Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, March 17, 2015.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    February 2015            March 2015             April 2015
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1  2  3  4  5  6  7    1  2  3  4  5  6  7             1  2  3  4
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Tuesday, March 17, 2015

11:00 am in 243 Altgeld Hall,Tuesday, March 17, 2015

#### The Taylor tower of Waldhausen's algebraic K-theory of spaces

###### Michael Ching   [email] (Amherst)

Abstract: Analytic functors from based spaces to spectra can be classified via modules over the Koszul duals of the E_n-operads. The goal of this talk is to outline this classification and then to consider how Waldhausen's algebraic K-theory functor A(X) fits into it. In particular we will show that the derivatives of A(X) have the structure of a module over the Koszul dual of E_3. This is joint work with Greg Arone. If there is time, I will describe work in progress with Andrew Blumberg to describe the structure on the derivatives of algebraic K-theory as a functor of associative or commutative ring spectra.

1:00 pm in 345 Altgeld Hall,Tuesday, March 17, 2015

#### The complexity of the homeomorphism relation between compact metric spaces

###### Joseph Zielinski   [email] (UIC Math)

Abstract: For equivalence relations $E$ and $F$ on Polish spaces $X$ and $Y$, respectively, $E$ is Borel reducible to $F$ when there is a Borel-measurable function from $X$ to $Y$ satisfying $x E y \Leftrightarrow f(x) F f(y)$. H. Becker and A.S. Kechris demonstrated that there are equivalence relations arising from Polish group actions that reduce all other such orbit equivalence relations. Moreover, J.D. Clemens, S. Gao, A.S. Kechris, J. Melleray, and M. Sabok, have variously shown that the natural relations of isometry between separable complete metric spaces, linear isometry between separable Banach spaces, and isomorphism of separable $C^*$-algebras share the same Borel-reducibility degree with these maximal orbit equivalence relations. We outline a proof that the relation of homeomorphism between metrizable compact spaces is also Borel bireducible with the complete orbit equivalence relations of Polish group actions.

1:00 pm in 347 Altgeld Hall,Tuesday, March 17, 2015

#### Finding the Stokes wave: from low steepness to almost highest wave

###### Sergey Dyachenko (Arizona)

Abstract: A Stokes wave is a fully nonlinear wave that travels over the surface of deep water. We solve Euler equations with free surface in the framework of conformal variables via Newton Conjugate Gradient method and find Stokes waves in regimes dominated by nonlinearity. By investigating Stokes waves with increasing steepness we observe peculiar oscillations occur as we approach Stokes limiting wave. Finally by analyzing Pade approximation of Stokes waves we infer that analytic structure associated with those waves has branch cut nature.

1:00 pm in 243 Altgeld Hall,Tuesday, March 17, 2015

#### Th singular fibers of the Hitchin map

###### Andre Oliveira (University of Trás-os-Montes e Alto Douro, Vila Real, Portugal.)

Abstract: The Hitchin map defines a remarkable fibration on the moduli space of Higgs bundles (closely related with moduli spaces of surface group representations). The generic fibers of this map are well known, but the same is not true for the singular ones. After sketching the main features of the fibration we study these singular fibers (in rank 2) and show that they are connected. This is joint work with Peter Gothen.

2:00 pm in 347 Altgeld Hall,Tuesday, March 17, 2015

#### The order-chaos phase transition for a general class of complex Boolean networks.

###### Shirshendu Chatterjee (City University of New York)

Abstract: We consider a model for heterogeneous gene regulatory networks that is an "annealed approximation" of Kauffmann's (1969) original random Boolean networks. In this model, genes are represented by the nodes of a random directed graph G_n on n vertices with specified degree distribution, and the interactions among the genes are approximated by an appropriate threshold contact process (in which a vertex with at least one occupied in-neighbor at time t will be occupied at time t+1 with probability q, and vacant otherwise) on G_n. We characterize the order-chaos phase transition curve segregating the chaotic and ordered random Boolean networks.

3:00 pm in 243 Altgeld Hall,Tuesday, March 17, 2015

#### Intersection numbers over relative Hilbert schemes and proof of S-duality conjecture for quintic threefold

###### Artan Sheshmani (Ohio State)

Abstract: I will talk about joint work with Gholampour on computing the generating series,  associated to the Hibert scheme of points relative to an effective divisor on a smooth quasi-projective surface. In particular we generalize the interesting work of Okounkov-Carlsson  who showed that in the case of absolute Hilbert schemes, such generating series are given as modular forms. We extend their constructions to the relative setting, and using localization and degeneration techniques, express the intersection numbers of the relative Hilbert scheme in terms of tangent bundle of the surface with logarithmic zeros and derive a similar nice formula as a modular form.  Then I will show how to use this result to prove the S-duality modularity conjecture for DT invariants of torsion sheaves supported on hyperplane sections of quintic threefold. I promise to use the blackboard this time.