Progress in artificial intelligence (AI), including machine learning (ML),
is having a large effect on many scientific fields at the moment, with much more to come.
Most of this effect is from machine learning or "numerical AI", 
but I'll argue that the mathematical portions of "symbolic AI" 
- logic and computer algebra - have a strong and novel roles to play
that are synergistic to ML. First, applications to complex biological systems
can be formalized in part through the use of dynamical graph grammars.
Graph grammars comprise rewrite rules that locally alter the structure of
a labelled graph. The operator product of two labelled graph
rewrite rules involves finding the general substitution 
of the output of one into the input of the next - a form of variable binding 
similar to unification in logical inference algorithms. The resulting models
blend aspects of symbolic, logical representation and numerical simulation.
Second, I have proposed an architecture of scientific modeling languages
for complex systems that requires conditionally valid translations of
one high level formal language into another, e.g. to access different
back-end simulation and analyses systems. The obvious toolkit to reach for
is modern interactive theorem verification (ITV) systems e.g. those
based on dependent type theory (historical origins include Russell and Whitehead).
ML is of course being combined with ITV, bidirectionally.
Much work remains to be done, by logical people.
 

This is part 1 of a 2 part talk. It will cover background on:
Sketch of background knowledge in typed formal languages, 
Curry-Howard-Lambek correspondence, 
current computerized theorem verification, ML/ITV connections;
scientific modeling languages based on rewrite rules
(including dynamical graph grammars),
with some biological examples.

Model theory of fields is a very successful topic. I believe it is fair to say that most of this subject consists of detailed studies to particular fields (the reals, complex, p-adic, etc...). Largeness is a field-theoretic notion introduced by Pop in the nineties. This notion now plays a very important role in Galois theory and is increasingly being studied for other purposes. A number of people, including myself, have long believed that largeness should play an important role in the model theory of fields. This is mainly because all known model-theoretically tame fields are large, so one hopes that a general model-theoretic study of large fields might have a unifying effect on the model theory of fields. Over the past year we have begun to develop a model theory of large fields and in particular we have proven the stable fields conjecture for large fields. The key tool is a new topology on the K-points of a variety over a large field K.

This talk will involve a little algebraic geometry, but I will try to make it accessible to those with minimal background.

We formulate the Kurepa hypothesis (KH) and its generalizations in determinacy and prove KH fails there. More generally, we show that if $\kappa < \Theta$ and $cof(\kappa) > \omega$, then the set of branches through a  $\kappa$-tree must be well-orderable and has cardinality less than or equal to \kappa. We also show there are no maximal almost disjoint families at $\omega_1$ and its appropriate generalizations at regular cardinals $\kappa < \Theta$. This is joint work with William Chan and Steve Jackson.

1-dependent theories better known as NIP theories are the first class of the hierarchy of n-dependent structures. The random n-hypergraph is the canonical object which is n-dependent but not (n-1)-dependent. Thus the hierarchy is strict. Recently, in a joint work with Chernikov, we proved the existence of strictly n-dependent groups for all natural numbers n and we started studying their properties. The connected component over A, inspired by the definition of the connected component of algebraic group, is the intersection of all A-type definable subgroups of bounded index. A crucial fact about (type)definable groups in 1-dependent theories is the absoluteness of their connected components: Namely given a definable group G and a small set of parameters A, we have that the connected component of G over A coincides with the one over the empty set. A
 
We will give examples of n-dependent groups and discuss a generalization of absoluteness of the connected component to n-dependent theories.