# Seminars & Events for Graduate Student Seminar

##### Grushko's Theorem

Grushko's Theorem states that $rank(G*H)=rank(G)+rank(H)$, where $rank$ is the minimum number of generators for a group, and $*$ denotes the free product. We will present Stallings' (topological) proof of Grushko's Theorem.

##### The Composition Problem in Measure Spaces

Let $X,Y$ be measurable spaces and $\eta : X \to Y$ be a measurable function. Under what conditions on $\eta$ is the composition with $f : Y \to C$ a well defined operation when $f$ is only specified almost everywhere? Does composition with $\eta$ induce a map between $L^p$ spaces? I will show how one generally would answer these questions , give an algebraic prespective on the problem(and on measure spaces in general), and give a complete solution when the map $\eta$ is multipilication on the $p$-adic integers.

##### GRH and polynomial-time primality testing

For a long time, a famous open problem was to figure out whether a number was prime quickly (in polynomial time). It's interesting to see how under the generalized riemann hypothesis, the problem becomes completely straightforward. I will introduce the relevant concept and present the simple proof. During the second half, I will present a provably polynomial time test, without reliance on GRH. If time permits, I will say some more about L-functions and their application to computing.

##### Ergodic Theory

I will be discussing two basic invariants of ergodic theory, entropy and ergodicity, through the problems they were invented to solve. This talk will be heavy on examples and low on rigour.

##### Differential Galois Theory

Algebraic groups, differential equations, and Galois theory: who isn't scared of at least one of those? It being the last GSS talk before Halloween, I'll do my best to mention all three. The ostensible purpose of doing so will be to explain why certain indefinite integrals cannot be written in terms of elementary functions. No knowledge of elementary functions will be presupposed.

##### Embedded Surfaces in 4-Manifolds

Given a 2-dimensional homology class in a closed 4-manifold, you can try to represent it by an embedded, closed, orientable surface. What is the minimum genus of such a surface? Come find out.

##### An overview of the l-adic and p-adic monodromy theorems

I will introduce (focusing on the case of elliptic curves) two essential theorems of arithmetic geometry that bind the geometry of an algebraic variety over a local field (or number field) to its arithmetic, Galois-theoretic properties. The l-adic monodromy theorem is in fact entirely elementary (one needn't even know what 'monodromy' means!) but will motivate the subtler p-adic theorem. Together these results will help us make sense of the Fontaine-Mazur conjecture, one of the fundamental open problems in arithmetic geometry.

##### Fibered Knots

A fibered knot is a knot whose complement can be filled "nicely" by copies of an oriented surface bounded by the disk, i.e., is filled by $S^1$ copies of $D^2$ (in fact, this fibration is globally trivial: $S^3-K\wedge S^1\times D^2$). By the time the pizza is all eaten, we should even be able to understand Milnor's construction of a fibration of the $(p,q)$ torus knot by surfaces of genus $(p-1)(q-1)/2$. You may care about fibered knots if you have ever been or will ever be interested in any of the following:

- hyperbolic structures
- algebraic knots and links
- unbranched cyclic covers
- open book decompositions

##### The Nielsen Realization Problem

The purpose of this talk will be to describe some problems in topology whose solutions rely on geometric techniques. I will introduce the mapping class group of a space $X$, which is the group of homeomorphisms of $X$ up to isotopy. I will then outline Steve Kerckhoff's solution to the Nielsen realization problem, which shows that any finite subgroup of $Mod_g$, the mapping class group of a genus $g$ surface, can be realized as the automorphism group of a Riemann surface of genus $g$. The proof relies heavily on an understanding of the geometry of Teichmüller space $T_g$, and ties in to many natural geometric questions about this space which I hope to describe.

##### Bend and break

Our goal is to explain the following theorem due to Mori: Given a compact complex manifold $X$ whose tangent bundle has lots of (holomorphic) determinantal sections (a Fano manifold), any pair of points on it lie in the image of a holomorphic map from the Riemann sphere $P^1$. Despite being a complex geometric statement, the only known proof of this result is by reduction to characteristic $p$. In this talk, we'll discuss what Fano manifolds are, explain the techniques that go into Mori's proof (reducing mod $p$, deformation theory of maps from curves), and the proof itself.

##### Models and Fields: A delicate Passage to Characteristic p

If a polynomial map from $C^n$ to itself is injective, then it is also surjective. The first proof of this elegant result actually used a passage to characteristic p. I'll introduce the relevant notions from model theory and explain the proof, which will naturally yield the following much stronger result: Every statement in the first order logic of rings is true in the algebraic closure of $F_p$ for almost all p iff it is true in C.

##### Lie Groups: Decomposition and Exponentiation

A manifold with a smooth group structure is called a Lie group. Most of the information about Lie groups is captured by the tangent space at the identity and its Poisson bracket. The map relating these two structures is the exponential map (which in the compact case is the same as the geodesic exponential map). In this talk I'll start from first principles, give a brief overview of some decompositions of Lie groups based on this algebraic/analytic interplay, and use these facts to prove some interesting theorems about the image of $\exp$, viz., 1) If $\exp$ is a homeomorphism, then the group is solvable; and 2) while $\exp$ need not be surjective, the connected component of the identity is the square of the image of $\exp$.

##### Morse theory

Morse theory gives the cell structure of a manifold in terms of the critical points of a 'random' real-valued function on this manifold. Besides being clever and pleasing to the eye, it has given us Bott periodicity, counts of geodesics, periodic orbits of dynamical systems, Heegard Floer homology, the foundations of Mirror Symmetry and many many more riches. I will explain the construction in detail, then sketch the applications.

##### Sum-product estimates via combinatorial geometry

Every two-dimensional drawing of any graph with $V$ vertices and $E\ge 4V$ edges necessarily has at least $E3/V2$ pairs of crossing edges. Also, for every set $A$ of real numbers, one of $A+A$ (the set of all pairwise sums of elements of $A$) or $A\cdot A$ (the set of all pairwise products) has size at least $|A|5/4$. What could these two theorems possibly have in common, besides the fact that Endre Szemerédi co-authored both? Surprisingly, quite a lot. We will see the proof of the first result, followed by a series of fascinating consequences which culminate in the second result. Of course, the Probablistic Method will make a crucial appearance.

##### Poisson summation formula

I will first talk about the classical Poisson summation formula and then about a vast generalization of it, namely the trace formula.

##### Mathai-Quillen's Thom form and Atiyah-Hirzebruch's Riemann-Roch theorem

After Hirzebruch's generalization of the classical Riemann-Roch formula, Grothendieck extended this result to a relative version. Then Atiyah and Hirzebruch gave a "differentiable analogue" of Grothendieck's theorem, which is called Atiyah-Hirzebruch's Riemann-Roch theorem. In spite of the original topological proof, I will present another one, in a more differential-geometric flavor, using Mathai-Quillen's construction of the Thom form.

##### Equivariant Cohomology

A cohomology theory is a way of associating a complex of abelian groups to a topological space. Study of such complexes gives geometric information about the structure of the space. Moreover if we have an action of a suitable group on a space we can associate another such complex to that space which carries more information. I will sketch main ideas of this construction and provide some examples.

##### Local entropy and projections of dynamically defined fractals

What's the smallest area a subset of the plane containing a unit line segment in every direction can have? Besicovitch showed that you can get 0 measure and Fefferman used the existence of this set to provide a solution to the ball multiplier problem. Kakeya sets have been important in understanding other phenomena like the Bochner-Riesz conjecture as the Hausdorff/Minkowski dimension of Kakeya sets relate to the two. Building on Fefferman's work, Bourgain improved our understanding introducing an ingenious bush argument . I will use the analytic story to motivate the corresponding problems for varieties over finite fields and recent work done there.