Polytopes are higher-dimensional generalizations of polygons in the plane. Not only are they intrinsically beautiful objects, they lie at the crossroads of diverse areas of mathematics. In this talk, I will discuss a quantity associated with polytopes, called the mixed volume, the fundamental Alexandrov—Fenchel Inequality for mixed volumes, and the many connections that these quantities have with other parts of mathematics.

A gentle yet rapid introduction to class field theory, its conjectured generalization – the Langlands program, and the role of L-functions therein. Our starting point is a density result due to Frobenius, which implies that the Galois (hence all) extensions of a number field are uniquely determined by the set of primes that split completely. The rest of the talk traces historically the quest of understanding this set culminating in its close connection with Artin L-functions, certain meromorphic functions on the complex plane reminiscent of the Riemann zeta function.

Given a partially ordered set (poset) P, a plane partition is a way of stacking boxes on each element that respect the partial order of P. In 1983, Proctor proved that the number of plane partitions of a rectangle poset is the same as that of its corresponding trapezoid poset. Proctor’s proof, however, is non-bijective. In this talk, I will describe a recent combinatorial bijection that comes from a surprising connection with K-theoretic Schubert calculus, in particular the operation of K-jeu-de-taquin on increasing Young tableaux.

Geometric representation theory uses geometric techniques such as sheaves and D-modules in order to study representations of symmetry objects such as Hecke algebras, quantum groups, quivers, or algebraic groups. In this talk, we will develop some geometric and representation-theoretic tools, and give an overview of several major results in geometric representation theory, including the Beilinson-Bernstein correspondence, Riemann-Hilbert correspondence, and Kazhdan-Lusztig conjectures.

Legendrian knots appear in many different guises in mathematics and physics: as wavefronts of light propagation, trajectories of a rolling coin that does not slip, generalizations of the derivative of smooth functions, and as graphs of critical points of a family of functions. I will discuss these classical examples and some recent developments in the field related to symplectic flexibility and microlocal sheaf theory.

The field of provability logic grows largely out of Godel’s second incompleteness result: the impossibility of a sufficiently powerful consistent formal system proving its own consistency. The main interest of this theorem, besides some philosophical musings, is the general method of proof which goes about formalizing within number theory itself the Godelian argument of constructing a sentence which is equivalent to its own unprovability. That is, number theory is sufficiently powerful to derive Godel’s first incompleteness theorem and show that the truth value of any Godel sentence is equivalent to the consistency of the theory (which, informally, is clear since if G is false then there must exist a proof of G, which is false). Godel’s second incompleteness result is intimately related to Lob’s theorem which expresses a sort of “epistemic modesty” that number theory has about its its own soundness. Lob’s theorem is the vital step in constructing GL or provability logic which succeeds in stripping number theory of all the extra junk about numbers while entirely capturing its self-referential behavior regarding provability captured within the language.

The cap set problem in combinatorics involves finding configurations with no valid moves in a generalization of the classic card game Set. In 2016, breakthrough work by Croot, Lev, and Pach and Ellenberg and Gijswijt made massive progress on the problem by an elementary proof that was totally different from previous methods. I will explain the problem and discuss some of the ideas in the proof.

In this talk, we will give some intuition for how probabilistic techniques can be useful in number theory by studying the function $\omega(n)$, which counts the number of distinct prime factors of $n$. After giving some motivation and basic results from analytic number theory, we will prove the Hardy-Ramanujan theorem, which gives a simple bound satisfied for $o(N)$ integers $n \le N$. From there, we will use basic probability theory to reinterpret Hardy-Ramanujan. Lastly, we will introduce a more general result about the distribution of $\omega(n)$: the Erdos-Kac theorem. Time permitting, we will give a heuristical proof of Erdos-Kac.

In this talk we'll discuss the circle of ideas introduced by Thurston to study 3-dimensional spaces: namely, how tools from geometry can be extremely powerful when studying topology. By the end of this talk, we'll state Thurston famous geometrization conjecture (now a theorem thanks to the work of Perelman).