Seminars archive

We start by exploring the theme of R-equivalence in algebraic groups. First introduced by Manin to study cubic surfaces, this notion proves to be a fundamental tool in the study of rationality of algebraic group varieties. A k-variety is said to be rational if its function field is purely transcendental over k. We exploit Merkurjev's fundamental computations of R-equivalence classes of adjoint classical groups and give a recursive construction to produce an infinite family of non-rational adjoint groups coming from quadratic forms. In a different direction, we address Serre's injectivity question which asks whether a principal homogeneous space under a linear algebraic group admitting a zero cycle of degree one in fact has a rational point. We give a positive answer to this question for any smooth connected reductive k-group whose Dynkin diagram contains connected components only of type A_n, B_n or C_n by relating Serre's question to the norm principles previously proved by Barquero and Merkurjev. The study of norm principles are interesting in their own right and we examine in detail the case of groups of type (non-trialitarian) D_n and get a scalar obstruction defined up to spinor norms whose vanishing will imply the norm principle for them. This in turn will also yield a positive answer to Serre's question for all connected reductive k-groups of classical type.

The relationship between elliptic curves and modular forms informs many modern mathematical discussions, including the solution of Fermat's Last Theorem and the Birch and Swinnerton-Dyer Conjecture. In this thesis we explore properties of elliptic curves, a particular family of modular forms called eta-quotients and the relationships between them. We begin by discussing elliptic curves, specifically considering the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic curves of everywhere good reduction and rational j-invariant. Using this, we determine the density of such real and imaginary fields. In the next chapter, we begin investigating the properties of eta-quotients and use this theory to prove a conjecture of Han related to the vanishing of coefficients of certain combinatorial functions. We prove the original conjecture that relates the vanishing of the hook lengths of partitions and the number of 3-core partitions to the coefficients of a third series by proving a general theorem about this phenomenon. Lastly, we will see how these eta-quotients relate to the Weierstrass mock modular forms associated with certain elliptic curves. Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass zeta-functions associated to modular elliptic curves encode the vanishing and non-vanishing for central values and derivatives of twisted Hasse-Weil L-functions for elliptic curves. We construct a canonical harmonic Maass form for the five curves proven by Martin and Ono to have weight 2 newforms with complex multiplication that are eta-quotients. The holomorphic part of this harmonic Maass form is referred to as the Weierstrass mock modular form. We prove that the derivative of the Weierstrass mock modular form for these five curves is itself an eta-quotient or a twist of one.

A typical problem in extremal graph theory is determining the maximal number of edges in a graph that does not contain a forbidden subgraph F. For example, such questions are partially answered by the Mantel, Turan and Erdos-Stone theorems. One way to generalize those theorems would be to determine how many copies of specific subgraphs an F-free graph can have. In my talk I will discuss our recent paper with S.Tsaturian and D.Gunderson about the number of cycles in a triangle-free graphs, possible generalizations and open questions related to this problem.

The nature of rational solutions to polynomial equations is one which is fundamental to Number Theory and more generally, to Mathematics. Given the straightforward nature of this problem, one may be surprised by the difficulty when it comes to producing solutions. The Hasse principle states that if an equation has local solutions everywhere then there is a global solution. Polynomials rarely satisfy this property. However Colliot-Thelene conjectures that another test on local solutions, the Brauer-Manin obstruction, exists for every rationally connected, smooth, projective, geometrically integral variety failing to satisfy the Hasse Principle. We wish to explore the existence of a Brauer-Manin obstruction to the Hasse principle for certain families of surfaces. The first of which is a cubic surface written down by Birch and Swinnerton-Dyer in 1975, Norm_{L/K}(ax+by+\phi z+\psi w) = (cx+dy)Norm_{K/k}(x+\theta y). The left-hand side of this equality is a cubic norm and the right-hand side contains a quadratic norm. They make a correspondence between this failure and the Brauer-Manin obstruction, recently discovered by Manin, in a few specific instances. Using techniques developed in the ensuing 40 years, we show that a much wider class of norm form cubic surfaces have a Brauer-Manin obstruction to the Hasse principle, thus verifying the Colliot-Thelene conjecture for infinitely many cubic surfaces. The second family is a general set of diagonal K3 surfaces, w^2=ax^6+by^6+cz^6+dw^6, defined as varieties in weighted projective space. This section focuses on the particular geometry of these surfaces, verifying that their Picard rank is generically 19. We conclude by computing the Galois cohomology group, H^1(Gal(\bar{k}/k),Pic\bar{X})\simeq (\mathbf{Z}/2\mathbf{Z})^3. The computation of this group is fundamental to determining the existence of a Brauer-Manin obstruction.

Over the last decade, differential privacy (DP) has emerged as the standard privacy notion for research in privacy-preserving data analysis and publishing. However, there is an ongoing debate about the meaning and value of DP. Some hail that the notion of DP offers strong privacy protection regardless of the adversary's prior knowledge while enabling all kinds of data analysis. Others offer criticisms regarding DP's privacy guarantee and utility limitations. In this talk, we focus on two issues. One is what does DP mean? More precisely, under what condition(s), the notion of DP delivers the promised privacy guarantee? We show that DP is based on the following Personal Data Principle: "Data privacy means giving an individual control over his or her personal data. Privacy does not mean that no information about the individual is learned, or no harm is done to an individual. Enforcing the latter is infeasible and unreasonable.'' Furthermore, the question of when DP is adequate is not just a technical question and depends on legal and ethical considerations. In the second part of the talk, we give a survey of the state of the art in publishing a summary of a relational dataset, ranging from publishing histograms for one-dimensional and two-dimensional datasets, to answering marginal queries for datasets with dozens of dimensions, and finally to finding frequent itemsets in transactional datasets with thousands or more of dimensions. Brief Bio: Ninghui Li is a Professor of Computer Science at Purdue University, where he has been a faculty member since 2003. His research interests are in security and privacy. He has published over 130 referred papers in these areas. Prof. Li is current on the editorial boards of Journal of Computer Security (JCS) and ACM Transactions on Internet Technology (TOIT). He was on the editorial board of IEEE Transactions on Dependable and Secure Computing (TDSC) from 2011 to 2015 and the VLDB Journal from 2007 to 2013. He recently served as Program Chair of 2014 and 2015 ACM Conference on Computer and Communications Security (CCS), ACM's flagship conference in the field of security and privacy.

This dissertation provides three results: (1) A Hasse principle for rational points of projective homogeneous spaces under unitary or special unitary groups associated to hermitian or skew hermitian spaces over function fields of p-adic curves; (2) A Springer-type theorem for isotropy of hermitian spaces over odd degree field extensions of function fields of p-adic curves; (3) Exact values of Hermitian u-invariants of quaternion or biquaternion algebras over function fields of p-adic curves.

In this talk I will give a tour of recent results and open problems about vector bundles of conformal blocks on the moduli space of curves. I will discuss how these results fit into the context of some of the open problems about the binational geometry of the moduli space.

Let E/Q be an elliptic curve and let Q(3^infty) denote the compositum of all cubic extensions of Q. While the group E(3^infty) is not finitely generated, one can show that its torsion subgroup is finite; this holds more generally for any Galois extension of Q that contains only finitely many roots of unity. I will describe joint work with Daniels, Lozano-Robledo, and Najman, in which we obtain a complete classification of the 20 torsion subgroups that can and do occur, along with an explicit description of the elliptic curves E/Q that realize each possibility (up to twists). This is achieved by determining the rational points on a corresponding set of modular curves and relies on several recent results related to the mod-n Galois representations attached to elliptic curves over Q.