The seminar is moved to Monday June 7, 2021

In 2011, Malliaris and Shelah proved that finite stable graphs satisfy a strengthened version of Szemeredi’s Regularity Lemma, with polynomial bounds and no irregular pairs. In 2017, Terry and Wolf proved an analogue of this for stable subsets of finite abelian groups, based on Green’s “arithmetic regularity lemma". Roughly speaking, they showed that stable subset of a finite abelian group can be approximated by a union of cosets of a subgroup whose index is bounded by a exponential function depending only on the stability constant and approximation error. These results for abelian groups were qualtitatively generalized to all finite groups by C., Pillay, and Terry, and then to finite subsets of arbitrary groups by Martin-Pizarro, Palacin, and Wolf. However, the generalizations for non-abelian groups used model-theoretic techniques involving ultraproducts, and thus produced no explicit quantitative bounds. In this talk, I will discuss a new proof of these results, which avoids the use of ultraproducts and yields effective bounds. These techniques also improve the bound in the abelian case from exponential to polynomial, and yield the Polynomial Freiman-Ruzsa Conjecture for finite stable subsets of arbitrary groups.
 

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.

Part II: Current and planned work.
Operator algebra semantics (and the relevant Fock spaces)
for scientific modeling languages based on rewrite rules,
derivation of simulation algorithms,
semantics-approximating transformations using ML;
semantics-preserving transformations using ITV ?,
relation to graph grammar pushout semantics,
verbs vs. nouns vs. expressions and logic of "eclectic types",
compositional and specialization hierarchies,
"Tchicoma" conceptual architecture.

Two classical results of Magidor are: 

(1) from large cardinals it is consistent to have reflection at $\aleph_{\omega+1}$, and

(2) from large cardinals it is consistent to have the failure of SCH at $\aleph_\omega$.

These principles are at odds with each other. The former is a compactness type principle. (Compactness is the phenomenon where if a certain property holds for every smaller substructure of an object, then it holds for the entire object.) In contrast, failure of SCH is an instance of incompactness. The natural question is whether we can have both of these simultaneously. We show the answer is yes

We describe a Prikry style iteration, and use it to force stationary reflection in the presence of not SCH.  Then we obtain this situation at $\aleph_\omega$. This is joint work with Alejandro Poveda and Assaf Rinot.

We present some ideas involved in the proof of the equiconsistency
of AD_\reals + Theta is regular and the existence of a strong,
pseudo-homogeneous ideal on P_{\omega_1}(\reals). Some variations of this
hypothesis are also shown to be equiconsistent with AD_\reals + Theta is
regular. This work is related to and partially answers a long-standing
conjecture of Woodin regarding the equiconsistency of AD_\reals + Theta is
regular and CH + the nonstationary ideal on \omega_1 is \omega_1-dense. We
put this result in a broader context of the general program of understanding
connections between canonical models of large cardinals, models of
determinacy, and strong forcing axioms (e.g. PFA, MM). This is joint work with G. Sargsyan and T. Wilson.