**Abstracts**

Alex Sutherland, UC Irvine

Max Engelstein, University of Minnesota, Twin Cities

Juliette Bruce, UC Berkeley

Ray Li, Stanford University

Billy Woods, University of Essex

Ben Blum-Smith, The New School

Vaidehee Thatte, King's College London

Sarah Arpin, University of Colorado, Boulder

Deewang Bhamidipati, UC Santa Cruz

Tim Hosgood, Stockholms universitet

Mahrud Sayrafi, University of Minnesota

Wojtek Wawrów, London School of Geometry and Number Theory

Matt Macauley, Clemson University

Anton Hilado, University of Vermont

Sunil Chetty, College of St. Benedict and St. John's University

## "What is... resolvent degree?"

Given a polynomial *f(z)*, how can we describe a root of *f *in terms of its coefficients? Trying to answer this question leads to an invariant known as resolvent degree. In this talk, we will address the original question, discuss what we know about resolvent degree, and use resolvent degree to connect questions about solving polynomials to other geometric and representation theoretic problems.

## "What are... Lojasiewicz inequalities in geometry?"

Lojasiewicz inequalities were originally introduced to study the structure of real analytic varieties. But since then, they've been co-opted by (geometric) analysts to study singularities in and the long time behavior of solutions to variational problems. We will give a gentle introduction to the Lojasiewicz inequalities, hopefully explaining what they have to do with Taylor series and why they carry so much geometric information.

## "What is... Castelnuovo-Mumford regularity?"

Broadly speaking, Castelnuovo–Mumford regularity should be thought of as a measure of how complicated a (homogenous) ideal in a polynomial ring can be. It turns out that understanding this complexity highlights the deep connections between the algebra of an ideal and the geometry of the corresponding algebraic variety.

## "What is... an error correcting code?"

Error correcting codes are sets of strings that are "far apart". They were pioneered by Claude Shannon in 1948 for communication across noisy channels, but have since found wide ranging applications and rich connections to mathematics. In this talk, I introduce error correcting codes, highlight some of their many applications and settings, and illustrate a few classical constructions.

## "What is... an Iwasawa algebra?"

## "What is... a Cohen-Macaulay ring?"

Slides for Ben's talk are available here.

## "What is... the defect?"

Classical examples of valuation fields are the field of $p$-adic numbers $\Q_p$ for a prime $p$ and the fraction field $k((X))$ of the power series ring $k[[X]]$ for a perfect field $k$ (such as $\F_p$ or $\C$). The valuation, obtained by looking at the exact power of $p$ (respectively $X$) dividing an element of the field, is discrete. Moreover, these fields are complete under the non-Archimedean metric induced by this valuation. Classical invariants of ramification theory capture important information about extensions of these fields.

This nice structure is lost when we allow non-discrete valuations and relax the conditions on the field $k$. For example, many interesting objects in the classical case are generated by a single element. However, in the general case, even finite generation is not guaranteed. In particular, we may encounter the mysterious phenomenon of the {\em defect} (or ramification deficiency). The occurrence of a non-trivial defect is one of the main obstacles to long-standing problems, such as obtaining resolution of singularities in positive characteristic.

In this talk, we will introduce the notion of the defect and discuss how the study of ramification theory leads to a better understanding of the defect.

## "What is... supersingular elliptic curve cryptography?"

## "What is... a scheme?"

## "What is... a quantum algorithm?"

Lots of people these days like talking about quantum computers, and quantum supremacy, and quantum cryptography, and other such quantum-y things. But what is the whole fuss about? I mean, has anybody even made a quantum computer yet? And, if they had, would they actually be better than “classical” computers for anything other than factoring really big numbers?

In this seminar, we will have a look at the very basics of quantum information theory (which requires absolutely no knowledge of anything remotely quantum physics-y, and needs only some simple linear algebra and probability theory). We will start by explaining qubits and quantum circuits, before studying a specific quantum algorithm, which is exponentially faster than the best possible classical algorithm, and solves a concrete computational problem.

## "What is... an exceptional collection?"

Using a single point as a toy example, we will take the scenic route for describing the derived category of coherent sheaves. Along the way, we will see familiar examples of free resolutions coming from commutative algebra and peek at strange resolutions over exterior algebras, until we realize that the real treasures are the exceptional collections that helped us make derived categories a little more explicit.

## "What is... an Euler system?"

One of the most notable open problems in all of mathematics is the Birch and Swinnerton-Dyer conjecture, which relates various analytic and algebraic data attached to an elliptic curve. Major progress towards this problem came from Kolyvagin's work, and one of the crucial ingredients was the notion of an Euler system introduced by him. In the talk, we will introduce this general notion and explain its relation to the BSD conjecture (as well as the more general Bloch-Kato conjecture), focusing on the example given by cyclotomic units.

## "What is... a Cayley diagram?"

A Cayley diagram is a visualization of a group presentation. These arise in fields such as geometry group theory and topology, but I will focus on their role in elementary abstract algebra. I hope to convince you that learning or teaching algebra without these tools is akin to learning or teaching calculus without graphs. This talk will be filled with colorful pictures and visualizations, and I promise that you will leave with new ideas that will forever change how you think about algebra.

## "What is... the Langlands program?"

## "What is... a Selmer group?"

In the theory of elliptic curves, understanding the behavior of rank is a central problem. In light of the Birch--Swinnerton-Dyer and Tate-Shafarevich Conjectures, there are three avenues for understanding rank of a given elliptic curve E/K: by the structure of the Mordell-Weil group E(K), by the vanishing of the associated L-function L(E/K, s), or by the structure of the associated Selmer group Sel(E/K). We will discuss some of the big ideas for attacking the rank problem over number fields via the Selmer group approach, as well as methods of comparing parallel tools in the L-function approach.

Slides for Sunil's talk are now available here.

For abstracts of talks given in previous semesters, click here.