## Spring 2023

Time & Location: All talks are in Gibson Hall 126A on Wednesday at 4:00 PM unless otherwise noted.

Organizer: Dipendranath Mahato

**Wednesday, September 13**

**Topic: ***Exploring Shape Reconstruction from Data Samples ***Rafal Komendarczyk - Tulane University**

**Abstract: **The challenge of figuring out unknown structures from limited data is a common problem in various scientific fields. Lately, the newly emerged field of TDA (Topological Data Analysis) has been focusing on something called "manifold reconstruction." In simpler terms, we're trying to recreate shapes from point cloud data. In this introductory talk, we will discuss various results in the manifold reconstruction as well as generalizations and open problems.

**Location: **Gibson Hall 126A**Time:** 4:00

**Wednesday, September 27**

**Topic: **An Overview of Various Time-Integration Methods for Solving ODEs and PDEs

Tommaso Buvoli - Tulane University

**Abstract:** Time integration methods are numerical algorithms for solving ordinary differential equations and spatially discretized partial differential equations. For the past fifty years they have proven indispensable for modeling a range of physical phenomena such as weather, plasmas, and fluids. In this talk I will provide a broad overview of time integration and present several different flavors of Euler's method, one of the simplest and well-known time integration methods. I will also discuss some of my recent work on parallel-in-time methods and on constructing new integrators using polynomials.

**Location: **Gibson Hall 126A**Time:** 4:00

**Wednesday, October 11**

**Topic: ***PROBLEMS WITH AN EXACT ANSWER*

**Victor Moll | Tulane University**

**Abstract:**

I have always been intrigued by mathematical problems with an exact answer. The talk will present a variety of problems that appeal to me.

My earliest memories deal with the quadratic formula to solve ax2 + bx + c = 0 and wondering what happens for higher degree polynomial equations? Once I heard that somehow degrees 5 and higher are impossible. What exactly does that mean and what can one do about it?

Then I heard that one can do things in more than one variable and solve equations like y2 = x3 + ax2 + bx + c exactly. Then I thought that perhaps the same methods can be used to solve y2 = cos x. How does one do this?

In the old days, people were interested in equations of the form y′′ + (a + b cos x)y = 0. Are there exact solutions for this?

The last example starts with the equation u′ = 0, for a function of one variable. This is easy. Then you move to two variables u = u(x,y) and try to solve ux = 0 exactly. Not so bad either. If you make a simple change of coordinates, you get ut + cux = 0. This is also easy. Then you make the problem more interesting (that is, not linear) by replacing c by u to get ut + uux = 0. This is bad. Then you try to fix it: it turns out that if you add a diffusive term and write ut + uux + uxx = 0 then you can convert this into a linear problem by a simple change of variables. Then you continue playing and look at ut + uux + uxxx = 0. This is the famous KdV equation. You can also make it linear, but for that one needs to know things like Jacobian varieties. Interesting.

It turns out that all the problems discussed above are different points of view of the same question: how do you know if your problem has an exact solution?

**Location: **Gibson Hall 126A**Time:** 4:00

**Wednesday, October 25**

**Topic: ***Using noise to estimate chaos*

**Punshon-Smith, Samuel - Tulane University**

**Abstract:** Chaos can be a tricky thing to understand. Indeed, one classical indicator of chaos, a positive Lyapunov exponent, is exceptionally hard to verify, even in relatively simple settings. In this talk I will discuss some of the obstacles to proving chaos and how adding a small amount of noise can help in estimating Lyapunov exponents for very general systems. I will discuss the many powerful applications of such estimates from my research.

**Location: **Gibson Hall 126A**Time:** 4:00

**Wednesday, November 08**

**Topic: ***The talk will be a surprise for us.*

Scott McKinley - Tulane University

**Abstract:** The talk will be a surprise for us.

**Location: **Gibson Hall 126A**Time:** 4:00

**Wednesday, December 06**

**Topic: ***Phylogenetic approach for estimating amounts of interlocus gene conversion in duplications*

**Xiang Xi | Tulane University**

**Abstract:** A variety of mutational mechanisms generate repeated sequences. After their formation, the evolutionary fates of individual repeated elements are intertwined. Interlocus gene conversion (IGC) is one source of this dependence that homogenizes repeats by copying sequence stretches from one repeat into the equivalent region of another. Such dependence structure is conventionally ignored in phylogenetic inference. This can potentially lead to misleading inferences about evolutionary history and process. Here, we introduce an approach for quantifying IGC. To measure the amount of IGC that occurred subsequent to a genome-wide duplication in the yeast lineage, we applied our procedure to 14 different protein-coding genes. Our finding of substantial levels of IGC in all 14 data sets suggests that IGC should not be ignored when the molecular evolution of multi-gene families is investigated.

**Location: **Gibson Hall 126A**Time:** 4:00