## Why algebraic topology and neuroscience?

Network science has proven itself an invaluable tool in modern neuroscience. However, network models intrinsically assume that their objects of study are

*dyadic*, i.e., that all pertinent information can be recorded in pairwise relations. This assumption is suspect in many complex systems, even those whose underlying structure is clearly a network. Methods from algebraic topology allow us to move beyond this limitation, providing a powerful mathematical tool set for understanding the mesoscale structure and dynamic properties of systems built on non-linear, higher-order interactions., as well as sophisticated but unexplored techniques for the study of networks. On the other hand, problems from neuroscience provide a steady stream of new questions that lie just beyond the frontiers of current methods in applied topology, practically begging for new collaborative efforts.

- Chad Giusti, Robert Ghrist and Danielle S Bassett, "Two's company, three (or more) is a simplex" in Journal of Computational Neuroscience, 2016

## What can the function of a system tell us about its structure?

Consider place cells in the hippocampus, which are known to have firing rates modulated by the spatial location of an animal in its environment; that is, these neurons have

*receptive fields*corresponding to locations in their environment. Because we understand their receptive fields, it is possible to use their activity to do things like decode position or recover a map of the animal's environment as it moves. However,*in the absence of a known behavioral correlate*, can we determine whether a population of neurons encode such information? Using tools from applied algebraic topology, we answer this question in the affirmative. We further study a range of biological network models through this lens, providing a blueprint for future applications.

- Chad Giusti, Eva Pastalkova, Carina Curto and Vladimir Itskov, "Clique topology reveals intrinsic geometric structure in neural correlations" in Proceedings of the National Academy of the Sciences USA, 2015
- Ann Sizemore, Chad Giusti, Danielle S. Bassett, "Classification of weighted networks through mesoscale homological features" in Journal of Complex Networks, 2016
- Chad Giusti, CliqueTop: a MATLAB package for topological analysis of symmetric matrices (though I recommend using Julia and Eirene)

## What can the structure of a system tell us about its function?

Suppose we know something about the structure of a neural system or that of the information it encodes. What can we conclude about its

However, in real biological networks where codes are sparse, we can restrict our attention to only maximal coactivity patterns. We show that any such sparse code can be realized by a one-layer feedforward network, so such networks are

*(combinatorial)**code,*i.e., collection of possible coactivation patterns? Two such structures which are surprisingly deeply related are neural populations with*convex*receptive field structures and*one-layer feedforward networks.*Applying the classical nerve theorem, we obtain a class of*local obstructions*in codes which preclude the existence of either structure.However, in real biological networks where codes are sparse, we can restrict our attention to only maximal coactivity patterns. We show that any such sparse code can be realized by a one-layer feedforward network, so such networks are

*universal simplicial complex approximators.*In the context of convex receptive fields, we show that only the absence of intersections of such maximal patterns obstruct such an architecture, and explicitly construct realizations when all such are present.

- Chad Giusti and Vladimir Itskov "A no-go theorem for one-layer feedforward networks" in Neural Computation, 2014
- Joshua Cruz, Chad Giusti, Vladimir Itskov and William Kronholm, "On open and closed convex codes", submitted

The human brain is a complex, multi-scale information processing system. At the macroscopic level, activity in anatomical brain regions correlates with behavioral function, suggesting that we can think of these regions as local processing units whose interactions underlie cognition. Understanding the architecture of this system is therefore a fundamental first step toward mechanistic models of cognitive processes. Diffusion imaging data, a proxy for the white matter tracts believed to form the "structure" of this network, provides a picture of this architecture. We studied the interaction between densely connected "local processing" units called

*cliques*and the interplay between strong and weak connections between cliques, represented by*cycles,*uncovering a range of non-local features which may support complex cognitive tasks.

- Ann Sizemore, Chad Giusti, Richard F. Betzel, Danielle S Bassett, "Closures and cavities in the human connectome", submitted

Densely packed granular media (gravel, for example) do not exhibit homogeneous distribution of internal forces. Rather, the inter-particle normal forces form a complex pattern of so-called

*force chains,*whose structure is conjectured to be responsible for many of the physical properties of the material. Due to the intricate nature of the networks, it is useful focus on intrinsically mesoscale statistical properties, which have intuitive physical interpretation. To this end, we developed the*Topological Compactness Factor*of the collection of force chains, which roughly measures rotational stability of the chain across its branch points, and demonstrated that this measurement can be used to recover the external pressure applied to the system.- Chad Giusti, Lia Papadopoulos, Eli Owens, Karen Daniels and Danielle S. Bassett, "Topological and geometric measurements of force chain structure" in Physical Review E, 2016

## What is the shape of the space of point clouds?

The space of point clouds -- that is, of configuration spaces -- is a classical subject of study in algebraic topology which finds its way into theory and applications throughout the sciences. One application of particular interest is that the limiting family of configurations in infinite-dimensional Euclidean space provide models for classifying spaces, and by studying the family as a whole one can extract new understanding of the structure of families of groups. Using the geometry of these models, we have computed the mod-two cohomology of both symmetric groups (another subject of classical study, here understood in a different way) and of alternating groups (the first such computation for a family of simple groups).

- Chad Giusti and Dev Sinha, "Mod-two cohomology of alternating groups", in preparation
- Chad Giusti, Paolo Salvatore and Dev Sinha, "Mod-two cohomology of symmetric groups as a Hopf ring" in Journal of Topology, 2012
- Chad Giusti and Dev Sinha, "Fox-Neuwirth cell structures and the cohomology of symmetric groups" in Configuration Spaces: Geometry, Combinatorics and Topology, 2012

## Can we approximate (spaces of) knots with simpler objects?

In his development of the theory of finite type invariants of knots, Vassiliev utilized a sequence of finite-dimensional polynomial approximations of the space of knots. Due to the intricate geometry both of the individual spaces and of the maps between them, these spaces have proved difficult to analyze directly. To put the study of these invariants on a more solid, geometric footing, I developed the spaces of

*plumbers' knots*, which decompose into combintorial cell complexes, and which fit together into a directed system through which one can follow cells. These spaces provide a new basis for undestanding "unstable" finite type invariants, as well as opening the door to computational approaches to Vasilliev theory.- A gallery of Plumbers' Knots
- Chad Giusti, "Plumbers' knots", under revision
- Chad Giusti, "Unstable Vassiliev theory", under revision
- My PhD dissertation: "Plumbers' knots and unstable Vassiliev theory".

## Funding

I would like to thank the following organizations for their generous (current or former) funding support.

My ORCID is orcid.org/0000-0003-2412-3622, my ArXiv page is here and my Google Scholar page is here.