This short course is an introduction to the nascent field of Fourier analysis on polytopes and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field.

Of the many applications of these techniques, we will focus on the following, as time permits:

1. The Fourier transform of a polytope, given its vertex description
2. Minkowski and Siegel's theorems in the geometry of numbers
3. Tilings and multi-tilings of Euclidean space by translations of a polytope
4. Discrete volumes of polytopes (Ehrhart theory)
5. The Fourier transform of a polytope, given its hyperplane description. Here we iterate the divergence theorem.

We assume familiarity with linear algebra, calculus and infinite series. Throughout, we introduce the topics gently, by giving examples and exercises.

This short course is an introduction to the nascent field of Fourier analysis on polytopes and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field.

Of the many applications of these techniques, we will focus on the following, as time permits:

1. The Fourier transform of a polytope, given its vertex description
2. Minkowski and Siegel's theorems in the geometry of numbers
3. Tilings and multi-tilings of Euclidean space by translations of a polytope
4. Discrete volumes of polytopes (Ehrhart theory)
5. The Fourier transform of a polytope, given its hyperplane description. Here we iterate the divergence theorem.

We assume familiarity with linear algebra, calculus and infinite series. Throughout, we introduce the topics gently, by giving examples and exercises.

This short course is an introduction to the nascent field of Fourier analysis on polytopes and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field.

Of the many applications of these techniques, we will focus on the following, as time permits:

1. The Fourier transform of a polytope, given its vertex description
2. Minkowski and Siegel's theorems in the geometry of numbers
3. Tilings and multi-tilings of Euclidean space by translations of a polytope
4. Discrete volumes of polytopes (Ehrhart theory)
5. The Fourier transform of a polytope, given its hyperplane description. Here we iterate the divergence theorem.

We assume familiarity with linear algebra, calculus and infinite series. Throughout, we introduce the topics gently, by giving examples and exercises.

This short course is an introduction to the nascent field of Fourier analysis on polytopes and cones. There is a rapidly growing number of applications of these methods, so it is appropriate to invite students, as well as professionals, to the field.

Of the many applications of these techniques, we will focus on the following, as time permits:

1. The Fourier transform of a polytope, given its vertex description
2. Minkowski and Siegel's theorems in the geometry of numbers
3. Tilings and multi-tilings of Euclidean space by translations of a polytope
4. Discrete volumes of polytopes (Ehrhart theory)
5. The Fourier transform of a polytope, given its hyperplane description. Here we iterate the divergence theorem.

We assume familiarity with linear algebra, calculus and infinite series. Throughout, we introduce the topics gently, by giving examples and exercises.

In this talk, we explore étale groupoids with a locally compact Hausdorff unit space , where itself may not be globally Hausdorff. For such groupoids, the essential -algebra offers a more suitable framework than the reduced -algebra , as it captures additional structural nuances. Specifically, arises as a proper quotient of .

We introduce the concept of essential amenability for groupoids, a condition that is strictly weaker than (topological) amenability yet sufficient to guarantee the nuclearity of . To establish this, we define a maximal version of the essential -algebra and show that any function with dense cosupport must be supported within the set of "dangerous arrows”, that is, arrows that cannot be topologically separated.

This essential amenability framework characterizes the nuclearity of and establishes its isomorphism to the maximal essential -algebra. Our results offer new insights into the interplay between groupoid structure and operator algebras, extending the utility of in non-Hausdorff settings. This is based on joint work with Diego Martinez.