This is a colloquium-style seminar. It runs on Wednesdays. The current organizers are Melkana Brakalova and Han-Bom Moon.
The seminar is broadcasted via zoom. Check the announcement emails for the link.
Date: October 1, 3:00 pm
Place: JMH 405 (RH)
Speaker: Ara Basmajian (CUNY)
Title: Counting problems, Hecke groups, and hyperbolic geometry
Abstract: The modular group is the group generated by the Mobius elements $z \mapsto \frac{-1}{z}$ and $z \mapsto z+1$ acting on the upper half-plane. These elements act as isometries with respect to the hyperbolic metric on the upper half-plane, and as a result this group, and its cousins the Hecke groups, is rich with interesting geometric, number theoretic and combinatorial questions.
In this talk we'll introduce hyperbolic geometry and the properties of the Hecke groups leading to various counting problems. The solutions of these counting problems involve using recursion relations, finding roots of polynomials, and elementary notions from linear algebra.
Date: October 22, 3:00 pm
Place: JMH 405 (RH)
Speaker: Anders Buch (Rutgers)
Title: Quantum Schubert Calculus
Abstract: A typical motivating question in classical algebraic geometry is to determine the list of all geometric objects of some type that satisfy a list of conditions. Surprisingly, while this task is usually extremely difficult, intersection theory provides tools that can often be used to predict the number of solution objects. More generally, when infinitely many objects satisfy the specified conditions, these objects are frequently parametrized by a moduli space, in which case one can ask for geometric invariants of this space. A particular example is a K-theoretic Gromov-Witten invariant, defined as the arithmetic genus of the moduli space of rational curves of a given degree in a flag manifold that pass through a list of orbit closures called Schubert varieties. The quantum K-theory ring of the flag manifold encodes all these Gromov-Witten invariants in its multiplicative structure. Much research by many mathematicians during the last 25 years has studied this ring with methods from multiple fields, ranging from algebraic geometry to combinatorics and physics. From a combinatorial perspective, the quantum K-theory ring has a basis of Schubert classes, and is determined by its structure constants relative to this basis. These structure constants are far generalizations of the classical Littlewood-Richardson coefficients from representation theory, as well as the structure constants of Schubert polynomials, which are central objects of study in algebraic combinatorics. In my talk I will first of all explain, in elementary geometric terms, what the quantum K-theory ring is, and how it can be used to count curves and compute geometric invariants of solution spaces. I will also discuss some of the main results and conjecture in the field, such as vanishing and sign properties of structure constants and combinatorial algorithms for computing Gromov-Witten invariants.
Date: October 29, 3:00 pm
Place: LL 510 (LC)
Speaker: Alvaro Lozano-Robledo (University of Connecticut)
Title: A short proof of Fermat's Last Theorem (for non-constant polynomials!)
Abstract: Fermat's last theorem was proposed by Fermat in a famous note written in a book's margins, around 1635. Since then, many, many mathematicians have tried to find Fermat's proof, but no short proof has ever been found. The first complete proof (Andrew Wiles' proof) was published in 1995, and it spans hundreds of pages of very advanced algebraic number theory, which is certainly not a proof that Fermat could have even dreamed of. In this talk, we will give a short proof of Fermat's last theorem... for (non-constant) polynomials. The main tool will the ABC Theorem (for non-constant polynomials), which will give us a chance to discuss the drama surrounding the ABC Conjecture for the integers.
Date: November 12, 3:00 pm
Place: JMH 405 (RH)
Speaker: Shree Rajkumar Saha (CUNY)
Title: Fitting the probability Gamma difference model with real return data and application of large deviation
Abstract: In this paper, we focus on our novel technique for jointly estimating the four parameters of the Gamma Difference distribution, which models the difference between two Gamma-distributed random variables. After verifying our method with real return data, we benchmark its performance against Klar (2014) on three dataset types. Leveraging the estimated parameters, we then use large deviation theory to estimate the tail probability of large losses over extended periods.
Date: November 19, 3:00 pm
Place: JMH 405 (RH)
Speaker: Matthew Junge (CUNY)
Title: How REU?
Abstract: What are math research experiences for undergraduates (REUs) like, and how can students and professors make the most of them? Expect insights and tips. I will also share recent student projects about a competitive stochastic growth model known as chase-escape.
Date: December 3, 3:00 pm
Place: JMH 405 (RH)
Speaker: Enrique Pujas (CUNY)
Title: TBA