Sieve methods are analytic tools that we can use to tackle problems in additive number theory. This talk will serve as a gentle introduction to the area. At the end we will discuss recent progress on a variation on the prime $k$-tuples conjecture which involves sums of two squares. No knowledge of sieves is required!

In 1983, Faltings proved Mordell's conjecture on the finiteness of $K$-points on curves of genus >1 defined over a number field $K$ by proving the finiteness of isomorphism classes of isogenous abelian varieties over $K$. The "first" major step from Mordell's conjecture to what Faltings did came 15 years earlier when Parshin showed that a certain conjecture of Shafarevich would imply Mordell's conjecture. In this talk, I'll focus on motivating and sketching Parshin's argument in an accessible manner and provide some heuristics on how to get from Faltings' finiteness statement to the Shafarevich conjecture.

The Riemann hypothesis (RH) is one of the great open problems in mathematics. It arose from the study of prime numbers in an analytic context, and—as often occurs in mathematics—developed analogies in an algebraic setting, leading to the influential Weil conjectures. RH for curves over finite fields was proven in the 1940’s by Weil using algebraic-geometric methods. In this talk, we discuss an alternate proof of this result by Stepanov (and Bombieri), using only elementary properties of polynomials. Over the decades, the proof has been whittled down to a 5 page gem! Time permitting, we also indicate connections to exponential sums and the original RH.
 

A remarkable theorem due to Khovanskii asserts that for any finite subset $A$ of an abelian group, the cardinality of the $h$-fold sumset $hA$ grows like a polynomial for all sufficiently large $h$. However, neither the polynomial nor what sufficiently large means are understood in general. We obtain an effective version of Khovanskii's theorem for any $A \subset \mathbb{Z}$ whose convex hull is a simplex; previously such results were only available for $d = 1$. Our approach also gives information about the structure of $hA$, answering a recent question posed by Granville and Shakan. The work is joint with Leo Goldmakher at Williams College.

The Dirichlet class number formula gives an expression for the residue at s=1 of the Dedekind zeta function of a number field K in terms of certain quantities associated to K. Among those is the regulator of K, a certain determinant involving logarithms of units in K. In the 1980s, Don Zagier gave a conjectural expression for the values at integers s $\geq$ 2 in terms of "higher regulators", with polylogarithms in place of logarithms. The goal of this talk is to give an algebraic-geometric interpretation of these polylogarithms. Time permitting, we will also discuss a similar picture for Hasse--Weil L-functions of elliptic curves.
 

I will explain how the representation theory of rational Cherednik algebras interacts with the commutative algebra of certain subspace arrangements arising from the reflection arrangement of a complex reflection group. Potentially, the representation theory allows one to study both qualitative questions (e.g., is the arrangement Cohen-Macaulay or not?) and quantitative questions (e.g., what is the Hilbert series of the ideal of the arrangement, or even, what are its graded Betti numbers?), by applying the tools (such as orthogonal polynomials, Kazhdan-Lusztig characters, and Dirac cohomology) that representation theory provides. This talk is partly based on joint work with Susanna Fishel and Elizabeth Manosalva.

The Ricci flow on a surface is an intrinsic evolution of the metric converging to a constant curvature metric within the conformal class. It can be seen as an infinite-dimensional gradient flow. We introduce a natural 'Langevin' version of that flow, thus constructing an SPDE with invariant measure expressed in terms of Liouville Conformal Field Theory.
Joint work with Hao Shen (Wisconsin).

Let $G$ be a connected reductive algebraic group, and let $G(\mathbb{F}_q)$ be its group of $\mathbb{F}_q$-rational points. Denote by $\mathrm{Irr}(G(\mathbb{F}_q))$ the set of (equivalence classes) of irreducible finite-dimensional representations. Deligne and Lusztig defined a finite subset $$\mathrm{Unip}(G(\mathbb{F}_q)) \subset \mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$$ 
of unipotent representations. These representations play a distinguished role in the representation theory of $G(\mathbb{F}_q)$. In particular, the classification of $\mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$ reduces to the classification of $\mathrm{Unip}(G(\mathbb{F}_q))$. 

Now replace $\mathbb{F}_q$ with a local field $k$ and replace $\mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q))$ with $\mathrm{Irr}_{\mathrm{u}}(G(k))$ (irreducible unitary representations). Vogan has predicted the existence of a finite subset 
$$\mathrm{Unip}(G(k)) \subset \mathrm{Irr}_{\mathrm{u}}(G(k))$$ 
which completes the following analogy
$$\mathrm{Unip}(G(k)) \text{ is to } \mathrm{Irr}_{\mathrm{u}}(G(k)) \text{ as } \mathrm{Unip}(G(\mathbb{F}_q)) \text{ is to } \mathrm{Irr}_{\mathrm{fd}}(G(\mathbb{F}_q)).$$
In this talk I will propose a definition of $\mathrm{Unip}(G(k))$ when $k = \mathbb{C}$. The definition is geometric and case-free. The representations considered include all of Arthur's, but also many others. After sketching the definition and cataloging its properties, I will explain a classification of $\mathrm{Unip}(G(\mathbb{C}))$, generalizing the well-known result of Barbasch-Vogan for Arthur's representations. Time permitting, I will discuss some speculations about the case of $k=\mathbb{R}$.

This talk is based on forthcoming joint work with Ivan Loseu and Dmitryo Matvieievskyi.