## Manish Mishra : Self-dual cuspidal representations

- Number Theory ( 248 Views )Let F be a non-archimedean local field (such as ℚ_p). The Langlands philosophy says that the arithmetic of F is intimately related to the category R(G) of smooth complex representations of G(F) where G denotes a reductive F-group (for example the general linear group). The building blocks of R(G) are the "supercuspidal" representations of G(F). I will define this term in the talk. The category R(G) comes equipped with an involution - the "contragradient" or the "dual". The supercuspidal representations of G(F) which are self-dual are of considerable interest in the subject. In this talk, I will talk about a joint work with Jeff Adler about the existence of supercuspidals and self-dual supercuspidals. Specifically, we show that G(F) always admits supercuspidal representations. Under some mild hypotheses on G, we determine precisely when G(F) admits self-dual supercuspidal representations. These results are obtained from analogous results for finite reductive groups which I will also talk about.

## Baiying Liu : On Fourier coefficients and Arthur parameters for classical groups

- Number Theory ( 183 Views )Recently, Jiang made a conjecture investigating the connection between two fundamental invariants of an automorphic representation \pi appearing in the discrete spectrum of quasi-split classical groups G(A). The first invariant is the wave front of \pi, WF(\pi), which is the set of maximal unipotent orbits of G, such that \pi admits a non-trivial Fourier coefficients with respect to them. The second invariant is the Arthur parameter \psi of \pi to which one can associate a unipotent orbit \underline{p}(\psi) of the dual group of G. The conjecture says that in any Arthur packet associated to \psi, the Barbasch-Vogan duality of the orbit \underline{p}(\psi) is a sharp upper bound for the wave front of the representations of the packet. This is an important conjecture that vastly generalizes Shahidi's conjecture which claims that in every tempered packet there exists a generic representation. In this talk, I will review this conjecture and present some recent progress towards it. This is a joint work in progress with Dihua Jiang.

## Hunter Brooks : Special Value Formulas for Rankin-Selberg p-adic L-Functions

- Number Theory ( 174 Views )We discuss special value formulas for a p-adic L-function L_p(f, \chi), where f is a fixed newform and \chi varies over the space of Hecke characters of a fixed imaginary quadratic field, as well as some recent applications. These formulas, first found by Bertolini, Darmon, and Prasanna, relate L_p(f, \mathbb{1}), a value which is outside the range of interpolation defining L_p, to arithmetic invariants of cycles on varieties fibered over modular curves.

## Jacek Brodzki : A generalised Julg-Valette complex for CAT(0)-cube complexes.

- Number Theory ( 148 Views )This talk will introduce a very natural and interesting differential complex associated with a CAT(0)-cube complex. The construction builds on ideas first introduced by Pytlik and Szwarc for the free group and extended by Julg and Valette in the case of groups acting on trees. We will extend ideas of Julg-Valette to show how this construction can be used to study K-amenability and K-homology of groups acting on CAT(0)-cube complexes. This talk is based on joint work with Erik Guentner and Nigel Higson.

## Jacob Tsimerman : Recovering elliptic curves from their p-torsion

- Number Theory ( 143 Views )(joint w/ B.Bakker) For an elliptic curve E over a field k, the p-torsion E[p] gives a 2-dimensional representation of the Galois group G_k over F_p. For k=Q and p>13, the Frey-Mazur conjecture famously states that one can recover the isogeny class of E from the representaiton E[p]. We state and prove a direct analogue of this question over function fields of complex algebraic curves. Specifically, for any complex algebraic curve C, let k(C) be its field of rational functions. Then there exists a constant A(C), such that for all primes p>A(C), isogeny classes of elliptic curves E over k(C) can be recovered from E[p]. Moreover, we show that A(C) can be made to depend only on the gonality of C, which can be thought of as the analogous notion of degree for number fields. The study of this question will lead us into the realm of moduli spaces and hyperbolic geometry. The use of the latter means that, unfortunately, these methods don't apply in finite characteristic.

## Ken Ono : Special values of modular shifted convolution Dirichlet series

- Number Theory ( 138 Views )Rankin-Selberg convolution L-functions are important functions in number theory. Their properties play a central role in many of deepest works on the Ramanujan-Petersson Conjecture. In a recent paper, Hoffstein and Hulse defined generalizations of these L-functions, the so-called "shifted-convolution" L-functions. They obtained the meromorphic continuation of the functions in many cases. Here we consider symmetrizations of these L-functions, and we exactly evaluate their special values at diagonal weights for all shifts. This is joint work with Michael Mertens.

## Albert Chau : Limits of the Kahler Ricci flow and non-negatively curved Kahler manifolds

- Number Theory ( 137 Views )In this talk I will discuss joint results with L.F.Tam on the limiting behavior of the Kahler Ricci flow and its application to the structure of non-negatively curved complete non-compact Kahler manifolds.

## Sug Woo Shin : Asymptotic behavior of supercuspidal characters

- Number Theory ( 135 Views )This is joint work with Julee Kim and Nicolas Templier. Irreducible smooth representations of a p-adic reductive group are said to be supercuspidal if they do not appear in any induced representation from a proper parabolic subgroup. While it is still an open problem to obtain a precise character formula for them (apart from some special cases), I will explain that we can prove a reasonable upper bound and a limit formula as the formal degree tends to infinity, for a large class of supercuspidal representations. An expected application is an equidistribution result as well as a low-lying zero statistics for L-functions in a new kind of families of automorphic representations.

## Caroline Turnage-Butterbaugh : The Distribution of the Primes and Moments of Products of Automorphic $L$-functions

- Number Theory ( 129 Views )The prime numbers are the multiplicative building blocks of the integers, and much thought has been given towards understanding their behavior. In this talk, we will examine prime numbers from two points of view. We will first consider questions on the distribution of the primes. In particular, we will illustrate how the breakthrough work of Maynard and Tao on bounded gaps between primes settles an old problem of Erdos and Turan. Secondly, we will explore the relationship between prime numbers and zeros of the Riemann zeta-function, as a way to motivate the study of the moments of the Riemann zeta function and more general L-functions. In particular, we consider arbitrary products of L-functions attached to irreducible cuspidal automorphic representations of GL(m) over the rationals. The Langlands program suggests essentially all L-functions are of this form. Assuming some standard conjectures, I will discuss how to estimate two types of moments: the continuous moment of an arbitrary product of primitive automorphic L-functions and the discrete moment (taken over fundamental discriminants) of an arbitrary product of primitive automorphic L-functions twisted by quadratic Dirichlet characters.

## Sol Friedberg : Higher theta functions

- Number Theory ( 129 Views )Higher theta functions are the residues of Eisenstein series on covers of the adelic points of classical groups. On the one hand, they generalize the Jacobi theta function. On the other, their Whittaker-Fourier coefficients are not understood, even for covers of $GL_2$. In this talk I explain how, using methods of descent, one may establish a series of relations between the coefficients of theta functions on different groups. In the first instance, this allows us to prove a version of Patterson's famous conjecture relating the Fourier coefficient of the biquadratic theta function to quartic Gauss sums. This is based on joint work with David Ginzburg.

## Hang Xue : Fourier--Jacobi periods on unitary groups

- Number Theory ( 122 Views )We will formulate a conjectural identity relating the Fourier--Jacobi periods on unitary groups and the central value of certain Rankin--Selberg $L$-functions. This refines the famous Gan--Gross--Prasad conjecture. We will give some examples supporting this conjecture.

## Jeff Streets : A parabolic flow of Hermitian metrics

- Number Theory ( 120 Views )I will introduce a parabolic flow of Hermitian metrics which is a generalization of Kahler-Ricci flow. This flow preserves the pluriclosed condition, and its existence and convergence properties are closely related to the underlying topology of the given complex manifold. I will classify static solutions to the flow on various classes of complex surfaces, and show that no static solutions exist on Class VII surfaces, an important first step in using this flow to classify these surfaces. Joint with G. Tian.

## Matthew Baker : Riemann-Roch for Graphs and Applications

- Number Theory ( 120 Views )We will begin by formulating the Riemann-Roch theorem for graphs due to the speaker and Norine. We will then describe some refinements and applications. Refinements include a Riemann-Roch theorem for tropical curves, proved by Gathmann-Kerber and Mikhalkin-Zharkov, and a Riemann-Roch theorem for metrized complexes of curves, proved by Amini and the speaker. Applications (mainly due to other researchers) include new proofs of some important results in Brill-Noether theory, a generalization of the Eisenbud-Harris theory of limit linear series, and new bounds for the number of rational points on algebraic curves over number fields.

## M. Haluk Sengun : Torsion Homology of Hyperbolic 3-Manifolds

- Number Theory ( 115 Views )Hyperbolic 3-manifolds have been studied intensely by topologists since the mid-1970's. When the fundamental group arises from a certain number theoretic construction (in this case, the manifold is called "arithmetic"), the manifold acquires extra features that lead to important connections with number theory. Accordingly, arithmetic hyperbolic 3-manifolds have been studied by number theorists (perhaps not as intensely as the topologists) with different motivations. Very recently, number theorists have started to study the torsion in the homology of arithmetic hyperbolic 3-manifolds. The aim of the first half of this introductory talk, where we will touch upon notions like "arithmeticity", "Hecke operators", will be to illustrate the importance of torsion from the perspective of number theory. In the second half, I will present new joint work with N.Bergeron and A.Venkatesh which relates the topological complexity of homology cycles to the asymptotic growth of torsion in the homology. I will especially focus on the interesting use of the celebrated "Cheeger-Mueller Theorem" from global analysis.

## Florent Krzakala : From spin glasses to Packing, Freezing and Computing problems

- Number Theory ( 112 Views )Over the last decades, the study of "spin glasses" in physics has stimulated a large amount of theoretical activity in physics, and led to several breakthroughs. While the original puzzle of spin glass materials is still not fully solved, their theoretical analysis has created powerful techniques as well as a rich conceptual framework, to study emergent properties of strongly disordered and interacting systems. In this talk, I will use these tools and discuss how apparently unrelated complex problems such as: how to pack many objects in a given volume, how to color a graph with a given number of colors, why a liquid is turning into a glass when the temperature is lowered, and why some computational (classical and quantum) problems are hard while other are easy actually (and surprisingly) do share many characteristics when looking at them through the (spin) glass.

## Chris Hall : Hilbert irreducibility for abelian varieties

- Number Theory ( 112 Views )If $K$ is the rational function field $K=\mathbb{Q}(t)$, then a polynomial $f$ in $K[x]$ can be regarded as a one-parameter family of polynomials over $\mathbb{Q}$. If $f$ is irreducible, then a basic form of Hilbert's irreducibility theorem states that there are infinitely many $t$ in $\mathbb{Q}$ for which the specialized polynomial $f_t$ is irreducible over $\mathbb{Q}$. In this talk we will discuss analogous theorems for an abelian variety $A/K$ regarded as a one-parameter family of abelian varieties over $K$. For example, we will exhibit $A$ which are simple over $K$ and for which there are only finitely many $t$ in $\mathbb{Q}$ such that the abelian variety $A_t$ is not simple over $\mathbb{Q}$.

## John Voight : Presentations for rings of modular forms

- Number Theory ( 111 Views )We give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, depending on the signature of the group. Our work can be seen as a generalization of the classical theorem of Petri: we give a presentation for the canonical ring of a stacky curve. This is joint work with David Zureick-Brown.