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for events the day of Thursday, April 21, 2016.

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Thursday, April 21, 2016

11:00 am in 241 Altgeld Hall,Thursday, April 21, 2016

Large gaps between zeros of Dedekind zeta-functions of quadratic number fields

Caroline Turnage-Butterbaugh (Duke University)

Abstract: Let $K$ be a quadratic number field with discriminant $d$. The Dedekind zeta-funciton attached to $K$ can be expressed by $\zeta_{K}(s)=\zeta(s)L(s,\chi_d)$ for $s\ne1$, where $\zeta(s)$ is the Riemann zeta-function, the character $\chi_d$ is the Kronecker symbol associated to $d$, and $L(s,\chi_d)$ is the corresponding Dirichlet $L$-function. Combining Hallís Method with the twisted second moment of $\zeta_K(1/2+it)$, we show that there are infinitely many gaps between consecutive zeros of $\zeta_K(1/2+it)$ which are greater than $2.866$ times the average spacing. This is joint work with Hung Bui and Winston Heap.

12:00 pm in Altgeld Hall 243,Thursday, April 21, 2016

Finitely generated groups with co-c.e. word problem (d'apres Morozov)

Paul Schupp (UIUC Math)

Abstract: Let $\mathcal{C}$ be the group of all computable permutations of the natural numbers. The general question is: What can one say about finitely generated subgroups of $\mathcal{C}$? While most groups studied in geometric group theory have computably enumerable word problems, one sees immediately that a finitely generated subgroup of $\mathcal{C}$ must have co-c.e. word problem, that is, the set of words equal to the identity in $G$ is the complement of a computably enumerable set. Andrey Morozov proved two important theorems about finitely generated subgroups of $\mathcal{C}$. We will discuss these theorems and interesting connections of the basic question to other groups.

12:30 pm in 464 Loomis Laboratory,Thursday, April 21, 2016

3d Gauge Theories and Their Gravity Duals

William Cottrell (Wisconsin Physics)

Abstract: In this talk we describe the general construction of supergravity duals to 2+1d †SYM gauge theories with 8 supercharges in M-Theory.† We will encounter an apparent mismatch between our sugra and field theory expectations and discuss also the role of curvature and gs corrections in reconciling these two perspectives.† Finally, the field theoretic interpretation of brane charge will be explained. † This will mainly go over† plus possibly some newer material.

2:00 pm in 243 Altgeld Hall,Thursday, April 21, 2016

Boundary Harnack principle for domains in fractal spaces

Janna Lierl (UIUC)

Abstract: A domain in ${\mathbb R}^n$ satisfies the boundary Harnack principle if any two non-negative harmonic functions that vanish continuously on a subset of the boundary have the same boundary decay rate. This talk is about the boundary Harnack principle for domains in fractal-type metric measure spaces. In this context, harmonic functions are defined as weak solutions of the Laplace equation for the generator of a local regular Dirichlet form. Under natural geometric assumptions, the boundary Harnack principle holds on domains that are inner uniform. This result is useful in the study of the Dirichlet heat kernel. I will present sharp two-sided estimates for the Dirichlet heat kernel on bounded inner uniform domains.

2:00 pm in Altgeld Hall 241,Thursday, April 21, 2016

Consecutive primes in tuples

Caroline Turnage-Butterbaugh (Duke University)

Abstract: Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$, then for an admissible $k$-tuple $\mathcal{H}(x)=\{gx+h_j\}_{j=1}^k$ of linear forms in $\mathbb{Z}[x]$, the set $\mathcal{H}(n)=\{gn+h_j\}_{j=1}^k$ contains at least $m$ primes for infinitely many $n \in \mathbb{N}$. Using this result, we will first show that $\mathcal{H}(n)$ contains at least $m$ consecutive primes for infinitely many $n \in \mathbb{N}$. As an application, we will consider questions concerning properties of strings of consecutive primes. In particular, we settle an old conjecture of Erd\H{o}s and Tur\í{a}n by producing strings of consecutive primes whose successive gaps form an increasing sequence. This is joint work with William Banks and Tristan Freiberg.

3:00 pm in 243 Altgeld Hall,Thursday, April 21, 2016

Implicitization of tensor product surfaces in the presence of basepoints

Eliana Duarte (UIUC Math)

Abstract: A tensor product surface is the image of a map $\lambda:\mathbb{P}^1\times \mathbb{P}^1\to \mathbb{P}^3$. Such surfaces arise in geometric modeling, and in this context, it is important to know their implicit equation. The goal for this talk is to explain how the implicitization problem for tensor product surfaces can be solved using syzygies and how the geometry of the base locus of $\lambda$ determines the syzygies that are used to compute the implicit equation.

4:00 pm in 245 Altgeld Hall,Thursday, April 21, 2016

Bernoulli numbers, Duflo isomorphism and the Kashiwara-Vergne conjecture

Anton Alekseev (University of Geneva)

Abstract: Bernoulli numbers were introduced by Jakob Bernoulli in the beginning of the 18th century to give formulas for sums of powers of integers. They proved useful in many fields of Mathematics including Number Theory, Analysis and Topology. One of their surprizing applications is in Lie theory. Two major results, the Kirillov character formula and the Duflo isomorphism theorem make use of Bernoulli numbers. To explain this unexpected link, we turn to the Kashiwara-Vergne conjecture on properties of the Campbell-Hausdorff series. This is a more complicated statement which implies the Duflo isomorphism theorem. The proof of the conjecture makes use of the generating series of multiple zeta values also known as the Drinfeld associator, and Bernoulli numbers are among the simplest coefficients of the associator. The main property of the Drinfeld associator (the pentagon equation) and the proof of the Kashiwara-Vergne conjecture are inspired by constructions from Quantum Field Theory.