### November 22,
Thomas Dreyfus, Université de Strasbourg

*Differential transcendence of solutions of difference equations (remote presentation)*

A function is said to be differentially algebraic if it satisfies a non trivial algebraic differential equation. It is said to be differentially transcendent otherwise. Example of differentially transcendent functions are known, for instance, the Gamma function, or the generating series of automatic sequences. All these functions have in common to satisfy a linear functional equation. In this framework, the difference Galois theory provides tools to prove the differential transcendence of the functions. This strategy has given many recent papers presenting results that get more and more general. In this talk we are going to present a new result for which the hypothesizes are very minimal. This is a joint work with B. Adamczewki and C. Hardouin.

### November 15,
Jonathan Kirby, University of East Anglia

* Local Definability of Holomorphic Functions (remote presentation)*

Given a collection $\mathcal{F}$ of complex or real analytic functions, one can ask what other functions are obtainable from them by finitary algebraic operations. If we just mean polynomial operations we get some field of functions.
If we include as algebraic operations such things as taking
implicit functions, maybe in several variables, we get a much
more interesting framework, which is closely related to the
theory of local definability in an o-minimal setting, starting
with suitable restrictions of the functions in $\mathcal{F}$.
O-minimality is a setting for tame topology of real- or
complex-analytic functions which does not allow for "bad"
singularities. However some more tame singularities can
occur. In this talk I will explain work showing what
singularities we have to consider to get a characterisation of
the locally definable functions in terms of complex analytic
operations.
Ax’s theorem on the differential algebra version of Schanuel’s
conjecture is important to give one counterexample, and also for
some applications to exponential and elliptic functions.

This is joint work with Gareth Jones, Olivier Le Gal, and Tamara Servi.

Slides
### November 8, Léo Jimenez, University of Notre Dame

*Strengthenings of C-algebraicity in differentially closed fields of characteristic zero*

In model theory, the notion of internality to a fixed family of
types plays an important role. During this talk, I will focus on
one of its differential algebraic manifestations: being
C-algebraic, where C is the field of constants of a differentially
closed field. An irreducible differential-algebraic variety is
C-algebraic if it is, roughly speaking, differentially birational
to an algebraic variety in C. I will discuss a new property,
uniform C-internality, and discuss examples, non-examples, and
applications.

Slides
Video
### November 1, Carsten Schneider, Johannes Kepler University

*An Algorithmic Difference Ring Theory for Symbolic Summation*

Inspired by Karr's pioneering work (1981) we developed an algorithmic
difference ring theory for symbolic summation
that enables one to rephrase indefinite nested sums and products in
formal difference rings.
An important outcome of this representation is that one obtains a
simplified expression where the arising sums and products are
algebraically independent among each other. In this talk the main ideas
of these
algorithmic constructions and crucial features of the underlying
difference ring theory are presented.
Combining such optimal representations in combination with definite
summation algorithms, like creative telescoping and recurrence solving
in the setting of difference rings, yield a strong summation toolbox for
practical problem solving. We will demonstrate this machinery
implemented in the summation package Sigma by concrete examples coming
from particle physics.

Slides
Video
### October 25, Fabian Immler, Carnegie Mellon University

*Formal mathematics and a proof of chaos*

Formal proof has been successfully applied to the verification of
hardware and software systems. But formal proof is also applicable to
mathematics: proofs can be checked with ultimate rigor and one can build
libraries of computer-searchable, formalized mathematics.

I will talk about formalization of mathematics and my formalization of
ordinary differential equations in the Isabelle/HOL theorem prover.
This underpins the formal verification of the computer-assisted part of
Tucker's proof of Smale's 14th problem, a proof that relies on numerical
bounds to certify chaos for the Lorenz system of ordinary differential
equations.

Slides

### October 18, Omar Leon Sanchez, University of Manchester

*Differentially large fields*

Recall that a field $K$ is large if it is existentially closed in
the field of Laurent series $K((t))$. Examples of such fields are
the complex, the real, and the p-adic numbers. This class of
fields has been exploited significantly by F. Pop and others in
inverse Galois-theoretic problems. In recent work with Tressl we
introduced and explored a differential analogue of largeness, that
we conveniently call "differentially large". I will present some
properties of such fields and characterise them using formal
Laurent series and to even construct “natural” examples (which
ultimately yield examples of DCFs and CODFs... acronyms that will
be explained in the talk). Time permitting I will mention some
applications to Parameterized PV theory.

Video
### October 11, Yi Zhou, Florida State University

*Algorithms on p-Curvatures of Linear Difference Operators*

In the study of factoring linear difference operators, we have
found p-curvature a powerful tool. I will talk about algorithms
for computing p-curvatures and the math behind them.

Slides
Video
### October 4, Anand Pillay, University of Notre Dame

*Finiteness theorems for Kolchin's constrained cohomology*

This is joint work with Omar Leon Sanchez. Working under a certain general assumption on the differential field $K$ (which includes the case
where $K$ is a closed order differential field in the sense of Michael Singer) we prove finiteness of the "constrained cohomology sets"
$H^{1}_{\partial}(K,G)$, for $G$ any linear differential algebraic
group over $K$. I will define everything and touch on some
applications.

Video
### September 13, Daniel Robertz, University of Plymouth

*Algorithmic Approach to Strong Consistency Analysis of
Finite Difference Approximations to PDE Systems
*

The most common numerical method for solving partial
differential equations is the finite difference method.
Consistency of a finite difference scheme with a given
PDE is a basic requirement for this method.
Earlier work by V. P. Gerdt and the speaker introduced
the notion of strong consistency that takes into account
the differential ideal and the difference ideal associated
with the PDE system and the approximating difference system,
respectively.
We present an algorithmic approach to strong consistency for
polynomially nonlinear PDE systems based on a new decomposition
technique for nonlinear partial difference systems that is
analogous to the differential Thomas decomposition. This is
joint work with Vladimir P. Gerdt (JINR, Dubna).

Slides
Video