##### Department of Mathematics,

University of California San Diego

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### Combinatorics Seminar

## Thao Do

#### MIT

## A general incidence bound in high dimensions

##### Abstract:

In this talk, I will present a general upper bound for the number of incidences with k-dimensional varieties in R$^d$ such that their incidence graph does not contain K$_{s,t}$ for fixed positive integers s,t,k,d (where s,t$>$1 and k$<$d). The leading term of this new bound generalizes previous bounds for the special cases of k=1, k=d-1, and k=d/2. Moreover, we find lower bounds showing that this leading term is tight (up to sub-polynomial factors) in various cases. To prove our incidence bounds, we define k/d as the dimension ratio of an incidence problem. This ratio provides an intuitive approach for deriving incidence bounds and isolating the main difficulties in each proof. If time permits, I will mention other incidence bounds with traversal varieties and hyperplanes in complex spaces. This is joint work with Adam Sheffer.

Host: Andrew Suk

### November 13, 2018

### 2:00 PM

### AP&M 6402

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