To any matroid $M$ on ground set $E$ we associate the characteristic polynomial $$p_M(t) = \sum_{A \subseteq E} (-1)^{|A|}t^{r(E) - r(A)}$$ where $r(A)$ denotes the rank of $A$. Let $r(E) = r.$ It is known that $$p_M(t) = a_0 t^r - a_1 t^{r-1} + \cdots + (-1)^r a_r$$ for $a_i \geq 0$ (i.e. coefficients alternate in sign). Moreover the coefficients are known to have some combinatorial interpretation. It was a long standing conjecture of Rota-Heron-Welsh that the sequence $(a_0, a_1, \cdots, a_r)$ is log-concave.

In Log-concavity of characteristic polynomials and the Bergman fan of matroids, Huh and Katz show that the Rota-Heron-Welsh conjecture is true for "realizable" matroids by making use of intersection theory and toric varieties. Adiprasito, Huh, and Katz prove the conjecture for general matroids in Hodge Theory for Combinatorial Geometries.

To any matroid $M$ on ground set $E$ we associate the characteristic polynomial $$p_M(t) = \sum_{A \subseteq E} (-1)^{|A|}t^{r(E) - r(A)}$$ where $r(A)$ denotes the rank of $A$. Let $r(E) = r.$ It is known that $$p_M(t) = a_0 t^r - a_1 t^{r-1} + \cdots + (-1)^r a_r$$ for $a_i \geq 0$ (i.e. coefficients alternate in sign). Moreover the coefficients are known to have some combinatorial interpretation. It was a long standing conjecture of Rota-Heron-Welsh that the sequence $(a_0, a_1, \cdots, a_r)$ is log-concave.

In Log-concavity of characteristic polynomials and the Bergman fan of matroids, Huh and Katz show that the Rota-Heron-Welsh conjecture is true for "realizable" matroids by making use of intersection theory and toric varieties. Adiprasito, Huh, and Katz prove the conjecture for general matroids in Hodge Theory for Combinatorial Geometries.

Edit: I have found a paper Matroid theory for algebraic geometers by Katz which gives a expository view of the work of Huh and Katz mentioned above as well as more background. It looks like a nice source which may be of interest to viewers of this question; so, I will share it here.

To any matroid $M$ on ground set $E$ we associate the characteristic polynomial $$p_M(t) = \sum_{A \subseteq E} (-1)^{|A|}t^{r(E) - r(S)}$$ where $r(A)$ denotes the rank of $A$. Let $r(E) = r.$ It is known that $$p_M(t) = a_0 t^r - a_1 t^{r-1} + \cdots + (-1)^r a_r$$ for $a_i \geq 0$ (i.e. coefficients alternate in sign). Moreover the coefficients are known to have some combinatorial interpretation. It was a long standing conjecture of Rota-Heron-Welsh that the sequence $(a_0, a_1, \cdots, a_r)$ is log-concave.

In Log-concavity of characteristic polynomials and the Bergman fan of matroids, Huh and Katz show that the Rota-Heron-Welsh conjecture is true for "realizable" matroids by making use of intersection theory and toric varieties. Adiprasito, Huh, and Katz prove the conjecture for general matroids in Hodge Theory for Combinatorial Geometries.

To any matroid $M$ on ground set $E$ we associate the characteristic polynomial $$p_M(t) = \sum_{A \subseteq E} (-1)^{|A|}t^{r(E) - r(A)}$$ where $r(A)$ denotes the rank of $A$. Let $r(E) = r.$ It is known that $$p_M(t) = a_0 t^r - a_1 t^{r-1} + \cdots + (-1)^r a_r$$ for $a_i \geq 0$ (i.e. coefficients alternate in sign). Moreover the coefficients are known to have some combinatorial interpretation. It was a long standing conjecture of Rota-Heron-Welsh that the sequence $(a_0, a_1, \cdots, a_r)$ is log-concave.

In Log-concavity of characteristic polynomials and the Bergman fan of matroids, Huh and Katz show that the Rota-Heron-Welsh conjecture is true for "realizable" matroids by making use of intersection theory and toric varieties. Adiprasito, Huh, and Katz prove the conjecture for general matroids in Hodge Theory for Combinatorial Geometries.