species concentration is maintained even when initial conditions are changed. Networks with ACR have been observed experimentally, for example, in E. coli EnvZ-OmpR and IDHKP-IDH systems. Another reaction network property that might be desirable is multistationarity-the capacity for two or more steady states, since it is often associated with the capability for cellular signaling and decision-making.

While the two properties seem to be opposite, having both properties might be favorable as a biochemical network may require robustness in its internal operation while maintaining flexibility as a signal-response mechanism. As such, our driving motivation is to explore what network structures can produce ACR and multistationarity. We show that it is highly atypical for both properties to coexist in very small and very large reaction networks without special structures. However, it is possible for them to coexist in certain classes of reaction networks. I will discuss in detail one such class of networks, which consists of multisite phosphorylation-dephosphorylation cycles with a ``paradoxical enzyme".

Most of us have been taught geometry from the perspective of equations and how those equations act on a given space. But in the 1870’s, Felix Klein’s Erlangen program was more concerned about the maps that preserved the geometric structures of a space rather than the equations themselves. In this talk, I will present some modern results from this perspective and show details of how to reconstruct the equations that preserve geometric structures.

Mammalian cortical networks are known to process sensory information utilizing feedforward and feedback connections along the cortical hierarchy as well as intra-areal connections between different cortical layers. While feedback and feedforward signals have distinct layer-specific connectivity motifs preserved across species, the computational relevance of these connections is not known. Motivated by predictive coding theory, we study how expected and unexpected visual information is encoded along the cortical hierarchy, and how layer-specific feedforward and feedback connectivity contributes to differential, context-dependent representations of information across cortical layers, by analyzing experimental recordings of neural populations and also by building a recurrent neural network (RNN) model of the cortical microcircuitry. Experimental evidence shows that information about identity of the visual inputs and expectations are encoded in different areas of the mouse visual cortex, and simulations with our RNNs which incorporate biologically plausible connectivity motifs suggest that layer-specific feedforward and feedback connections may be the key contributor to this differential representation of information.