Abstract. The problem of classifying representations of a quiver essentially asks: what are the different ways a diagram of vector spaces and linear maps can behave, up to changes of basis? A theorem of Gabriel shows that this problem has its simplest answer when the underlying diagram is one of the Dynkin diagrams, in which case the indecomposable representations match up with positive roots of the associated root system.

Given the appearance of Dynkin diagrams and root systems, it's tempting to ask: is there a more direct connection between representations of a quiver and the associated Lie algebra itself? This opens up a deep area of study based on reconstructing Lie algebras and useful bases for them in terms of moduli spaces of quiver representations.

In this course, we'll first lay out the context of quiver representations and Gabriel's theorem. Then, we'll scratch the surface of the deeper theory using the Ringel-Hall algebra, which recovers the universal enveloping algebra of the positive part of a semisimple Lie algebra.

I'll introduce most of the above concepts during the course, though prior exposure to Lie algebras will be helpful.

Abstract. Schubert calculus is the study of intersection theory on Grassmannians and other related objects. The subject was first introduced to answer questions in enumerative geometry, but has since found many connections to other areas of combinatorics and representation theory. In this minicourse, I will give an overview of Schubert calculus on the Grassmannian, solve some enumerative geometry problems, and make the connection to symmetric polynomials and Young tableaux. Time permitting, we will also discuss generalizations to other varieties such as the full flag variety, and other cohomology theories such as equivariant cohomology and K-theory

Abstract. Enumerative geometry is the study of counting problems in geometry, such as: How many circles pass through 3 points in the plane? How many conics are tangent to five given conics? How many lines are there on a cubic surface? This minicourse will give an introduction to enumerative geometry, starting with the Chow ring and its applications to solving classical problems.

The second part of this minicourse will introduce Gromov-Witten theory, a field with connections to enumerative geometry and physics; Gromov-Witten invariants of X are numbers roughly corresponding to the number of algebraic curves in X satisfying certain conditions. We’ll see how Gromov-Witten theory can be used to answer the question: how many rational curves of degree d pass through 3d-1 general points in P^2?

My goal is to provide an accessible introduction to these topics, and I will try to introduce necessary concepts as we go along (though some exposure to algebraic geometry and/or (co)homology will be helpful).

Abstract. A vanishing theorem broadly refers to a theorem which gives conditions for coherent cohomology groups to vanish. Such theorems have tremendous applications to various branches of algebraic geometry, for instance in the Minimal Model Program.

In this minicourse, I will present some of the fundamental vanishing theorems in char 0 algebraic geometry: Kodaira vanishing, Kawamata-Viehweg vanishing, and a few more. If time permits, I'll also talk about multiplier ideals and some of their applications.

While basic knowledge of sheaf cohomology will be assumed, I'll introduce the necessary background from intersection theory in the first lecture.

Abstract. In this course, we will investigate equivariant analogues of algebraic objects such as abelian groups, rings, modules and algebras. We will begin by discussing the algebra of Mackey functors, which are equivariant analogues of abelian groups. We will go on to discuss Green functors and (incomplete) Tambara functors, which are equivariant analogues of commutative rings. We will include many interesting examples, and emphasize the similarities and differences between ordinary algebra and equivariant algebra. There are minimal prerequisites for this course, so all are encouraged to participate!

Abstract. Hochschild homology groups of an algebra A over a commutative ring k arise as the homology groups of the explicit complex HH(A/k), they were introduced already in 1950s. If the algebra A is commutative and smooth, then by the Hochschild-Kostant-Rosenberg theorem the n-th homology group is identified with the n-th Kähler forms module. It turns out that one can write down an explicit differential on HH(A/k) which coincides with the usual de Rham differential for smooth and commutative algebras.

We will discuss why one can call this structure the circle action and we will derive the cyclic, negative cyclic, and periodic cyclic homology from this action. Also, we will discuss the topological version of the Hochschild homology and its relation to arithmetics.

Abstract. Even if one is interested only in smooth algebraic varieties, it soon becomes necessary to understand singularities. Singularities of algebraic surfaces provide a rich class of examples which has motivated much of the study of singularities in higher dimension. In this course, we will cover resolution of singularities for algebraic surfaces, classes of algebraic surface singularities (quotient, rational, Du Val, etc.), cohomological approaches to singularities via the study of various cycles on resolutions, and applications to commutative algebra via complete ideals and unique factorization. For much of the course the only prerequisite is a first course in algebraic geometry, but some familiarity with sheaf cohomology will be useful for the second half of the course.

Abstract. ropical geometry is the study of algebraic varieties over the min-plus semiring. Tropical geometry has deep connections to other areas of mathematics (including enumerative algebraic geometry, algebraic statistics, and dynamics), and has been rediscovered independently many times. Only recently have efforts been made to unify tropical results from different fields into a single theory of tropical geometry.

In this course I will introduce some of the fundamental results in tropical geometry with a focus on computing examples. Specifically, I plan to cover: tropical hypersurfaces, tropical linear spaces, tropical polytopes and some applications of tropical geometry. I am also happy to talk about other tropical geometry topics after the minicourse!

Abstract. Symplectic geometry has its roots in classical mechanics where it was mainly used to study the time evolution of planetary systems. We have come a long way since then, and with huge development in this area, symplectic geometry has become a central branch of differential geometry and topology. This mini-course will aim to introduce this subject with minimal prerequsites (basic knowledge of manifolds). After introducing the notion of a symplectic structure on manifolds, we will see why symplectic manifolds have no local invariants (Darboux Theorem) unlike Riemannian manifolds. Then, we will introduce Hamiltonian group actions on symplectic manifolds and the associated moment map which will naturally lead to the idea of Marsden-Weinstein reduction. If time permits, we will also try to introduce the Atiyah-Guillemin-Sternberg convexity theorem which characterizes the image of the moment map for torus actions and is linked to toric geometry.

Abstract. Systems arising in biology and chemistry frequently involve the interaction of components evolving on vastly different timescales, resulting in rich dynamics. In this minicourse, we describe mathematical techniques used to analyze such systems and study example applications arising in the recent literature. We will begin with a rapid review of some basic notions in nonlinear dynamics and bifurcation theory. Following this, we will proceed to briefly describe the main theorems of Fenichel theory (geometric singular perturbation theory) and moreover discuss desingularization of nilpotent singularities via the blowup method; this will be accompanied by an extended example analyzing a model for an autocatalytic chemical process (Gucwa, Szmolyan 2008). Next, we will study canard solutions arising in 2D and 3D systems, focusing specially on the geometry of slow manifolds near a folded node singularity. To complement these ideas, we discuss recent applications towards explaining the effects of experimental manipulations on early afterdepolarizations in cardiomyocytes (Vo, Bertram 2009). Finally, time permitting, we will describe interesting noise-induced effects on relaxation oscillations, namely coherence resonance and self-induced stochastic resonance (DeVille et al 2005).

Abstract. In these talks we will discuss notions of ``derivatives'' for integers. A main motivation for this pursuit is Mason-Stothers theorem, which is an polynomial analogue of the abc conjecture, and for which its proof relies on being able to take derivatives of polynomials. We will discuss basic aspects of Buium's theory of p-derivations and arithmetic differential equations, as well as a recent preprint by Pasten which has a different interpretation of derivatives on integers.