Virtual Logic Seminars Worldwide
There are two fairly complete virtual logic seminar lists maintained by
and by Anton Bernshteyn.
Two particularly popular, completely virtual logic seminars:
UW Logic Seminar Schedule
All talks in the academic year 2020-21 will be virtual.
Links are provided below.
The regular "virtual" meeting time is Mondays at 3:30PM (Central Time)
during the spring semester 2021 (joining the
Midwest Computability Seminar) roughly every other week,
and with some joint meetings Tuesdays at 4PM (Central Time) with the
The recurring Zoom link for the Midwest Computability Seminar is:
Meeting ID: 997 5433 2165,
The recurring Webex link for the Midwest Model
Theory Seminar is:
Meeting number: 126 507 5533,
Password: (stable, but ask me if you don't have it,
it is different from the password in fall 2020!)
- 1/25/2021 1PM (local UW logic seminar),
Joel David Hamkins,
University of Oxford, England
Title: Can there be natural instances of nonlinearity
in the hierarchy of consistency strength?
Abstract: It is a mystery often mentioned in the foundations of mathematics
that our best and strongest mathematical theories seem to be linearly ordered
and indeed well-ordered by consistency strength. Given any two of the familiar
large cardinal hypotheses, for example, generally one of them proves the
consistency of the other. Why should this be? The phenomenon is seen as
significant for the philosophy of mathematics, perhaps pointing us toward
the ultimately correct mathematical theories. And yet, we know as a purely
formal matter that the hierarchy of consistency strength is not well-ordered.
It is ill-founded, densely ordered, and nonlinear. The statements usually used
to illustrate these features are often dismissed as unnatural or as
Gödelian trickery. In this talk, I aim to overcome that criticism —
as well as I am able to — by presenting a variety of natural hypotheses that
reveal ill-foundedness in consistency strength, density in the hierarchy of
consistency strength, and incomparability in consistency strength.
The talk should be generally accessible to university logic students,
requiring little beyond familiarity with the incompleteness theorem
and some elementary ideas from computability theory.
Discussion and commentary can be made on the speaker's web site at
- 1/26/2021 4PM (Midwest Model Theory
University of Cambridge, England
Title: NIP approximate groups and arithmetic
Abstract: I will speak about joint work with Anand Pillay on the structure of
finite approximate groups satisfying a local NIP assumption. Our results can
be seen as a unification of the model-theoretic study of "tame" arithmetic
regularity with work of Hrushovski and Breuillard, Green, and Tao on the
structure theory of approximate groups.
- 2/1/2021 3:30PM (Midwest Computability Seminar),
Swansea University, Wales
Title: The structure of Weihrauch degrees - what we
know and what we don't know
Abstract: The Weihrauch degrees are a popular setting for classifying the
computational content of mathematical theorems. Understanding their structure
is useful as a technical tool in concrete classifications. Moreover, their
structure tells us something about how degrees of non-computability look like
in principle. In this talk, I'll summarize what is already known about the
structure of the Weihrauch degrees, and try to draw attention to some open
problems. For example, we know that they form a distributive lattice, which
is not a Heyting algebra and which is not complete. We have further natural
algebraic operations, and we know of a few that they are definable in terms
of others. The Medvedev degrees embed into the Weihrauch degrees as a
lattice, as do the many-one degrees (but in a weird way).
- 2/8/2021 3:30PM (local UW logic seminar),
University of Ljubljana, Slovenia
Title: Synthetic mathematics with an excursion into
Abstract: According to Felix Klein, “synthetic geometry is that which studies
figures as such, without recourse to formulae, whereas analytic geometry
consistently makes use of such formulae as can be written down after the
adoption of an appropriate system of coordinates”. To put it less eloquently,
the synthetic method axiomatizes geometry directly by construing points and
lines as primitive notions, whereas the analytic method builds a model,
the Euclidean plane, from the real numbers.
Do other branches of mathematics posses the synthetic method, too? For
instance, what would “synthetic topology” look like? To build spaces out of
sets, as topologists usually do, is the analytic way. The synthetic approach
must construe spaces as primitive and axiomatize them directly, without any
recourse to sets. It cannot introduce continuity as a desirable property of
functions, for that would be like identifying straight lines as the
It is indeed possible to build the synthetic worlds of topology, smooth
analysis, measure theory, and computability. In each of them, the basic
structure – topological, smooth, measurable, computable – is implicit by
virtue of permeating everything, even logic itself. The synthetic worlds
demand an economy of thought that the unaccustomed mind finds frustrating at
first, but eventually rewards it with new elegance and conceptual clarity.
The synthetic method is still fruitfully related to the analytic method by
interpretation of the former in models provided by the latter.
We demonstrate the approach by taking a closer look at synthetic computability,
whose central axiom states that there are countably many countable subsets of
the natural numbers. The axiom is validated and explained by its interpretation
in the effective topos, where it corresponds to the familiar fact that the
computably enumerable sets may be computably enumerated. Classic theorems of
computability may be proved in a straightforward manner, without reference to
any notion of computation. We end by showing that in synthetic computability,
Turing reducibility is expressed in terms of sequential continuity of maps
between directed-complete partial orders.
- 2/9/2021 4PM (Midwest Model Theory
Stevens Institute of Technology, Hoboken, New Jersey
Title: Fields interpretable in the free group
Abstract: In this talk I will show that no infinite field is interpretable in
the first-order theory of nonabelian free groups.
- 2/15/2021 3:30PM (Midwest Computability Seminar),
Becher, University of Buenos Aires, Argentina
Title: Normal numbers and perfect necklaces
Abstract: The most famous example of a normal number is Champernowne's
constant 0.123456789101112… Although the definition is very simple,
the original proof of normality requires quite some work. In this talk,
I present "perfect necklaces", a combinatorial object that yields a simple
proof of Champernowne's normality result. And with a class of them, the
"nested perfect necklaces", I explain M. Levin's constant, the number with
the fastest known speed of convergence to normality.
- 2/22/2021 3:30PM (local UW logic seminar),
José ("Goyo") Mijares,
California State University-Los Angeles
Title: Coideals and the Local Ramsey Property
Abstract: We will talk about characterizations of the local Ramsey property
relative to several types of coideal; in different contexts, including
infinite games, block sequences of vectors and topological Ramsey spaces.
- 2/23/2021 4PM (Midwest Model Theory
University of Chicago, Illinois
Title: Continuous combinatorics and natural
Abstract: The theory of graph quasirandomness studies several asymptotic
properties of the random graph that are equivalent when stated as properties
of a deterministic graph sequence and was one of the main motivations for the
theory of dense graph limits, also known as theory of graphons. Since the
theory of graphons can itself be used to study graph quasirandomness and can
be generalized to a theory of dense limits of models of a universal first-order
theory, a natural question is whether a general theory of quasirandomness is
In this talk, I will briefly introduce the general theory of dense limits of
combinatorial objects (often associated with the name continuous combinatorics)
and talk about the notion of natural quasirandomness, a generalization of
quasirandomness to the same general setting of universal first-order theories.
The main concept explored by our quasirandomness properties is that of unique
coupleability that roughly means that any alignment of two limit objects on
the same ground set "looks random".
This talk is based on joint work with Alexander A. Razborov.
- 3/1/2021 3:30PM (Midwest Computability Seminar),
Eisenträger, Pennsylvania State University, University Park
Title: A topological approach to undefinability in
algebraic extensions of the rationals
Abstract: In 1970, Matiyasevich proved that Hilbert’s Tenth Problem over the
integers is undecidable, building on work by Davis-Putnam-Robinson. Hilbert’s
Tenth Problem over the rationals is still open, but it could be resolved by
giving an existential definition of the integers inside the rationals. Proving
whether such a definition exists is still out of reach. However, we will show
that only “very few” algebraic extensions of the rationals have the property
that their ring of integers are existentially or universally definable.
Equipping the set of all algebraic extensions of the rationals with a natural
topology, we show that only a meager subset has this property. An important
tool is a new normal form theorem for existential definitions in such
extensions. As a corollary, we construct countably many distinct computable
algebraic extensions whose rings of integers are neither existentially nor
universally definable. Joint work with Russell Miller, Caleb Springer, and
- 3/8/2021 3:30PM (local UW logic seminar),
Daniel Erman, UW
Title: Ultraproducts in commutative algebra
Abstract: I’ll explain how an ultraproduct of polynomial rings in an unbounded
number of variables turns out to be surprising, and how this was used to
answer some old questions in commutative algebra. This is joint work with
Steven Sam and Andrew Snowden.
- 3/9/2021 4PM (Midwest Model Theory
University of Denver, Colorado
Title: Fraïssé classes with simply
characterized big Ramsey degrees
Abstract: Analogues of the infinite Ramsey theorem to infinite structures have
been studied since the 1930’s, when Sierpiński gave a coloring of pairs
of rationals into two colors such that, in any subset of the rationals forming
a dense linear order, both colors persist. Such a coloring is called
“unavoidable” since both colors persist in any infinite substructure
isomorphic to the original (in this case the rationals as a linear order).
In the 1970’s, Galvin showed that two is the optimum number for pairs of
rationals, while Erdős, Hajnal and Pósa extended Sierpiński’s
result to colorings of edges in the Rado graph. These results instigated a
steady stream of results for the next several decades, a pinnacle of which was
the work of Laflamme, Sauer, and Vuksanović finding the exact number of
colors for unavoidable colorings of finite graphs inside the Rado graph, as
well as other Fraïssé structures with finitely many binary
relations, including the generic tournament. This exact number is called the
“big Ramsey degree”, a term coined by Kechris, Pestov, and
In this talk, we will provide a brief overview of the area of big Ramsey
degrees on Fraïssé limits. Then we will present recent joint work
with Rebecca Coulson and Rehana Patel characterizing the big Ramsey degrees
for some seemingly disparate Fraïssé classes. We formulate an
amalgamation property, which we call the Substructure Free Amalgamation
Property, and show that every Fraïssé relational class with
finitely many relations satisfying SFAP has big Ramsey degrees which are
characterized in a manner as simply as those of the Rado graph.
A more general property for disjoint amalgamation classes, which we call
SDAP+, also ensures the same simple characterization of big Ramsey
degrees. One of the novelties of our approach is that we build trees of
quantifier-free 1-types with special nodes coding the vertices in a given
enumerated Fraïssé limit. Then we use the method of forcing to do
an unbounded search for a finite object, which produces in ZFC the exact big
Ramsey degrees for these structures. SDAP+ holds for unrestricted
relational structures, relational structures with forbidden 3-irreducible
substructures, and others, producing new lines of results while recovering
in a streamlined manner several previous results, including those of Laflamme,
Sauer, and Vuksanović.
- 3/15/2021 3:30PM (Midwest Computability Seminar),
Chris Conidis, City University of New York-College of Staten
Title: The reverse mathematics of Noether's
Abstract: We will survey some recent results in the reverse mathematics of
Noetherian algebra, and in particular Noether's Decomposition Lemma, which
states that a Noetherian ring has only finitely many minimal prime ideals.
Such an analysis naturally leads one to formulate a combinatorial principle,
namely, the Tree-Antichain Principle, which states that every binary-branching
tree with infinitely many splittings has an infinite antichain.
- 3/22/2021 3:30PM (local UW logic seminar),
Bartošová, University of Florida, Gainesville
Title: Extensions of and by compact groups and their
universal minimal flows
Abstract: We will explore how the dynamical notion of universal minimal flow
interacts with the topological group notion of group extension of or by a
compact group. As an application, for a locally compact group of automorphisms
of a countable first-order structure with a compact open normal subgroup,
we completely describe the underlying space of its universal minimal flow.
- 3/29/2021 3:30PM (local UW logic seminar),
University of California-San Diego
Title: Big Ramsey degrees for binary free amalgamation
classes with a finite constraint set
Abstract: The infinite Ramsey theorem states that whenever one colors the
n-element subsets of the natural numbers using finitely many colors, one can
find an infinite subset whose n-element subsets all receive a single color.
When considering colorings of finite substructures of more interesting
countable structures, the situation becomes more complex. For example, one
can color the pairs of the rational linear order using two colors so that on
any infinite subset which itself is a rational linear order, both colors of
pairs appear. However, two colors is the worst possible; given such a coloring
with three colors, one can pass to an infinite, rationally ordered subset
where only two colors appear. Combining these results, we say that 2 is the
big Ramsey degree of the two-element linear order. In this talk, we give a
survey on some recent work involving big Ramsey degrees, culminating in a
discussion of recent joint work with Balko, Chodounský, Dobrinen,
Hubička, Konečný and Vena which characterizes the precise
big Ramsey degrees for many classes of binary Fraïssé structures.
- 4/5/2021 3:30PM (Midwest Computability Seminar),
Knight, University of Notre Dame, Indiana
Title: Describing structures and classes of
Abstract: The talk will recall some definitions and results on Scott
complexity of individual countable structures, Borel complexity of classes of
structures, and Borel cardinality, for comparing classification problems for
different classes. I will try to indicate how methods from computability may
sometimes be used in this connection, even for results that do not mention
computability. I will state some results on torsion-free Abelian groups,
due to Downey-Montalbán, Hjorth, Thomas, and Paolini-Shelah. Finally,
I will mention a project with Turbo Ho and Russell Miller, saying what we
would like to do, but have not done.
- 4/12/2021 3:30PM (local UW logic seminar),
University of San Diego, California
Title: Avoiding large cardinals in category
Abstract: Category theory appears to study "all" objects of a given sort, such
as the category of all groups. In ZFC, such categories are proper classes;
but this prohibits constructions like functor categories. Thus, category
theorists often use instead the category of groups (say) of rank below some
inaccessible. But this raises new issues, such as whether arithmetical
theorems proven using category theory (e.g. Wiles's proof of Fermat's Last
Theorem) are still true without large cardinals. McLarty has argued
convincingly that they are, but at the expense of requiring familiar
category-theoretic notions to be encoded in unnatural-looking ways.
This situation can be improved by using a "synthetic" language for categories.
Instead of using set theory as the implicit foundation for all mathematics,
we can regard ordinary category-theoretic proofs as implicitly formalized
in a type-theoretic language, which contains an apparently "inaccessible"
universe of sets. Then we encapsulate McLarty's encodings (which are
essentially a variant of "indexed category theory") in a single theorem
interpreting this type-theoretic language into a set theory containing a
much weaker universe.
- 4/19/2021 3:30PM (Midwest Computability Seminar),
Greenberg, Victoria University of Wellington, New Zealand
Title: The strength of Borel Wadge comparability
Abstract: Wadge’s comparability lemma says that the Borel sets are almost
linearly ordered under Wadge reducibility: For any two Borel sets A and B,
either A is a continuous pre-image of B, or B is a continuous pre-image of the
complement of A. Wadge’s proof uses Borel determinacy, which is not provable
in second-order arithmetic (H. Friedman). Using deep and complex techniques,
Louveau and Saint-Raymond showed that Borel Wadge comparability is provable
in second-order arithmetic, but did not explore its precise proof-theoretic
strength. I will discuss recent work aiming to clarify this.
One of the main technical tools we use is Montalbán’s “true stage”
machinery, originally developed for iterated priority constructions in
computable structure theory, but more recently used by Day and Marks for
their resolution of the decomposability conjecture.
Joint work with Adam Day, Matthew Harrison-Trainor, and Dan Turetsky.
- 4/20/2021 4PM (Midwest Model Theory
Title: Effective non-local-finiteness in flat strongly
Abstract: I'll talk about a connection
between model theory and recursive model theory. In answering Zilber's
trichotomy conjecture, Hrushovski built a strongly minimal theory with a
geometric property called flatness. Flatness precludes the existence of a
definable group, but it does much more. I'll discuss what the assumption of
flatness tells us about being able to build recursive models of strongly
The needs of the recursion-theoretic constructions translate into two purely
model-theoretic questions. I'll discuss the solution to these questions and
try to highlight the connections with recursive model theory.
Joint work with Omer Mermelstein.
- Monday, Apr. 26th: no logic seminar due to hybrid
- 5/3/2021 3:30PM (Midwest Computability Seminar),
Nies, University of Auckland, New Zealand
- 5/4/2021 4PM (Midwest Model Theory
Purdue University, West Lafayette, Indiana
- 8/18/2020 3PM (Midwest Computability Seminar),
Title: Redundancy of information: lowering effective
Abstract: A natural way to measure the similarity between two infinite binary
sequences X and Y is to take the (upper) density of their symmetric difference.
This is the Besicovitch distance on Cantor space. If d(X,Y) = 0, then we say
that X and Y are coarsely equivalent. Greenberg, Miller, Shen, and Westrick
(2018) proved that a binary sequence has effective (Hausdorff) dimension 1
if and only if it is coarsely equivalent to a Martin-Löf random sequence.
They went on to determine the best and worst cases for the distance from a
dimension t sequence to the nearest dimension s>t sequence. Thus, the
difficulty of increasing dimension is understood.
Greenberg et al. also determined the distance from a dimension 1 sequence to
the nearest dimension t sequence. But they left open the general problem of
reducing dimension, which is made difficult by the fact that the information
in a dimension s sequence can be coded (at least somewhat) redundantly.
Goh, Miller, Soskova, and Westrick recently gave a complete solution.
I will talk about both the results in the 2018 paper and the more recent work.
In particular, I will discuss some of the coding theory behind these results.
No previous knowledge of coding theory is assumed.
- 8/25/2020 3PM (Midwest Model Theory Seminar),
University of California-Los Angeles
Title: Model-theoretic tree properties
Abstract: The first model-theoretic tree properties were introduced by Shelah
as a by-product of his analysis of forking in stable theories. Since then,
other tree properties have appeared and, together, these combinatorial
dividing lines (TP, TP1/SOP2, TP2,
SOP1, etc.) serve as the basis for a growing body of research in
model theory. I'll survey the work done in this area (and try to justify the
idea that it can be understood as an area) by explaining three of the core
ingredients in the theory developed so far: generalized indiscernibles,
dividing at a generic scale, and amalgamation.
- 9/1/2020 3PM (Midwest Computability Seminar),
University of California-Berkeley
Title: Part 1 of Martin's Conjecture for order
Abstract: Martin's Conjecture is an attempt to make precise the idea that the
only natural functions on the Turing degrees are the constant functions,
the identity, and transfinite iterates of the Turing jump. The conjecture is
typically divided into two parts. Very roughly, the first part states that
every natural function on the Turing degrees is either eventually constant
or eventually increasing and the second part states that the natural functions
which are increasing form a well-order under eventual domination, where the
successor operation in this well-order is the Turing jump.
In the 1980's, Slaman and Steel proved that the second part of Martin's
Conjecture holds for order-preserving Borel functions. In joint work with
Benny Siskind, we prove the complementary result that (assuming analytic
determinacy) the first part of the conjecture also holds for order-preserving
Borel functions (and under AD, for all order-preserving functions).
Our methods also yield several other new results, including an equivalence
between the first part of Martin's Conjecture and a statement about the
Rudin-Keisler order on ultrafilters on the Turing degrees.
In my talk, I will give an overview of Martin's Conjecture and then describe
our new results.
- 9/8/2020 3PM (Midwest Model Theory Seminar),
Western Illinois University, Macomb (visiting University of Vienna, Austria)
Title: Quantifier elimination for o-minimal groups
expanded by a valuational cut
Abstract: We let R be an o-minimal expansion of a group in a language in which
Th(R) eliminates quantifiers, and we let C be a valuational cut in R. We show
that if nonforking in certain Morley sequences is symmetric, then the theory
of R expanded by a predicate for C and a small number of constants eliminates
quantifiers. This is a generalization of results on o-minimal fields with
convex subrings satisfying some extra conditions such as T-convexity or
o-minimality of the residue field. This is joint work with C. F. Ealy.
- 9/15/2020 3PM (Midwest Computability Seminar),
University of Notre Dame, Indiana
Title: Noncomputable coding, density, and
Abstract: We introduce the
operations in order to construct sets of arbitrary intrinsic density from any
Martin-Löf random. We then show that these operations are useful more
generally for working with other notions of density as well, in particular,
for viewing Church and MWC stochasticity as a form of density.
- 9/22/2020 3PM (local UW logic seminar),
University of Lyon 1, France
Title: Invariant measures on the space of linear
orders on an ℵ0-categorical structure
Abstract: Let M be an ℵ0-categorical structure and denote by
LO(M) the compact space of linear orders on M. We investigate the probability
measures on LO(M) invariant under the natural action of the automorphism
group of M and prove, under rather general model-theoretic assumptions,
that either M has a definable linear order or LO(M) carries a unique
invariant measure (which can be easily and explicitly described). For
many structures M, the space LO(M) is the universal minimal flow of the
group Aut(M), and our work is in part motivated by a general unique
ergodicity question of Angel, Kechris, and Lyons in topological
dynamics. Our proof uses techniques from model theory, representation
theory, and probability theory, but no special knowledge will be assumed
in the talk. I will also provide some background and motivation. This is
joint work with Colin Jahel.
- 9/29/2020 3PM (Midwest Computability Seminar),
Drake University, Des Moines, Iowa
Title: Effective dimension and the intersection of
random closed sets (video and slides)
Abstract: The connection between the effective dimension of sequences and
membership in algorithmically random closed subsets of Cantor space was first
identified by Diamondstone and Kjos-Hanssen. In this talk, I highlight joint
work with Adam Case in which we extend Diamondstone and Kjos-Hanssen's result
by identifying a relationship between the effective dimension of a sequence
and what we refer to as the degree of intersectability of certain families of
random closed sets (also drawing on work by Cenzer and Weber on the
intersections of random closed sets).
As we show, (1) the number of relatively random closed sets that can have a
non-empty intersection varies depending on the choice of underlying probability
measure on the space of closed subsets of Cantor space - this number being the
degree of intersectability of a given family of random closed sets - and
(2) the effective dimension of a sequence X is inversely proportional to the
minimum degree of intersectability of a family of random closed sets, at least
one of which contains X as a member. Put more simply, a sequence of lower
dimension can only be in random closed sets with more branching, which are
thus more intersectable, whereas higher dimension sequences can be in random
closed sets with less branching, which are thus less intersectable, and the
relationship between these two quantities (that is, effective dimension and
degree of intersectability) can be given explicitly.
- 10/6/2020 3PM (local UW logic seminar),
Title: Complexity profiles and the generic Muchnik
Abstract: The generic Muchnik degrees, introduced by Schweber, give a way of
comparing the computability-theoretic content of uncountable structures.
Though obscured slightly by the need for some set-theoretic machinery, I hope
to highlight how this notion really gives an easy and natural way to talk
about computable structure theory for uncountable structures.
I will focus on the tool of complexity profiles.
Complexity profiles are a way of measuring, for two structures A generic
Muchnik reducible to B, which subsets of A can be defined using B.
The complexity profile of A on itself is the natural analog of considering
the relatively intrinsically Σksets in A.
Using complexity profiles, I will compare three generic muchnik degrees:
Cantor space < Baire space < the Borel-complete degree. In particular,
I will describe some dichotomy theorems regarding simple expansions of these
and describe how to build degrees strictly between them.
(Joint work with Joseph S. Miller, Noah Schweber, and Mariya Soskova.)
- 10/13/2020 3PM (Midwest Computability Seminar),
University of Warsaw, Poland
Title: Reverse mathematics of combinatorial principles
over a weak base theory
Abstract: Reverse mathematics studies the strength of axioms needed to prove
various mathematical theorems. Often, the theorems have the form
∀X ∃Y ψ(X,Y) with X and Y denoting subsets of N and ψ
arithmetical, and the logical strength required to prove them is closely
related to the difficulty of computing Y given X. In the early decades of
reverse mathematics, most of the theorems studied turned out to be equivalent,
over a relatively weak base theory, to one of just a few typical axioms,
which are themselves linearly ordered in terms of strength. More recently,
however, many statements from combinatorics, especially Ramsey theory,
have been shown to be pairwise inequivalent or even logically incomparable.
The usual base theory used in reverse mathematics is RCA0, which is intended
to correspond roughly to the idea of "computable mathematics". The main two
axioms of RCA0 are: comprehension for computable properties of
natural numbers and mathematical induction for c.e. properties. A weaker
theory, in which induction for c.e. properties is replaced by induction for
computable properties, has also been introduced, but it has received much less
attention. In the reverse mathematics literature, this weaker theory is known
In this talk, I will discuss some results concerning the reverse mathematics
of combinatorial principles over
RCA*0. We will focus
mostly on Ramsey's theorem and some of its well-known special cases: the
chain-antichain principle CAC, the ascending-descending chain principle ADS,
and the cohesiveness principle COH.
The results I will talk about are part of a larger project joint with Marta
Fiori Carones, Katarzyna Kowalik, Tin Lok Wong, and Keita Yokoyama.
- 10/20/2020 3PM (Midwest Model Theory Seminar),
Title: Using Ultraproducts to Compare Continuous
Abstract: We revisit two research programs that were proposed in the 1960's,
remained largely dormant for five decades, and then become hot areas of
research in the last decade.
The monograph "Continuous Model Theory" by Chang and Keisler, Annals of
Mathematics Studies (1966), studied structures with truth values in
[0,1], with formulas that had continuous functions as connectives, sup and
inf as quantifiers, and equality. In 2008, Ben Yaacov, Berenstein, Henson, and
Usvyatsov introduced the model theory of metric structures, where equality
is replaced by a metric, and all functions and predicates are required to be
uniformly continuous. This has led to an explosion of research with results
that closely parallel first-order model theory, with many applications to
analysis. In my forthcoming paper "Model Theory for Real-valued Structures",
the "Expansion Theorem" allows one to extend many model-theoretic results
about metric structures to general [0,1]-valued structures - the structures in
the 1966 monograph but without equality.
My paper "Ultrapowers Which are Not Saturated", J. Symbolic Logic 32
(1967), 23-46, introduced a pre-ordering M ≤ N on all first-order structures,
that holds if every regular ultrafilter that saturates N saturates M, and
suggested using it to classify structures. In the last decade, in a remarkable
series of papers, Malliaris and Shelah showed that that pre-ordering gives a
rich classification of simple first-order structures. Here, we lay the
groundwork for using the analogous pre-ordering to classify [0,1]-valued and
- 10/27/2020 3PM (Midwest Computability Seminar),
University of Notre Dame, Indiana
Title: Fickleness and bounding lattices in the
recursively enumerable Turing degrees
Abstract: The ability for a recursively enumerable Turing degree d to bound
certain important lattices depends on the degree's fickleness. For instance,
d bounds L7 (or the 1-3-1 lattice) if and only if d's fickleness is
> ω (≥ ωω, respectively). We work towards
finding a lattice that characterizes the > ω2 levels of
fickleness and seek to understand the challenges faced in finding such a
lattice. The candidate lattices considered include those that are generated
from three independent points, and upper semilattices that are obtained by
removing the meets from important lattices.
- 11/3/2020 3PM (local UW logic seminar),
University of Vienna, Austria
Title: Independent families in the countable and the
Abstract: Independent families on ω are families of infinite sets of
integers with the property that for any two disjoint finite subfamilies A and B,
the set ⋂ A \ ⋃ B is infinite. Of particular interest are the
sets of the possible cardinalities of maximal independent families, which we
refer to as the spectrum of independence. Even though we do have the tools to
control the spectrum of independence at ω (at least to a large extent),
there are many relevant questions regarding higher counterparts of independence
in generalized Baire spaces, which remain wide open.
- 11/10/2020 3PM (Midwest Computability Seminar),
University of Leeds, England
Title: Randomness notions and reverse mathematics
Abstract: There are many notions of algorithmic randomness in addition to
classic Martin-Löf randomness, such as 2-randomness, weak 2-randomness,
computable randomness, and Schnorr randomness. For each notion of randomness,
we consider the statement "For every set Z, there is a set X that is random
relative to Z" as a set-existence principle in second-order arithmetic, and
we compare the strengths of these principles. We also show that a well-known
characterization of 2-randomness in terms of incompressibility can be proved
, which is non-trivial because it requires avoiding the use
This work is joint with André Nies.
- 11/17/2020 3PM (local UW logic seminar),
Jaap van Oosten,
Utrecht University, Netherlands
Title: Partial combinatory algebras: Variations on a
topos-theoretic theme (video
Abstract: We show a number of constructions in the theory of partial
combinatory algebras which highlight the interplay between topos theory
and recursion theory.
- 11/24/2020 3PM (Midwest Computability Seminar),
Wellesley College, Massachusetts
Title: Complexity of root-taking in power series
fields and related problems
Abstract: In earlier work with Knight and Solomon, we bounded the computational
complexity of the root-taking process over Puiseux and Hahn series, two kinds
of generalized power series. But it is open whether the bounds given are
optimal. By looking at the most basic steps in the root-taking process for
Hahn series, we together with Hall and Knight became interested in the
complexity of problems associated with well-ordered subsets of a fixed ordered
abelian group. Here we provide an overview of the results so far in both
- 12/1/2020 3PM (local UW logic seminar),
University of Birmingham, England
Title: Equality of mathematical structures
Abstract: Two groups are regarded to be the same if they are isomorphic, two
topological spaces are regarded to be the same if they are homeomorphic, two
metric spaces are regarded to be the same if they are isometric, two categories
are regarded to be the same if they are equivalent, etc. In Voevodsky's
Univalent Foundations (HoTT/UF), the above become theorems: we can replace
"are regarded to be the same" by "are the same". I will explain how this works.
I will not assume previous knowledge of HoTT/UF or type theory.
- 12/8/2020 3PM (Midwest Computability Seminar),
Linda Brown Westrick,
Pennsylvania State University, State College
Title: Luzin's (N) and randomness reflection
Abstract: We show that a computable real-valued function f has Luzin's property
(N) if and only if it reflects
-randomness, if and
only if it reflects
relative to Kleene's O, and if and only if
it reflects Kurtz randomness relative to Kleene's O. Here a function f is
said to reflect a randomness notion R if whenever f(x) is R-random, then x is
R-random as well. If additionally f is known to have bounded variation,
then we show f has Luzin's (N) if and only if it reflects weak-2-randomness,
and if and only if it reflects Kurtz randomness relative to 0'. This links
classical real analysis with algorithmic randomness.
Joint with Arno Pauly and Liang Yu.
Archived UW Logic Seminar Pages