Zoom link: https://videoconf-colibri.zoom.us/j/91864016501?pwd=J0qrDNWOva0DeM1XTOuUiDN3ToRd3b.1
 
The classical theory of type A crystals provides a graphical framework for proving Schur positivity results. In this talk we will discuss a new category of "square root crystals" (introduced implicitly in work of Yu) that can be used to establish instances of Grothendieck positivity. For example, Buch's combinatorial interpretation of the coefficients expanding products of symmetric Grothendieck functions has a simple description in terms of the tensor product for this category. We will also discuss some shifted analogues and applications to a conjectural formula of Cho-Ikeda for K-theoretic Schur P-functions.

Venue: office 2, 3rd floor of the Mathematics Department Building

Venue: office 2, 3rd floor of the Mathematics Department Building

Venue: office 2, 3rd floor of the Mathematics Department Building

Venue: office 2, 3rd floor of the Mathematics Department Building

Venue: office 2, 3rd floor of the Mathematics Department Building

Let F be a finite field and let m, d and r be positive integers. Consider the following
question: What is the maximum number of common zeros over F that a system of r linearly
independent homogeneous polynomials of degree d in m + 1 variables can have?
Because of homogeneity, we will disregard the trivial zero (viz., the origin) and regard two
zeros as equivalent if they are proportional to each other, i.e., if one is obtained from another
upon multiplying all coordinates with a nonzero scalar. In other words, we look for zeros in
the m -dimensional projective space over the field F.
This question was first raised by M. Tsfasman in the case of a single homogeneous
polynomial, i.e., when r = 1. It was then settled by J.-P. Serre (1991). Later Tsfasman
together with M. Boguslavsky formulated a remarkable conjecture in the general case, and
this was shown to hold in the affirmative in the next case of r = 2 by Boguslavsky (1997).
Then about two decades later, it was shown that the conjecture is valid conjecture is valid if
the number of polynomials is at most the number of variables, i.e., r ≤ m + 1, but the
conjecture can be false in general. Newer conjectures were then formulated and although
there has been considerable progress concerning them, the general case is still open.
These questions are intimately related to the study of maximal sections of Veronese varieties
by linear subvarieties of the ambient projective case, and also to the study of the an important
class of linear error correcting codes, called projective Reed-Muller codes.
In this talk, we will outline these developments and explain the above connections.
An attempt will be made to keep the prerequisites at a minimum.

Venue: office 2, 3rd floor of the Mathematics Department Building

Venue: office 10, 3rd floor of the Mathematics Department Building

Venue: office 2, 3rd floor of the Mathematics Department Building

I will give a brief introduction to Algebraic Machine Learning (AML), a purely algebraic method that does not use statistics, search or optimization. Instead, AML relies on semantic embeddings in semilattices and subdirect decompositions. AML is capable of learning from data and also from just a problem statement in the form of a set of formulas; for example, AML can learn to find Hamiltonian Cycles from the problem statement. With the same algorithm, AML can learn from data (e.g. classifying medical images) with a test accuracy that rivals that of deep multilayer perceptrons. I will also give an introduction to Atomized Semilatices, a mathematical method developed to operate and compute AML models.

Venue: office 10, 3rd floor of the Mathematics Department Building

Venue: office 2, 3rd floor of the Mathematics Department Building

Venue: office 10, 3rd floor of the Mathematics Department Building

Venue: office 2, 3rd floor of the Mathematics Department Building