October 25, 2018: *Denis Saveliev*: Hindman's finite sums theorem and its application to topologizations of algebras

16.00--17.00

In the first part of my talk, I'll give a brief overview of Hindman's finite sums theorem, a famous result in algebraic Ramsey theory, including some of its prehistory and a few of its further generalizations. I'll outline a modern technique used in proving these and close results, which is based on idempotent ultrafilters in ultrafilter extensions of semigroups.

The second, main part of my talk will contain an application of a generalization of Hindman's theorem to the topologizability problem in algebra. First I'll recall the problem and related classical results concerning groups and rings as well as some newer results. Then I'll define a wide class of universal algebras called polyrings; instances of such algebras include Abelian groups, rings, modules, algebras over a ring, differential rings, and other classical structures. Generalizing earlier results, I'll show that the Zariski topologies of all infinite polyrings are nondiscrete. Actually, I'll prove the following, much stronger fact:

**Main Theorem**. If $K$ is an infinite polyring, $n$ a natural number, and a map $F$ of $K^n$ into $K$ is defined by a term in $n$ variables, then $F$ is a closed nowhere dense subset of the space $K^{n+1}$ with its Zariski topology. In particular, $K^n$ is a closed nowhere dense subset of $K^{n+1}$.

In a certain sense, the theorem shows that Zariski topologies of polyrings, although generally even non-Hausdorff, admit a reasonable notion of topological dimension. My proof of this theorem essentially uses a stronger, multidimensional version of Hindman's finite sums theorem established by Bergelson and Hindman. As a corollary, I'll prove a result on the Hausdorff topologizability of polyrings. In conclusion, if time will permit, I'll briefly discuss some related problems.

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November 30, 2018: *Matias Menni*: The Unity and Identity of Decidable objects and double negation sheaves

15:30--16:30

We give sufficient conditions on a topos for the existence of a Unity and Identity for the subcategories of decidable objects and of double negation sheaves, making them adjointly opposite. Typical examples of such toposes include many ‘gros’ toposes in Algebraic Geometry, simplicial sets and other toposes of ‘combinatorial’ spaces in Algebraic Topology, and certain models of Synthetic Differential Geometry.

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June 28, 2019: *Makoto Fujiwara*: Constructivism and weak logical principles in arithmetic

16:00--17:00

Brouwer’s intuitionistic mathematics is an origin of modern constructive mathematics which allows only constructive reasoning in the proofs.
In the 1930s, Brouwer’s student A. Heyting introduced so-called Heyting arithmetic (HA) with an informal semantics, called the Brouwer-Heyting-Kolmogorov(BHK) interpretation, which works as a criterion for constructive reasoning since.
Note that HA is based on intuitionstic logic and one can obtain classical arithmetic (PA) just by adding the law of excluded middle (A or not A) into the axioms of HA.
Though the sentences provable in HA are valid in the (informal) sense of the BHK interpretation, it seems not that all the sentences valid in the BHK interpretation are provable in HA.
Thus some weak fragment of the law of excluded middle can be contained in the arithmetic which exactly obey the BHK interpretation.
Based on this situation, I have studied the structure (derivability relation) of weak logical principles over Heyting arithmetic and its extension in all finite types.
The weak logical principles are closely related to various kinds of realizability, which can be seen as formal treatments of the BHK interpretation.
In fact, to show that one logical principle A does not imply another principle B, one typically uses appropriate forms of realizability to show that A has a certain semi-constructive interpretation which B does not.
In this talk, I would present the structure of the weal logical principles and discuss about the relation with the variations of realizability.
If time permits, I would also talk briefly about a recent attempt to establish a general method to use Kripke models which separate intermediate logics for the separation of the logical principles over Heyting arithmetic.

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