Update current_projects.md (#205)

Author: quigleyjd

mathexlab/current_projects.md

@@ -20,7 +20,7 @@ redirect_from:
 - [Aseel Farhat](https://math.virginia.edu/people/af7py/)
 - [J.D. Quigley](https://math.virginia.edu/people/mbp6pj/)
 
-### Projects Spring 2025:
+### Projects Fall 2025:
 1. **Differential Geometry and the Shape of Data** ([Bakhyt Aitzhanova](https://math.virginia.edu/people/axh7cj/) and [Josh Turner](https://math.virginia.edu/people/rbh3vx/)): In this project, we’ll be working with 3D geometric data and at the same time exploring some basic ideas from differential geometry. The main goal is to understand key concepts—like curvature—from two angles: how they are defined in mathematics and how they can be computed in practice. Looking at both sides helps build intuition and also shows how these ideas lead to useful algorithms for real-world problems. We’ll also develop some important tools from calculus and linear algebra, but with a focus on building intuition and developing a good visual understanding. Throughout the project, you’ll see both the mathematical background and plenty of practical examples and applications. We’ll also take a look at newer developments in digital geometry processing and discrete differential geometry. Some of the topics we’ll cover include: curves and surfaces, curvature, simplicial homology, differential forms, geodesics, and numerical linear algebra. On the application side, we’ll see how to approximate curvature, smooth curves and surfaces, parameterize surfaces, design vector fields, and compute geodesic distances.
 2. **Random surfaces and random permutations** ([Leonid Petrov](https://math.virginia.edu/people/lap5r/)): Imagine you have a 100x100x100 room, and you stack some number of 1x1x1 unit cubes in its corner. If you do this at random, what would be the shape of the pile? We will investigate the mathematical structure of such random 3D stepped surfaces, which leads to beautiful limit shapes. We will use basic combinatorics, coding, and we will potentially 3D print our results. We will also explore cutting edge research directions in this area, in particular, connections of random surfaces to random permutations. Most of the models we will explore are hands-on and computational, and we will be able to visualize and manipulate them.
 3. **Discrete dynamical systems** ([Oliver Wang](https://math.virginia.edu/people/dfh3fs/)): Discrete dynamics studies the behavior of functions as we iterate them. For instance, if f(x)=x+2, and we let f^n(x) denote f(f(f(...f(x)))) (where we've iterated f n-many times) then, for every fixed x, f^n(x) approaches infinity, as n increases. If f(x)=x^2, then f^n(x) approaches 0 for -1<x<1 and it approaches infinity for x<-1 and x>1. When x=-1 or 1, f^n(x)=1. We will study the behavior of f^n for more complicated functions. We may also explore some complex dynamics where we see fractals arise.