# Research

## Graduate Seminar Series (Leader: Nathan Hayford <nhayford (at) usf.edu>document.write('<a href="mai' + 'lto:' + 'nhayford' + '&#64;' + 'usf.edu' + '">Nathan Hayford</a>');)

### Friday, March 24, 2023

Title: Numerical Integration of Hamiltonian Monte Carlo
Speaker: Lorenzo Nagar, Basque Center for Applied Mathematics
Time: 1:00pm–2:00pm
Place: CMC 109

#### Abstract

Hamiltonian Monte Carlo (HMC) has been widely recognized as a powerful tool for sampling in molecular simulation and Bayesian Statistics. HMC combines a deterministic proposal — using Hamiltonian dynamics — with stochastic Monte Carlo in order to generate correlated samples from a target distribution. The accuracy of numerical integration of the Hamiltonian equations of motions affects the acceptance rate of Monte Carlo trials, and thus is crucial for performance of HMC. In this talk, after giving a brief overview on multi-stage splitting integration schemes used in the HMC context, we present a novel Adaptive Integration Approach (we call it s-AIA) that detects the optimal multi-stage integrator in terms of the best conservation of energy for harmonic forces, being able to achieve a competitive sampling efficiency in HMC Bayesian inference applications.

### Friday, March 10, 2023

Title: Idempotents in a quandle ring and its applications, Part II
Speaker: Dipali Swain
Time: 1:00pm–2:00pm
Place: CMC 109

### Friday, March 3, 2023

Title: Idempotents in a quandle ring and its applications
Speaker: Dipali Swain
Time: 1:00pm–2:00pm
Place: CMC 109

#### Abstract

For a quandle, its associated quandle ring is a very interesting structure. Unlike Groups, Quandle is a non-associative algebra, binary operation of which imitates the three Reidemeister Moves found in Knot theory. When we associate a ring to such an algebraic structure, the most naturally occurring object is an idempotent. We study this object and investigate if such a collection of objects itself forms a quandle and hence try to create an invariance of knots/links. In this talk, I will give a brief introduction of quandles and an associated quandle ring. I will also talk about some popular knot invariants that arise from this particular structure.

### Friday, February 24, 2023

Title: Introduction to Ramsey Number, and Some Elementary Yet Beautiful Examples
Speaker: Boyoon Lee
Time: 1:00pm–2:00pm
Place: CMC 109

#### Abstract

Ramsey theory can be roughly understood to be the statement on the existence of regular structure in a seemingly chaotic environment.

In this talk, designed for people who have little to no experience in Ramsey theory, we will go over what the Ramsey number is, how hard it is to track down any of them while the question of finding one seems quite unassuming/innocuous.

We will warm up with a very easy case of a particular Ramsey number $$r(3,3)=6$$.

Then, we will borrow a pure genius argument of L. Lovász to reach the conclusion of Ramsey number $$r(3,3,3)=17$$ and be amazed.

The meaning of $$r(a,b,c,\dotsc)$$ is going to be given at the talk.

### Friday, February 17, 2023

Title: Designs, Nibbles, and Approximate Structures, Part II
Speaker: Eion Mulrenin
Time: 1:00pm–2:00pm
Place: CMC 109

### Friday, February 10, 2023

Title: Designs, Nibbles, and Approximate Structures
Speaker: Eion Mulrenin
Time: 1:00pm–2:00pm
Place: CMC 109

#### Abstract

For integers $$1 < t < k < n$$ and a positive integer $$\lambda$$, a family $$F$$ of $$k$$-subsets (sets of size $$k$$) of $$\{1,\dotsc,n\}$$ is called a $$t-(n,k,\lambda)$$ design provided that every $$t$$-subset of $$\{1,\dotsc,n\}$$ is contained in precisely $$\lambda$$ sets in the family $$F$$. A simple double counting argument gives necessary divisibility conditions for the existence of $$t-(n,k,\lambda)$$ designs, and it was asked c. 1850 (the so-called "existence conjecture") whether these were also sufficient. In 1963, P. Erdös and H. Hanani conjectured the asymptotic existence of "approximate designs" for $$\lambda=1$$ and $$k$$ and $$t$$ fixed. The conjecture was verified in 1985 by V. Rodl, and in his solution, he introduced his seminal "nibble" technique, which has since become a powerful tool in extremal combinatorics. The existence conjecture was ultimately verified in 2014 by P. Keevash and in 2016 by D. Kuhn et al., with both proofs making copious use of nibble techniques. In this talk, we will introduce and prove some basic facts about combinatorial designs and sketch a proof of the Erdös-Hanani conjecture using nibble methods.

### Friday, February 3, 2023

Title: It’s Raining Coding Theory
Speaker: Panith Thiruvenkatasamy
Time: 1:00pm–2:00pm
Place: CMC 109

#### Abstract

I’m planning to discuss Dr. Giacomo Micheli’s and Dr. Alessandro Neri’s paper titled “New Lower Bounds for Permutation Codes using Linear Block Codes”. We will be discussing in brief the main theorems in the paper and the proofs for those theorems, and I plan to end the talk by mentioning some questions to ponder upon that were inspired by the paper.

### Friday, January 27, 2023

Title: An Introduction to Tamo-Barg Codes, Part II
Speakers: Clifton Garrison, Logan Nott, Phillip Waitkevich
Time: 1:00pm–2:00pm
Place: CMC 109

### Friday, January 20, 2023

Title: An Introduction to Tamo-Barg Codes
Speakers: Clifton Garrison, Logan Nott, Phillip Waitkevich
Time: 1:00pm–2:00pm
Place: CMC 109

#### Abstract

Prior to the discovery of Tamo-Barg codes, it was common for codes to recover a single erased coordinate by using all the other coordinates. In 2013, Itzhak Tamo and Alexander Barg created an optimal locally recoverable code, which is a code that can recover a single erasure using at most r other coordinates instead of all the other coordinates. This talk will introduce coding theory and the construction of Tamo-Barg Codes.