The empty set is one of the most overlooked main characters of mathematics. Better known by its full legal title “\(\emptyset\)”, this trivial guy has confounded careless mathematicians for over a century. Since most mathematics works with nonempty objects, it’s common to forget to ask the simple question: what if there aren’t any objects? We’ll take a look at how \(\emptyset\) appears in various fields of math (including examples from topology, algebra, logic, and category theory), with special attention toward how \(\emptyset\) often requires special cases in definitions and theorems. The main takeaway for the audience: always check your “trivial” cases.

The calculus of finite differences is an analogue of ordinary calculus for sequences instead of functions. It is used in both pure and applied math, providing theorems and methods for evaluating infinite sums based on discrete analogues of derivatives, integrals, and the Fundamental Theorem of Calculus.

Note: Free refreshments. The talk starts at 5:40.

Harvard University bestowed upon Kurt Gödel an honorary doctorate “for the discovery of the most significant mathematical truth of the century.” John von Neumann regarded him as the greatest “logician since Aristotle,” the only mathematician who was “absolutely irreplaceable.” His friend Einstein liked to say that eh went to the Institute of Advanced Studies “um das Privileg zu haben, mit Gödel zu Fuss nach Hause gehen zu dürfen.” This talk reports on progress made toward using animated logic puzzles, AI, and digital pedagogy to introduce a new generation to Gödel’s Theorems. 

Four 10-minute talks will be presented. Possible topics will include research questions, background for statements of interesting theorems and conjectures, summaries of papers, and advertisements for reading courses.

Abstract: In this paper, we present axiomatic characterizations of some simple risk-sharing (RS) rules, such as the uniform, the mean-proportional and the covariance-based linear RS rules. These characterizations make it easier to understand the underlying principles behind how risks are distributed under these simple rules. Such principles typically include maintaining some degree of anonymity regarding participants’ data and incident-specific data, adopting non-punitive processes and ensuring the equitability of risk sharing. By formalizing key concepts of reshuffling property, source-anonymous contributions and aggregated contributions, along with their generalizations, we develop a comprehensive framework that expresses these principles clearly and defines the relevant rules. To illustrate, we demonstrate that the uniform RS rule, a simple mechanism in which risks are shared equally, is the only RS rule that satisfies both the reshuffling property and source-anonymous contributions. These straightforward axiomatic characterizations serve as a foundation for exploring similar principles in two other broader classes of risk-sharing rules for which we baptize the $q$-proportional RS rules and the $(q_1,q_2)$-based linear RS rules. Furthermore, this framework allows us to introduce novel particular RS rules, such as the scenario-based RS rules. This is joint work with Jan Dhaene and Rodrigue Kazzi, both from KU Leuven.

Speaker’s bio: Emil is a Fellow of the Society of Actuaries and holds a Ph.D. from the University of Wisconsin in Madison. He joined the University of Connecticut in 2007 but had a brief stint (2013-2015) as professor and director of the actuarial science program at Michigan State University. His prime research of interest is actuarial science that cover topics in copula models, dependencies, post-retirement asset management, and some related to risk measures and capital allocation. In recent years, his research work has evolved around the applications of data science and statistical modeling to actuarial and insurance problems. The quality of his research has been recognized through awards that include the E. A. Lew Award, the Halmstad Memorial Prize, the Hachemeister Prize, and most recently, the Robert I. Mehr award. See https://emiliano-valdez.scholar.uconn.edu/ for more details

At this meeting, Prof. Conrad will answer any question you have about math (except homework questions). This is a chance to find out more about any problems, concepts, examples, historical events, etc. in math that you’ve heard or read about but don’t understand as well as you’d like.

The Riemann hypothesis is often presented as the most important unsolved problem in mathematics.  Why is that?

In this talk the Riemann hypothesis will be described, including the context that led to it. Then examples will be given that illustrate a range of problems where the Riemann hypothesis can be applied once it gets solved and where the problems themselves don’t sound related to the Riemann hypothesis at all.  

It will be assumed that the audience is familiar with basic complex analysis.

Join us in the Logic Colloquium!

Zeynep Soysal (Rochester):

The Metalinguistic Construal of Mathematical Propositions

In this talk I will defend the metalinguistic solution to the problem of mathematical omniscience for the possible-worlds account of propositions. The metalinguistic solution says that mathematical propositions are possible-worlds propositions about the relation between mathematical sentences and what these sentences express. This solution faces two types of problems. First, it is thought to yield a highly counterintuitive account of mathematical propositions. Second, it still ascribes too much mathematical knowledge if we assume the standard possible-worlds account of belief and knowledge on which these are closed under entailment. I will defend the metalinguistic construal of mathematical propositions against these two types of objections by drawing upon a conventionalist metasemantics for mathematics and an algorithmic model of belief, knowledge, and communication.

All welcome!

https://logic.uconn.edu/calendar/

An ideal is called basic if it does not contain a smaller ideal with the same integral closure. In this talk, we investigate when a monomial ideal is basic. Using methods from convex geometry, we provide a clear characterization of basic monomial ideals, highlighting the similarities between the algebraic and geometric structure. We also discuss a method of calculating analytic spread with convex geometry.

Have you ever seen a mosquito and wanted to shoo it away? Certainly, if you used a nuclear bomb, this would solve the issue, despite there being many simpler ways to approach the problem. We will discuss mathematical equivalents of this: ways to prove elementary facts using overly complicated methods.

Note: Free refreshments. The talk starts at 5:40.

Composition of functions usually depends on the order in which it is done. In this talk we will discuss all the ways that two polynomials can commute. It turns out that there are only two interesting ways this can happen, and one of them has an unexpected relationship to trigonometry.