All talks are both in-person and over Zoom, at 2pm, on Fridays; here is a view of the public ICS file:
Friday Week 5, Oct 6: Salman Parsa, SoC in Lincoln Park Campus
Title: Reeb Spaces and the Borsuk–Ulam Theorem
Abstract: I will talk about Reeb spaces and their discretization called mapper. These are applied tools in topological data analysis. After showing a few examples, I will use the Reeb space to prove a partial extension of the well-known Borsuk-Ulam theorem for maps from \(2\)-sphere into \(\mathbb{R}\). This extension says that there are always two antipodal points \(S^2 \ni x, -x\) such that \(f(x) = f(-x)\) and the two points are connected in the preimage. The proof uses the concept of the Reeb graph which is a 1-dimensional Reeb space. I also consider the relationship between excess homology of the Reeb space of \(f: S^n \to \mathbb{R}^{n - 1}\) and the existence of the analogous extensions of the Borsuk–Ulam theorem for maps into \(\mathbb{R}^m, n > 2\).
Friday Week 9, Nov 3: Stefan Mitsch, SoC in Lincoln Park Campus
Title: The cyber-physical systems proof workhorse: quantifier elimination in the first-order theory of real closed fields
Abstract: Cyber-physical systems are characterized by interaction between discrete computational processes and their effects in continuous differential equation models of physics, biology, chemistry, etc. In proving universal and existential properties about such systems, many questions of interest reduce to questions in first-order real arithmetic. Conveniently, the first-order theory of real closed fields is decidable due to a seminal result of Alfred Tarski. Algorithms for quantifier elimination, however, are highly non-trivial and not proof-producing, which makes it difficult to construct an unbroken chain of proof arguments. This talk presents a formalization of quadratic virtual term substitution, a quantifier elimination procedure that is complete for low-degree polynomials.
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