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Title: Introduction to Bootstrap Speaker: Tejasvi Channagiri Time: 1:00pm–2:00pm Place: CMC 109

A central problem of statistics is estimating a parameter (e.g., mean) of an unknown distribution given a sample. We usually also want to know the uncertainty of the estimate, such as its standard error. For simple cases like the sample mean there are closed-forms for the standard error. However, in more complex situations, closed-forms may be unavailable. The bootstrap is a computational method for determining standard errors of estimates that is widely applicable and easy to implement. Here we introduce the bootstrap, and present examples of its applications and limitations. No background is required aside from some elementary probability.

Title: Random matrices and the Combinatorics of Ribbon Graphs, Part II Speaker: Nathan Hayford Time: 1:00pm–2:00pm Place: CMC 109

Title: Random matrices and the Combinatorics of Ribbon Graphs Speaker: Nathan Hayford Time: 1:00pm–2:00pm Place: CMC 109

Random matrices have long been a source of interest in mathematics, with applications ranging from number theory to data analysis to quantum gravity. Another unexpected place random matrices have appeared is in the combinatorial enumeration of ribbon graphs. A number of conjectures related to this problem were posed by W. Tutte in the 60’s; the answers to a number of these conjectures came (surprisingly) from a group of theoretical physicists, who used random matrix techniques to approach these conjectures. In this talk, I will try and outline some of the techniques used, and show how random matrices are able to distinguish ribbon graphs by “genus”.

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

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.

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

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

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.

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

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.

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

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

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.

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

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.

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

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

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.