Harry Richman (UW)

PDL C-38

**Preseminar 2-2:30**

**Title**:Continuity over p-adic numbers

**Abstract**:In calculus class we learn the importance of continuous functions and how to recognize them, when using the real topology. The principle of continuity is useful, for example, in defining the value of 10^x for any real exponent x when starting from rational powers of 10. Can we use the same idea to define 10^x for any p-adic number x? We explain how to answer this nicely using Mahler expansions, which are a discrete analogue of Taylor expansions. (About p-adic 10^x, the answer is yes for some p's and no for others!)

**Seminar 2:30-3:30**

**Title**:Derangements and the p-adic incomplete gamma function

**Abstract**:A derangement on a finite set is a permutation with no fixed points. A classic combinatorics problem, studied by Euler in 1779, is to count how many derangements there are on a set of n elements. In this talk we will address the question: how many derangements are there on a set of (-1) elements? We claim there is a (unique) reasonable p-adic answer to this question, which is connected to a p-adic analogue of the incomplete gamma function. Based on joint work with Andrew O'Desky, arXiv:2012.04615.

The talk will also be available via Zoom: https://washington.zoom.us/j/689897930

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