Let $X$ be a surface germ in $R^n$ (a closed connected two-dimensional germ at the origin) definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic). There are two natural metrics on $X$: the inner metric, where the distance between two points in $X$ is the length of a shortest path in $X$ connecting these points, and the outer metric, where the distance between two points of $X$ is their distance in $R^n$. A set $X$ is called normally embedded if its inner and outer metrics are equivalent.
There are three natural equivalence relations associated with these metrics. Two sets $X$ and $Y$ are inner (outer) Lipschitz equivalent if there is an inner (outer) bi-Lipschitz homeomorphism $h:X\to Y$. The sets $X$ and $Y$ are ambient Lipschitz equivalent if the homeomorphism $h$ can be extended to a bi-Lipschitz homeomorphism of the ambient space. The ambient equivalence is stronger than the outer equivalence, and the outer equivalence is stronger then the inner equivalence. There are finitely many ambient Lipschitz equivalence classes in any definable family.
Classification of surface germs with respect to inner Lipschitz equivalence was done by Lev Birbrair in 1999, but classification with respect to outer Lipschitz equivalence is still an open problem. Classification of surface germs in $R^4$ with respect to ambient Lipschitz equivalence, even when they are ambient topologically trivial, includes all of the knot theory.
In this talk, an introduction to Lipschitz geometry of definable surface germs, and a review of recent progress in their outer Lipschitz classification, will be given.

Let $X$ be a surface germ in $R^n$ (a closed connected two-dimensional germ at the origin) definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic). There are two natural metrics on $X$: the inner metric, where the distance between two points in $X$ is the length of a shortest path in $X$ connecting these points, and the outer metric, where the distance between two points of $X$ is their distance in $R^n$. A set $X$ is called normally embedded if its inner and outer metrics are equivalent.
There are three natural equivalence relations associated with these metrics. Two sets $X$ and $Y$ are inner (outer) Lipschitz equivalent if there is an inner (outer) bi-Lipschitz homeomorphism $h:X\to Y$. The sets $X$ and $Y$ are ambient Lipschitz equivalent if the homeomorphism $h$ can be extended to a bi-Lipschitz homeomorphism of the ambient space. The ambient equivalence is stronger than the outer equivalence, and the outer equivalence is stronger then the inner equivalence. There are finitely many ambient Lipschitz equivalence classes in any definable family.
Classification of surface germs with respect to inner Lipschitz equivalence was done by Lev Birbrair in 1999, but classification with respect to outer Lipschitz equivalence is still an open problem. Classification of surface germs in $R^4$ with respect to ambient Lipschitz equivalence, even when they are ambient topologically trivial, includes all of the knot theory.
In this talk, an introduction to Lipschitz geometry of definable surface germs, and a review of recent progress in their outer Lipschitz classification, will be given.

Let $X$ be a surface germ in $R^n$ (a closed connected two-dimensional germ at the origin) definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic). There are two natural metrics on $X$: the inner metric, where the distance between two points in $X$ is the length of a shortest path in $X$ connecting these points, and the outer metric, where the distance between two points of $X$ is their distance in $R^n$. A set $X$ is called normally embedded if its inner and outer metrics are equivalent.
There are three natural equivalence relations associated with these metrics. Two sets $X$ and $Y$ are inner (outer) Lipschitz equivalent if there is an inner (outer) bi-Lipschitz homeomorphism $h:X\to Y$. The sets $X$ and $Y$ are ambient Lipschitz equivalent if the homeomorphism $h$ can be extended to a bi-Lipschitz homeomorphism of the ambient space. The ambient equivalence is stronger than the outer equivalence, and the outer equivalence is stronger then the inner equivalence. There are finitely many ambient Lipschitz equivalence classes in any definable family.
Classification of surface germs with respect to inner Lipschitz equivalence was done by Lev Birbrair in 1999, but classification with respect to outer Lipschitz equivalence is still an open problem. Classification of surface germs in $R^4$ with respect to ambient Lipschitz equivalence, even when they are ambient topologically trivial, includes all of the knot theory.
In this talk, an introduction to Lipschitz geometry of definable surface germs, and a review of recent progress in their outer Lipschitz classification, will be given.

An elliptic curve is an algebraic curve which comes endowed with a group structure. Certain elliptic curves have more endomorphisms than expected, these elliptic curves are said to have "complex multiplication". A singular modulus is a numerical invariant (known as the $j$-invariant) of an elliptic curve with complex multiplication.

The arithmetic properties of these numbers are of great interest, in particular, there are important results concerning the differences of singular moduli, as well as the (lack of) multiplicative dependencies of singular moduli. In joint work with Vahagn Aslanyan and Guy Fowler we study the multiplicative dependencies that can arise among differences of singular moduli, and give a fairly explicit characterization. This result is a special case of a form of the Zilber-Pink conjecture, and uses methods from algebraic number theory as well as o-minimality.

The Gamma function extends factorials to complex numbers and thus appears in many different mathematical contexts. Though Gamma is a transcendental holomorphic function, it satisfies several important functional equations. It is natural to ask what other algebraic properties the Gamma function may possess. In this talk, I will present recent work with Sebastian Eterović in which we prove certain systems of equations involving addition, multiplication, and the Gamma function must have infinitely many solutions in the complex numbers. Similar results have been obtained for periodic functions such as exp, the modular j function, and others. We combine techniques established for these functions with new ideas in order to study Gamma, a non-periodic function. An immediate corollary is that the Gamma function has infinitely many periodic points of every period.

Since the work of Farah, Hart, and Sherman in "Model Theory of Operator Algebras" I, II, and III in the early 2010's, there has been a large body of results stemming from the application of model theory to studying operator algebras (and in the past few years, the converse as well). In this talk, I will give a bit of history and survey a few of these results in the case of tracial von Neumann algebras. Notably, I will highlight some structural results related to the classification of von Neumann algebras and their first-order theories. Both operator algebraists and logicians are welcome!

Let $S$ be a Shimura variety such that every connected component of the space of complex points of $S$ arises as the quotient of a Hermitian symmetric domain by a torsion-free arithmetic group. In the 1970s, Borel proved that any holomorphic map from a complex algebraic variety $V$ into such a Shimura variety $S$ is algebraic. In this talk, I'll discuss joint work with Anand Patel, Ananth Shankar, and Xinwen Zhu on a $p$-adic version of this result.

Since Dehn proposed the word problem for groups in 1912, it has been a central area of research in group theory. Classically, the word problem of a group is the set of words equal to the identity of the group, and we analyze them using Turing reductions. In this talk, we consider the word problem of a group as a computably enumerable equivalence relation (ceer), namely, two words are equivalent if and only if they are equal in the group. We compare ceers using computable reductions: $E$ is reducible to $F$ if there is a computable function $f$ so that $iEj$ if and only if $f(i)Ff(j)$.

We will discuss some recent results and see that the landscape of word problems as ceers differs from the classical theory. For instance, in the classical setting, any Turing degree can be realized as a word problem by constructing a countable group and then embedding it into a finitely presented group via the Higman embedding theorem. However, we prove that in the ceer setting, there is a group $G$ such that any non-trivial free product $G*H$ has a strictly more complicated word problem. We also show that there are ceer degrees that are not realized by any word problem of a group.

This is a joint work with Uri Andrews and Luca San Mauro.

The model theory for metric structures, is an alternative setting created to overcome the limitations of first order logic to study structures as metric spaces, C*-algebras, Banach spaces, etc. It is based in a continuous logic, where formulas can take values between 0 and 1, and many motions of the usual model that can be translated in terms of approximation to a formula or value. Following the work done by Berenstein and Henson, we will discuss an application of this model theory to get an axiomatization for atomless probability spaces and prove that such theory admits quantifier elimination as it is defined in this context.