Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, November 13, 2018.

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Tuesday, November 13, 2018

11:00 am in 345 Altgeld Hall,Tuesday, November 13, 2018

The parametrized Tate construction

JD Quigley (Notre Dame Math)

Abstract: The Tate construction is a powerful tool in classical homotopy theory. I will begin by reviewing the Tate construction and surveying some of its applications. I will then describe an enhancement of the Tate construction to genuine equivariant homotopy theory called the ``parametrized Tate construction" and discuss some of its applications, including C_2-equivariant versions of Lin's Theorem and the Mahowald invariant, blueshift for Real Johnson-Wilson spectra (joint work with Guchuan Li and Vitaly Lorman), and trace methods for Real algebraic K-theory (work-in-progress with Jay Shah).

12:00 pm in 243 Altgeld Hall,Tuesday, November 13, 2018

Taut sutured handlebodies as twisted homology products

Margaret Nichols (University of Chicago)

Abstract: A basic problem in the study of 3-manifolds is to determine when geometric objects are of ‘minimal complexity’. We are interested in this question in the setting of sutured manifolds, where minimal complexity is called ‘tautness’. One method for certifying that a sutured manifold is taut is to show that it is homologically simple - a so-called ‘rational homology product’. Most sutured manifolds do not have this form, but do always take the more general form of a ‘twisted homology product’, which incorporates a representation of the fundamental group. The question then becomes, how complicated of a representation is needed to realize a given sutured manifold as such? We explore some classes of relatively simple sutured manifolds, and see one class is always a rational homology product, but that the next natural class contains examples which require twisting. We also find examples that require twisting by a representation which cannot be ‘too simple’.

1:00 pm in 347 Altgeld Hall,Tuesday, November 13, 2018

Blow-up solutions for ODEs from the viewpoint of dynamical systems: theory and applications

Kaname Matsue (Kyushu University, Japan )

Abstract: A geometric treatment of blow-up solutions for autonomous ODEs including their asymptotic behavior is concerned. My idea is based on applications of “compactifications” of phase spaces so that the infinity corresponds to the boundary of embedded spaces. I then show that dynamics on center-stable manifolds of “invariant sets at infinity” with appropriate time-scale desingularizations can characterize dynamics of blow-up solutions as well as their rigorous blow-up rates not only of so-called “type-I” but also of other types (so-called “type-II” for example). This approach not only characterize typical monotonous blow-up behavior but also oscillatory blow-ups in terms of stable manifolds of “periodic orbits at infinity” by means of standard machineries in dynamical systems. Potentially, the above geometric treatment reveals a common mechanism among finite-time singularities (such as finite-time extinction, compacton traveling waves and quenching) in ordinary differential equations. Finally, I will talk about algebraic, geometric, analytic and numerical aspects and prospectives of the present study.

2:00 pm in 243 Altgeld Hall,Tuesday, November 13, 2018

Special four cycles in triple systems

Zoltan Furedi (Alfréd Rényi Institute of Mathematics, Budapest, Hungary)

Abstract: A special four-cycle $F$ in a triple system consists of four triples inducing a $C_4$. This means that $F$ has four special vertices $v_1,v_2,v_3,v_4$ and four triples in the form $v_iv_{i+1}w_i$ where the $w_j$'s are not necessarily distinct but disjoint from $\{v_1,v_2,v_3,v_4\}$ (indices are understood $\pmod 4$). There are seven non-isomorphic special four-cycles, their family is denoted by $\cal{F}$. Our main result implies that the Turán number ${\rm ex}(n,{\mathcal{F}})=\Theta(n^{3/2})$. In fact, we prove more, ${\rm ex}(n,\{F_1,F_2,F_3\})=\Theta(n^{3/2})$, where the $F_i$'s are specific members of $\mathcal{F}$.

We also study further generalizations, many cases remain unsolved.

3:00 pm in 243 Altgeld Hall,Tuesday, November 13, 2018

Severi degrees via representation theory

Yaim Cooper (IAS)

Abstract: The Severi degrees of $\mathbb{P}^1 \times \mathbb{P}^1$ can be computed in terms of an explicit operator on the Fock space $F[\mathbb{P}^1]$. We will discuss this and variations on this theme. We will explain how to use this approach to compute the relative Gromov-Witten theory of other surfaces, such as Hirzebruch surfaces and $E \times\mathbb{P}^1$. We will also discuss operators for calculating descendants. Joint with R. Pandharipande.