Donald Knutson proposed the conjecture, later disproven and refined by Savitskii, that for every irreducible character of a finite group, there existed a virtual character such their tensor product was the regular character. In this talk, we disprove both this conjecture and its refinement. We then introduce the Knutson Index as a measure of the failure of Knutson's Conjecture and discuss its algebraic properties.

Automated Market Makers (AMMs) are a new prototype of 
trading venues which are revolutionising the way market participants 
interact. At present, the majority of AMMs are Constant Function 
Market Makers (CFMMs) where a deterministic trading function 
determines how markets are cleared. A distinctive characteristic of 
CFMMs is that execution costs for liquidity takers, and revenue for 
liquidity providers, are given by closed-form functions of price, 
liquidity, and transaction size. This gives rise to a new class of 
trading problems. We focus on Constant Product Market Makers with 
Concentrated Liquidity and show how to optimally take and make 
liquidity. We use Uniswap v3 data to study price and liquidity 
dynamics and to motivate the models.

For liquidity taking, we describe how to optimally trade a large 
position in an asset and how to execute statistical arbitrages based 
on market signals. For liquidity provision, we show how the wealth 
decomposes into a fee and an asset component. Finally, we perform 
consecutive runs of in-sample estimation of model parameters and 
out-of-sample trading to showcase the performance of the strategies.

Hypergraphs for multiscale cycles in structured data

Iris Yoon

Understanding the spatial structure of data from complex systems is a challenge of rapidly increasing importance. Even when data is restricted to curves in three-dimensional space, the spatial structure of data provides valuable insight into many scientific disciplines, including finance, neuroscience, ecology, biophysics, and biology. Motivated by concrete examples arising in nature, I will introduce hyperTDA, a topological pipeline for analyzing the structure of spatial curves that combines persistent homology, hypergraph theory, and network science. I will show that the method highlights important segments and structural units of the data. I will demonstrate hyperTDA on both simulated and experimental data. This is joint work with Agnese Barbensi, Christian Degnbol Madsen, Deborah O. Ajayi, Michael Stumpf, and Heather Harrington.

Minmax Connectivity and Persistent Homology 

Ambrose Yim

We give a pipeline for extracting features measuring the connectivity between two points in a porous material. For a material represented by a density field f, we derive persistent homology related features by exploiting the relationship between dimension zero persistent homology of the density field and the min-max connectivity between two points. We measure how the min-max connectivity varies when spurious topological features of the porous material are removed under persistent homology guided topological simplification. Furthermore, we show how dimension one persistent homology encodes a relaxed notion of min-max connectivity, and demonstrate how we can summarise the multiplicity of connections between a pair of points by associating to the pair a sub-diagram of the dimension one persistence diagram.

For geometric variational problems one often only has weak, rather than strong, compactness results and hence has to deal with the problem that sequences of (almost) critical points $u_j$ can converge to a limiting object with different topology.

A major challenge posed by such singular behaviour is that the seminal results of Simon on Lojasiewicz inequalities, which are one of the most powerful tools in the analysis of the energy spectrum of analytic energies and the corresponding gradient flows, are not applicable.

In this talk we present a method that allows us to prove Lojasiewicz inequalities in the singular setting of almost harmonic maps that converge to a simple bubble tree and explain how these results allow us to draw new conclusions about the energy spectrum of harmonic maps and the convergence of harmonic map flow for low energy maps from surfaces of positive genus into general analytic manifolds.