# Seminars & Events for Department Colloquium

##### A Lefschetz principle in non-archimedean geometry

PLEASE CLICK ON COLLOQUIUM TITLE FOR COMPLETE ABSTRACT: The "Lefschetz Principle" is the informal idea that for geometric questions about algebraic varieties over fields of characteristic 0, it is often sufficient to assume the ground field is the complex numbers (where analytic tools are available).

##### The 15-theorem & the 290-theorem

The citation for Manjul Bhargava's recent Fields Medal mentions his improvement of my proof (partly with Will Schneeberger) of the 15- theorem, and his proof (with John Hanke) of the 290-theorem. I shall talk about the history of universal quadratic forms, which was started in 1770 by Lagrange's four squares theorem, and culminated about 20 years ago in these two theorems. I'll also talk about some related problems, some of which are still open.

##### Riemann Sums and Mobius

[$S =$] square-free natural numbers. An Hilbert-Schmidt operator, [$\mathcal{A}$] , associated to the Möbius function has the property that [$\mathcal{A}: \bigcup_{0<r<\infty} l^r(S) \to \bigcap_{0<r<\infty} l^r(S),$] injectively. If [$0<r<2$] and [$\xi \in l^r(S)$] , the series [$f_\xi = \sum_{n\in S} \mathcal{A} \xi(n) \cos 2\pi n x$] converges uniformly to [$f_\xi \in \mathcal{R}_0$] , i.e., a periodic, even, continuous function with equally spaced Riemann sums, [$\sum_{j=0}^{N-1} f_\xi\left(\frac{j}{n}\right) = 0$] , [$N = 1,2,\dots$] . If [$\mathcal{A} \xi_\lambda = \lambda \xi_\lambda$] , [$\xi_\lambda(1) = 1$] , then [$\xi_\lambda$] is multiplicative.

##### Branching laws and period integrals for non-tempered representations

Question about decomposition of representations of groups over local fields to subgroups, such as from SO(n) to SO(n-1), has been of considerable interest in the recent past with impressive works of Waldspurger, Moeglin-Waldspurger, and Raphael Beuzart-Plessis. Corresponding global question for unitary groups has been partly settled by Wei Zhang. These questions have so far been considered only for tempered representations (of the group and the subgroup). In this lecture, I will discuss the situation with non-tempered representations with many examples.

##### Computing Fukaya categories

The Fukaya category has become a central tool in symplectic topology. In this talk, I will begin with some motivating examples, and explain methods for making computations of the Fukaya categories sufficiently explicit in order to extract information about Lagrangian submanifolds. Mirror symmetry lurks in the background of most of these computations, and the rough idea is that the complexity of the answer is related to the extent to which the mirror deviates from being an honest (commutative) space.

##### Automorphic L-functions and descent method

Automorphic L-functions are fundamental invariants attached to a given cuspidal automorphic representation of a reductive group G. In addition to arithmetic applications, automorphic L-functions are used to characterize various types of Langlands functorial transfers. In this lecture, I will discuss some basic cases of the theory, including the automorphic descent endoscopy correspondence via automorphic integral transforms, and report my recent work on the twisted version of automorphic descent.

##### Knot invariants via the cotangent bundle

In recent years, symplectic geometry has emerged as a key tool in the study of low-dimensional topology. One approach, championed by Arnol'd, is to study the topology of a smooth manifold through the symplectic geometry of its cotangent bundle, building on the familiar concept of phase space from classical mechanics. I'll describe how one can use this approach to construct an invariant of knots called knot contact homology. I'll then try to give some sense of what this invariant measures, including recently-discovered relations to representations of the knot group, and to mirror symmetry and topological string theory.

##### Stabilization of control systems: From the water clocks to the regulation of rivers

A control system is a dynamical system on which one can act by using controls. For these systems a fundamental problem is the stabilization issue: Is it possible to stabilize a given unstable equilibrium by using suitable feedback laws? (Think to the classical experiment of an upturned broomstick on the tip of one's finger.) On this problem, we present some pioneer devices and works (Ktesibios, Watt, Maxwell, Lyapunov...), some more recent results, and an application to the regulation of the rivers La Sambre and La Meuse in Belgium. A special emphasize is put on positive or negative effects of the nonlinearities.

##### Torsion homology

Arithmetic groups --- that generalize the modular group --- can have'a lot' of torsion in their homology. Starting from the examples of congruence subgroups of the modular and Bianchi groups where homology reduces to abelianization, I will explain what 'a lot' means. I will then try to explain how this interacts with more classical question of number theory and geometry. This is joint work with Akshay Venkatesh and Mehmet Haluk Sengun.

##### Quantitative transversality in symplectic geometry

I will survey some applications of Donaldson's technique of quantitative transversality of "approximately holomorphic" functions in symplectic geometry. I will explain the basic terms and present the main ideas of the technique. Donaldson used it to show that the Poincare dual of any sufficiently large multiple of an integral symplectic form is represented by a symplectic submanifold. Another application is joint work with E. Giroux in which we prove the existence of Lefschetz fibrations on certain symplectic manifolds (I will discuss this particular result in more detail in my talk in the symplectic geometry seminar).

##### Some unitary representations of Thompson’s groups F and T

In a ``naive'' attempt to create algebraic quantum field theories on the circle, we obtain a family of unitary representations of Thompson's groups T and F for any subfactor. The Thompson group elements are the ``local scale transformations'' of the theory. In a simple case the coefficients of the representations are polynomial invariants of links. We show that all links arise and introduce new ``oriented'' subgroups $\overrightarrow F <F$ and $\overrightarrow T< T$ which allow us to produce all *oriented* knots and links.

##### Generalization of Selberg’s 3/16 theorem for finitely generated subgroups of SL(2,Z)

A celebrated theorem of Selberg in 1965 states that for congruence subgroups of SL(2,Z) there are no exceptional eigenvalues below 3/16. We will discuss how Selberg’s theorem can be generalized to finitely generated subgroups of SL(2,Z) which are of infinite index. This talk is based on joint work with Dale Winter

##### The Monge ampere and related Hessian equations: Some non local versions.

*****Please note this week's colloquium will be on Tuesday in room 214. **An important family of equations in geometry and other applications are those involving symmetric functions of the eigenvalues of the Hessian. We will describe some possible non local versions, and their properties: existence regularity, etc

##### Heuristics for boundedness of ranks of elliptic curves

The set of rational points on an elliptic curve E over Q has the structure of an abelian group, and in 1922 Mordell proved that this group is finitely generated. We present heuristics that suggest that there is a uniform upper bound on its rank as E varies over all elliptic curves over Q. This is joint work with Jennifer Park, John Voight, and Melanie Matchett Wood.

##### On the Satisfiability Conjecture

The random K-SAT model gives a model of random Boolean formulas and is perhaps the canonical random constraint satisfaction problem. The Satisfiability Conjecture posits that the probability of a satisfying assignment undergoes a sharp transition at a critical density of constraints. Heuristics developed in statistical physics predict the location of the transition as well as much more. I will survey what is known and predicted and describe recent progress establishing the conjecture for large enough K. This is joint work with Jian Ding and Allan Sly.

##### Complex multiplication, from Abel to Zagier

According to Hilbert, the theory of complex multiplication, which brings together number theory and analysis, is not only the most beautiful part of mathematics but also of all science. "Complex multiplication" refers to a lattice in the complex numbers (or an elliptic curve) which admits endomorphisms by a ring larger than the integers. We will begin with Kronecker's "Jugendtraum" -- the use of complex multiplication to solve Hilbert's twelfth problem. This will lead us into a discussion of Heegner points. We will conclude with some recent work on the modular curve "at infinite level", and the unexpected role that complex multiplication plays in its geometry.

##### Universally defined cycles

The Franchetta conjecture (now a theorem) says basically that any line bundle on the universal family of curves of genus at least 2 restricts to a multiple of the canonical bundle on each fiber. We formulate a generalization of this statement for surfaces and provide a characterization of the Chern classes: polynomials in the Chern classes are the only "universally defined'' cycles.

##### From molecular dynamics to kinetic theory and hydrodynamics

Please note that this colloquium talk is on Tuesday, not Wednesday.

##### Celebrating the work of Prof. Edward Nelson (1932-2014)

In place of the regular departmental colloquium and pre-colloquium presentations, we shall have three talks focused on different aspects of Prof. Edward Nelson's work:

12:15 - 1:15 (Fine Hall Common Room, 3rd fl.)**"Ed Nelson's Work in Analysis, Especially Related to Quantum Field Theory" **

Barry Simon, California Institute of Technology

4:30 - 5:30 pm (Fine Hall 314) **"Questioning the Infinite"**

Greg Lawler, University of Chicago**"Ed Nelson's stochatic mechanics" **Eric Carlen, Rutgers University

Presentations moderated by professor emeritus Simon Kochen.

##### Zeta(3) in arithmetic and geometry

Euler proved in 1735 that zeta(2) = $\pi^2 /6$, and also computed the special values of zeta(n) at all positive even integers. Yet it took almost another 250 years for Apery proved that zeta(3) was irrational. In this talk, we shall talk about zeta(3) as well as its p-adic version, and the connection of these numbers to both arithmetic and geometry.