Update current_projects.md (#204)

Author: quigleyjd

mathexlab/current_projects.md

@@ -8,19 +8,19 @@ redirect_from:
 
 <h2 class="mb-3">Current mentors and projects</h2>
 
-### Mentors for Spring 2025:
+### Mentors for Fall 2025:
 
 - [Bakhyt Aitzhanova](https://math.virginia.edu/people/axh7cj/)
-- [Brandon Shapiro](https://brandontshapiro.github.io/)
-- [Michael Wills](https://sites.google.com/view/michael-wills/)
+- [Leonid Petrov](https://math.virginia.edu/people/lap5r/)
+- [Josh Turner](https://math.virginia.edu/people/rbh3vx/)
+- [Oliver Wang](https://math.virginia.edu/people/dfh3fs/)
 
 
 ### Faculty Advisors:
-- [Evangelia Gazaki](https://sites.google.com/view/valiagazakihomepage/home)
-- [Sara Maloni](https://sites.google.com/view/sara-maloni)
-- [J.D. Quigley](https://quigleyjd.github.io)
+- [Aseel Farhat](https://math.virginia.edu/people/af7py/)
+- [J.D. Quigley](https://math.virginia.edu/people/mbp6pj/)
 
 ### Projects Spring 2025:
-1. **Compass and Straightedge Constructions** ([Evangelia Gazaki](https://sites.google.com/view/valiagazakihomepage/home), [Michael Wills](https://sites.google.com/view/michael-wills/)): This project will examine compass and straightedge constructions in the plane. We will use these tools to carry out explicit constructions (bisecting an angle, drawing a regular hexagon) as mathematicians have been doing for over 2000 years. Even as more and more constructions were discovered, some (such as constructing a square with the same area as a given circle) remained elusive. This project will also examine the seemingly distant branch of math known as _Galois theory_, and how it was used in the 1800's to show the mathematical impossibility of these famous open problems.
-2. **Computer graphics and Robotics** ([Bakhyt Aitzhanova](https://math.virginia.edu/people/axh7cj/)): We will learn a standard method for solving the forward and inverse kinematic problems for a given robot "arm". The forward kinematic problem is a fundamental concept in robotics and computer graphics that involves determining the position and orientation of the hand of a robot arm based on its joint parameters. Unlike forward kinematics, where the hand position is computed from known joint parameters, inverse kinematics works in reverse: given the hand position, we find the joint parameters. The inverse kinematic problem is more challenging because it often requires solving a system of non-linear polynomial equations.
-4. **Zome and polyhedra** ([Brandon Shapiro](https://brandontshapiro.github.io/)): The Zome toolkit makes it easy to build toy models of polyhedra, 3 dimensional solid shapes like cubes and pyramids built out of polygons on the outside. Using the Zome pieces, we can show why there are only 5 polyhedra whose faces are all the same shape, and also why in the fourth dimension there are only 6 shapes with the same kind of property. We will explor the geometry of shapes we can build using Zome, such as angles, counting faces of solid shapes, symmetries, knots, and/or many other possibilities.
+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.