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Mathematical Logic at Fudan

## Fudan Logic Seminar 2022

### December 14

Yu Wei 魏宇

Time: 15:00 - 18:00. Location: HGW2403.

#### Do you know my name?: Quantifier-free epistemic term-modal logic with assignment operator

In standard epistemic logic, the names and the existence of agents are usually assumed to be common knowledge implicitly. This is unreasonable for various applications in computer science and philosophy. In this talk, inspired by term-modal logic and assignment operators in dynamic logic, we introduce a lightweight modal predicate logic where names can be non-rigid, and the existence of agents can be uncertain. The language can handle various de dicto/de re distinctions in a natural way. We characterize the expressive power of our language (which can, for example, express the meta-level notion “existence” and something just like Barcan formula & Converse Barcan formula without any quantifiers), obtain complete axiomatizations of the logics over several classes of varying-domain/constant-domain epistemic models, and show their (un)decidability. This is joint work with Yanjing Wang and Jeremy Seligman.

### November 29

Wei Li 李伟

Time: 14:00 - 17:00. Tencent Meeting: 615 646 792.

#### On the Luroth Problem for Partial Differential Fields

In 1875, J. Luroth proved the well-known Luroth's theorem, which states that every subfield of the rational function field in one variable is a simple extension. Geometrically, the Luroth's theorem means a unirational curve is always rational. Around 1940s, Ritt and Kolchin proved the ordinary differential version of Luroth's theorem, and thus unirational differential curves rational. But as Kolchin pointed out, the Luroth's theorem ceases to hold for partial differential fields. A natural question arises: under what conditions are unirational partial differential curves rational?
In this talk, we present our partial differential Luroth's theorem in both theoretical and algorithmic aspects. We first give a necessary and sufficient condition for a subfield of a partial differential rational function field to be a simple extension. This result generalizes Ritt and Kolchin’s classical differential Luroth's theorem. We then give an algorithm to decide whether a given finitely generated differential subfield admits a Luroth generator, and in the affirmative case, to compute a Luroth generator. As an application, we solve the problem of deciding whether a unirational partial differential curve is rational.

### October 18

Mark van Atten

Time: 15:30 - 17:00. Zoom Meeting: 847 0772 1155, passcode: 200433.

#### Intuitionistic inductive definitions

I reconstruct Brouwer's and Heyting's views on inductive definitions, and compare them with (a) the neutral constructive one, and (b) the operationalist one of Lorenzen and Heinzmann. This leads to the following conclusions about the intuitionistic view: (1) The clauses must be understood neither as propositions nor as permissions, but as instructions. Put differently, inductive definitions are governed by the grammar of the imperative. (2) An extremal clause then is redundant, and there is no concern over nonstandard iterations of the clauses. (3) The permissive aspect of the definition is no part of the meaning of the rules of which it consists, but lies in our freedom to undertake the construction of the defined object or not.

### September 14

Haosui Duanmu 端木昊随

Time: 15:00 - 18:00. Location: HGW2401. Zoom Meeting: 857 7614 6788, passcode: 200433.

#### Applications of Nonstandard Analysis in Economics, Probability and Statistics

Nonstandard analysis, a powerful machinery derived from mathematical logic, has had many applications in various areas of mathematics such as probability theory, stochastic processes, mathematical physics, functional analysis, and mathematical economics. Nonstandard analysis allows construction of a single object — a hyperfinite probability space — which satisfies all the first order logical properties of a finite probability space, but which can be simultaneously viewed as a measure-theoretical probability space via the Loeb construction. As a consequence, the hyperfinite/measure duality has proven to be particularly in porting discrete results into their continuous settings. We present two applications of this novel approach:
(1) Extending the stable matching lemma to the infinite markets setting;
(2) Existence of Walrasian equilibrium for production economy models that are specific to climate change.
Finally, we breifly discuss other applications of nonstandard analysis to Markov processes, statistical decision theory and economic dynamics.

### July 3

Zhixing You 游志兴

Time: 15:00 - 18:00. Zoom Meeting: 979 6001 0897, passcode: 213140.