Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, December 1, 2015.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, December 1, 2015

11:00 am in 243 Altgeld Hall,Tuesday, December 1, 2015

Uniqueness of smooth structures on spheres.

Zhouli Xu (U Chicago)

Abstract: In this talk, I will report recent progress that the 61-sphere has a unique smooth structure. Following results of Moise, Kervaire-Milnor, Browder and Hill-Hopkins-Ravenel, we show that the only odd dimensional spheres with a unique smooth structure are in dimension 1, 3, 5 and 61. Following recent work of Isaksen, we also show that in dimensions from 5 through 61, the only spheres with a unique smooth structure are in dimension 5, 6, 12, 56 and 61. Recent work of Behrens-Hill-Hopkins-Mahowald shows that the next sphere with a unique smooth structure, if exists, is in dimension at least 126. The computation of the stable homotopy groups of spheres at the prime 2 is essential to this result. I will review classical techniques and explain our new technique. This work is joint with Guozhen Wang.

12:00 pm in 345 Altgeld Hall,Tuesday, December 1, 2015

The abelianization of automorphism groups of right-angled Artin groups

Javier Aramayona (Madrid)

Abstract: Automorphism groups of right-angled Artin groups form an interesting class of groups, as they ``interpolate" between the two extremal cases of Aut(Fn) and GL(n,Z). In this talk we will discuss some conditions on a simplicial graph which imply that the automorphism group of the associated right-angled Artin group has (in)finite abelianization. As a direct consequence, we obtain families of such automorphism groups that do not have Kazhdan's property (T). This is joint work with Conchita Martinez-Perez.

12:30 pm in 322 Loomis Laboratory,Tuesday, December 1, 2015

Large Field Inflation and Gravitational Entropy

Albion Lawrence (Brandeis Physics)

Abstract: Inflationary models which produce potentially observable primordial gravitational waves via quantum fluctuations are highly sensitive to quantum gravity.  One recent attempt to bound a class of such models is by appealing to the covariant entropy bound and arguing that large numbers of species and/or a large axion decay constant leads to a violation of this bound during inflation. I will discuss the proper implementation of this bound in terms of physical, renormalized couplings, in which case the arguments based on this bound lose their force. 

1:00 pm in 347 Altgeld Hall,Tuesday, December 1, 2015

Necklace solitary waves on bounded domains

Gadi Fibich   [email] (Tel Aviv University (University of Maryland))

Abstract: The critical power for collapse appears to place an upper bound on the amount of power that can be propagated by intense laser beams. In various applications, however, it is desirable exceed this limit and deliver more power. In this talk I will present new solitary waves of the two-dimensional nonlinear Schrodinger equation on bounded domains, which have a ``necklace'' structure. I will consider their structure, stability, and how to compute them. In particular, I will show that these solitary waves can stably propagate more than the critical power for collapse.

1:00 pm in 345 Altgeld Hall,Tuesday, December 1, 2015

Minimal idempotent ultrafilters and polynomial Ramsey Theory

Joel Moreira   [email] (Ohio State Univ)

Abstract: Ultrafilters with certain algebraic and topological properties - so-called minimal idempotent ultrafilters - have found remarkable applications in Ramsey theory, allowing for surprisingly short and clean proofs of some classical results such as van der Waerden's theorem on arithmetic progressions, and Rado's classification of partition regular systems of linear equations. By combining the power of ultrafilter methods with the polynomial extension of van der Waerden's theorem of Bergelson and Leibman, we obtain a far-reaching polynomial generalization of Rado's theorem, unifying several results in arithmetic Ramsey theory.

1:00 pm in 241 Altgeld Hall,Tuesday, December 1, 2015

Mathematicians Collaboration Problem

Amin Bahmanian   [email] (Illinois State University)

Abstract: In a mathematics workshop with $mn$ mathematicians in $n$ different areas, each area consisting of $m$ mathematicians, we want to create a collaboration network. For this purpose, we would like to schedule daily meetings between groups of size three, so that (i) two people of the same area meet one person of another area, (ii) each person has exactly $r$ meetings each day, and (iii) each pair of people of the same area have exactly $\lambda$ meetings with each person of another area by the end of the workshop. We prove a more general theorem. In particular we show that meetings can be scheduled if: $3\mid rm$, $2\mid rnm$ and $r\mid 3\lambda(n-1)\binom{m}{2}$.

2:00 pm in 347 Altgeld Hall,Tuesday, December 1, 2015

Random Perturbations of Periodically Driven Nonlinear Oscillators

Navaratnam Sri Namachchivaya   [email] (UIUC, Department of Aerospace Engineering)

Abstract: This talk will develop a unified approach to study the dynamics of single degree of freedom systems excited by both periodic and random perturbations. The near resonant dynamics of such systems, in the presence of weak noise, is not well understood. We will study this problem in depth with the aim of discovering a common geometric structure in the phase space, and determining the effects of noisy perturbations on the passage of trajectories through the resonance zones. This is a joint work with Nishanth Lingala and Ilya Pavlyukevich.

3:00 pm in 243 Altgeld Hall,Tuesday, December 1, 2015

Counting spaces of polynomials with specified root multiplicities over finite fields

Boris Xu (UIUC Math)

Abstract: In his recent colloquium talk, Benson Farb discussed the space $Poly_n$ of monic square-free polynomials of a fixed degree $n$, which has $q^n-q^{n-1}$ points over a finite field of order $q$. This count may be obtained through a calculation in the Grothendieck ring of varieties and is related, via the Grothendieck trace formula, to cohomology of the configuration space $Poly_n$ over $\mathbb{C}$. I will explain how to generalize these methods to counting $Poly_n(\lambda)$, the space of polynomials of degree $n$ whose root multiplicities partition $n$ by a fixed partition $\lambda$.

4:00 pm in 245 Altgeld Hall,Tuesday, December 1, 2015

National lab careers for mathematics graduates

Evan VanderZee, Meghan Galiardi, Sean Shahkarami, Zoi Rapti   [email] (Argonne National Lab, UIUC Math, UIUC Math, UIUC Math)

Abstract: Informal discussion about careers in the national lab system for mathematics graduates. Dr. VanderZee received his PhD from the U of Illinois in 2010, and we are pleased to welcome him back for this event. Meghan Galiardi and Sean Shahkarami are current PhD students who have done internships at Sandia and Argonne, respectively. Dr. Zoi Rapti is a professor in the department who worked for a while at Los Alamos National Lab earlier in her career.

4:00 pm in 149 Henry building,Tuesday, December 1, 2015

Taking the long way home: Orbits of plane partitions

Oliver Pechenik (UIUC)

Abstract: Plane partitions are piles of cubes stacked in the corner of a room. P. Cameron and D. Fon-der-Flaass (1995) studied a simple action on such piles, whose dynamics are nonetheless quite mysterious. In particular, repeating this action will always eventually return the original pile, but sometimes the voyage is much longer than expected. To understand the Grothendieck rings of algebraic vector bundles over Grassmannians and other spaces, H. Thomas and A. Yong (2009) introduced a suite of combinatorial algorithms on certain grids of numbers. In particular, there is a beautiful K-theoretic promotion operator, which again has some mysteriously large orbits. We'll see how these two mysteries are in fact the same mystery, and use this relation to explain special cases of both actions. (Based on joint work with Kevin Dilks and Jessica Striker)