Department of

Mathematics


Seminar Calendar
for events the day of Thursday, December 7, 2017.

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Thursday, December 7, 2017

11:00 am in 241 Altgeld Hall,Thursday, December 7, 2017

Real and rational systems of forms

Simon Myerson (University College London)

Abstract: Consider a system $f$ consisting of $R$ forms of degree $d$ with integral coefficients. We seek to estimate the number of solutions to $f=0$ in integers of size $B$ or less. A classic result of Birch (1962) answers this question when the number of variables is of size at least $C(d) R^2$ for some constant $C(d)$, and the zero set $f = 0$ is smooth. We reduce the number of variables needed to $C'(d)R$, and give an extension to systems of Diophantine inequalities $|f| < 1$ with real coefficients. Our strategy reduces the problem to an upper bound for the number of solutions to a multilinear auxiliary inequality.

12:00 pm in 243 Altgeld Hall,Thursday, December 7, 2017

Least Dilatation of Pure Surface Braids

Marissa Loving (Illinois Math)

Abstract: The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. For the n=1 case, much is known about this group including upper and lower bounds on the least dilatation of its pseudo-Anosovs due to Dowdall and Aougab—Taylor. I am interested in the least dilatation of pseudo-Anosov pure surface braids for n>1 punctures. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n

12:30 pm in 222 Loomis,Thursday, December 7, 2017

Conformal Truncation: A New Method for Strongly-Coupled QFTs

Zuhair Khandker (University of Illinois at Urbana Champaign)

Abstract: I will present a new numerical method for studying strongly-coupled QFTs in any dimension. The method harnesses conformal symmetry, but in a manner applicable to general, non-conformal QFTs. The input is information about the UV CFT from which the QFT originates. The output is the physical QFT spectrum, along with real-time, infinite-volume correlation functions. So far, we have used the method to study 2D phi^4 theory. I will present new results for correlation functions at any coupling, including the Zamolodchikov c-function along the full RG-flow. I will also present a non-trivial cross-check of our numerical results: for a critical value of the coupling, phi^4 theory flows to the Ising model, and we match known analytical predictions.

2:00 pm in 241 Altgeld Hall,Thursday, December 7, 2017

A primer on the circle method for forms in many variables

Simon Leo Rydin Myerson (UCL)

Abstract: We give an introduction to the circle method in the form used by Birch (1962) to treat nonsingular systems of forms of the same degree. Given a suitably nice system f consisting of R forms of degree d with integral coefficients, this will give an asymptotic formula as B becomes large for the number solutions to f=0 in integers of size up to B.

3:00 pm in 243 Altgeld Hall,Thursday, December 7, 2017

A Counterexample to the Weitzenböck Conjecture in Characteristics p > 2 (Part 2)

Stephen Maguire (UIUC Math)

Abstract: Weitzenböck’s Theorem states that a representation $\mu: \mathbb{G}_a \to \mathrm{GL}(V_n)$ has a finitely generated ring of invariants $k[X]^{\mathbb{G}_a}$ if the field $k$ is an algebraically closed field of characteristic zero. In this talk, we produce a representation $\mu: \mathbb{G}_a \to \mathrm{GL}(V_6)$ over an algebraically closed field $k$ of characteristic $p > 2$ such that the ring of invariants $k[x_1,\dots,x_6]^{\mathbb{G}_a}$ is not a finitely generated $k$-algebra. In order to do this, we reduce this problem to a curve counting problem, and then use this reduction to further reduce this problem to a problem about the support of a bi-graded ring.

4:00 pm in 1 Illini Hall,Thursday, December 7, 2017

Sofic groups and sofic entropy

Anton Bernshteyn (UIUC Math)

4:00 pm in 245 Altgeld Hall,Thursday, December 7, 2017

An Introduction to Dynamic Materials

Suzanne Weekes (Worcester Polytechnic Institute)

Abstract: I will give an overview of work on wave propagation through dynamic materials (DM). DM are spatio-temporal composites - materials whose properties vary in space and in time. Mathematically, we formulate the problem as linear, hyperbolic partial differential equations with spatio-temporally varying coefficients. The variability in the material constituents leads to effects that are unachievable through static (spatial-only) design. For example, with dynamic laminates we are able to screen portions of the material from the effects of a wave disturbance. With checkerboard geometry in space-time, we create pulse compression and energy accumulation, and recent work shows that these effects are structurally stable.