Modelling dementia

Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are
devastating conditions with poorly understood mechanisms and no known cure.

Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. In this talk, I will show how we can use mathematical modelling to gain insight into this process and, doing so, gain some understanding on how the brain works. In particular, by looking at protein dynamics on the neuronal network, we can unravel some of the universal features associated with dementia that are driven by both the network topology and protein kinetics. By further coupling this approach with functional models of the brain, we will show that we can explain important aspects of function loss during disease development.

Counting things:
enumerative algebraic geometry from string theory

For centuries mathematicians have generalised statements like “there is a unique line through any 2 points”, but with increasing technical
difficulties. It was not until the late 1990s that new ideas from
mathematics and physics allowed rigorous definitions to be made of these “curve counting problems”.

I will outline two different ways to count curves, assuming only a bit
of undergraduate complex analysis. The “MNOP conjecture” states that the two definitions give equivalent information. Its recent proof by
Pandharipande and Pixton has enabled the solution of various counting problems, such as the “KKV conjecture” from string theory, expressing all cuTBArve counting problems on “K3 surfaces” in terms of modular forms.