Spring Semester
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Applications of Substructural Logic
Speaker: Eben Blaisdell
Date: May 1, 2023
Location: DRLB TBD
Abstract: (No abstract provided)
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A Taste of the Diophantine Applications of O-Minimality
Speaker: Matthew Stevens (Zoom)
Date: March 19, 2023
Location: TBD
Abstract: In recent years, o-minimality has been successfully applied to a number of problems in diophantine geometry via the so-called Pila-Zannier strategy. The aim of this talk is to introduce this ingenious method by sketching the Pila-Zannier strategy proof of the multiplicative Manin-Mumford conjecture, which will be stated during the talk. In the process, we will introduce the Pila-Wilkie counting theorem, which informally states that "transcendental" sets have relatively few rational points. Prerequisites: The very basics of classical algebraic geometry and Galois theory, and enough familiarity with model theory to know what a definable set is. I will briefly review the necessary ideas from o-minimality near the beginning of the talk; hence, while previous experience with o-minimality might be helpful, those who did not attend the previous seminar talk on o-minimality should still be able to follow this talk.
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Approaching the Elephant
Speaker: Marc Muhleisen
Date: February 27, 2023
Location: DRL 2N36
Abstract: A topos arises as the category of sheaves on a Grothendieck site (by definition). But what exactly is a Grothendieck site? Every source appears to give a slight variation on the definition. For instance, some authors speak of covering families while others restrict to covering sieves. Often certain singleton sets are taken as coverings, but is it the identities, isomorphisms, or split epis? Must a Grothendieck site have fiber products (or else how would you formulate the sheaf condition)? I will explain why none of the answers to these questions make a difference insofar as what categories arise as sheaves on a site, or in other words, why the notion of topos is independent of the exact definition of Grothendieck topology. This phenomenon is in part what led Johnstone to the namesake of his topos theory compendium, writing “however you approach [the elephant], it is still the same animal.”
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O-Minimality and Tame Topology
Speaker: Julian Gould
Date: February 6, 2023
Location: DRL 4N49
Abstract: Julian will give us an overview of o-minimality as a realization of Grothendieck’s vision of tame topology.
Fall Semester
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Substance and Form: Categorical Foundations of Set Theory
Speaker: Marc Muhleisen
Date: December 6, 2022
Location: DRL 4N49
Abstract: Sets and membership are often considered the most fundamental concepts in math, a belief reflected in the wildly successful axioms of ZFC. On the one hand, these axioms concisely express our deepest convictions about sets; on the other, they can encode and prove a great deal of theorems in essentially every mathematical discipline. And yet, to what extent does your work rely on no set being an element of itself (i.e. the axiom of foundation)? And tell me, are the 7-adic integers an element of the 7-sphere? These silly questions (where in both cases the answer is ambiguous and doesn’t remotely matter) underlie serious criticisms of a good foundational framework. On the one hand, the ZFC axioms do not directly reflect the tools of many “working mathematicians”; on the other, there is a massive disconnect between what a given set might be used for and what its elements are.
Abstract (continued): Enter Lawvere, who in the mid-1960s proposed a new axiomatization called the “Elementary Theory of the Category of Sets” (ETCS). He takes sets and functions as the fundamental concepts, using universal properties to build sets rather than extensionality. In this talk, we will sketch Lawvere’s approach to set theory, contrast ETCS with ZFC in a couple toy examples, and prove some groundwork theorems like Peano’s postulates in the ETCS framework. Time permitting, we’ll also sketch a proof that ETCS characterizes the category of sets up to equivalence. Some references for this talk are Lawvere’s original paper, Leinster’s brief article “Rethinking Set Theory,” and Todd Trimble’s nLab entry “fully formal ETCS.”
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You Could Have Invented Linear Logic
Speaker: Eben Blaisdell
Date: November 22, 2022
Location: DRL 4N49
Abstract: While linear logic can fairly be described as a logic of resources, it also forms a "computational core" of logic. This talk will serve mainly as a paced introduction to the fundamentals of linear logic, seeking to motivate each (intuitionistic) connective via both the resource interpretation and lattices. Based on time and interest we will see connections to modal logic, positive and negative data types, calling conventions, and more.
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An Introduction to Forking and Dividing in Model Theory
Speaker: Krishan Canzius
Date: November 8, 2022
Location: Meet in Math Grad Lounge @ 2 PM
Abstract: The concepts of forking and dividing are ubiquitous in model theory but are often poorly motivated. I will attempt to provide motivation in two ways: measure and independence.
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Continuous Logic for Metric Structures
Speaker: Jin Wei
Date: October 25, 2022
Location: Meet in Math Grad Lounge @ 2 PM
Abstract: A metric can be viewed as a strengthened equality: not only does it decide whether two elements are equal but also quantitatively measures the failure of an equality. Replacing equality in a structure with a metric has nice results, for instance, Hilbert spaces are not axiomatizable in regular logic but will be under this change. Doing so will require a logic that permits continuum many truth values. In this talk, I will define continuous logic for metric structures, discuss its semantic and syntactic properties, and show soundness and completeness for a Hilbert-style proof system of this logic. This talk is primarily based on works of Itaï Ben Yaacov and others.
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Models of Homotopy Type Theory Part 2
Speaker: Julian Gould
Date: October 11, 2022
Location: Meet in Math Grad Lounge @ 2 PM
Abstract: Julian will talk about models of Homotopy Type Theory.
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Models of Homotopy Type Theory Part 1
Speaker: Julian Gould
Date: September 27, 2022
Location: Meet in Math Grad Lounge @ 2 PM
Abstract: Julian will talk about models of Homotopy Type Theory.