Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, January 19, 2016.

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Tuesday, January 19, 2016

11:00 am in 345 Altgeld Hall,Tuesday, January 19, 2016

Tensor triangular geometry of the stable motivic homotopy category

Kyle Ormsby (Reed)

Abstract: In Balmer's framework of tensor triangular geometry, the prime thick tensor ideals in a tensor triangulated category $\mathcal{C}$ form a space which admits a continuous map to the Zariski spectrum $\mathrm{Spec}^h(\mathrm{End}^\bullet_u(1))$ of homogeneous prime ideals in the graded endomorphism ring of the unit object. (Here the grading is induced by an element $u$ of the Picard group of $\mathcal{C}$.) If $\mathcal{C}$ is the stable motivic homotopy category and $u$ is the punctured affine line, then this endomorphism ring is the Milnor-Witt K-theory ring $K^{MW}_*(F)$ of the base field $F$. I will describe work by my student, Riley Thornton, which completely determines $\mathrm{Spec}^h(K^{MW}_*(F))$ in terms of the orderings of $F$. I will then comment on work in progress which uses the structure of this spectrum to study the thick subcategories of the stable motivic homotopy category.

2:00 pm in 347 Altgeld Hall,Tuesday, January 19, 2016

Estimates on Functional Integrals of Quantum Mechanics and Non-Relativistic Quantum Field Theory

Gonzalo Bley   [email] (University of Virginia)

Abstract: We provide a unified method for obtaining upper bounds for certain functional integrals appearing in quantum mechanics and non-relativistic quantum field theory, functionals of the form $E[exp(A_T)]$, the (effective) action $A_T$ being a function of particle trajectories up to time $T$. The estimates in turn yield rigorous lower bounds for ground state energies, via the Feynman-Kac formula. The upper bounds are obtained by writing the action for these functional integrals in terms of stochastic integrals. The method is first illustrated in familiar quantum mechanical settings: for the hydrogen atom, for a Schr\"{o}dinger operator with $1/|x|^2$ potential with small coupling, and, with a modest adaptation of the method, for the harmonic oscillator. We then present our principal applications of the method, in the settings of non-relativistic quantum field theories for particles moving in a quantized Bose field, including the optical polaron and Nelson models. This is joint work with Lawrence Thomas from the University of Virginia.

3:00 pm in 243 Altgeld Hall,Tuesday, January 19, 2016

Organizational Meeting

3:00 pm in 241 Altgeld Hall,Tuesday, January 19, 2016

Maximal $k$-edge-colorable subgraphs, Vizing's Theorem, and Tuza's Conjecture

Gregory J. Puleo   [email] (UIUC Coordinated Science Lab)

Abstract: If $M$ is a maximal matching in a graph $G$ and $F$ is the set of vertices not covered by $M$, it is easy to see that $F$ is an independent set. We generalize this observation to maximal $k$-edge-colorable subgraphs of $G$, where a maximal matching can be viewed as a maximal 1-edge-colorable subgraph. In particular, we prove that if $F$ is the set of vertices having degree less than $k$ in $M$, then each vertex $v \in F$ satisfies $d_F(v) \leq d_M(v)$. This implies Vizing's Theorem on edge-coloring as well as a special case of Tuza's Conjecture on packing and covering of triangles.

4:00 pm in 243 Altgeld Hall,Tuesday, January 19, 2016

Organizational meeting

5:00 pm in 245 Altgeld Hall,Tuesday, January 19, 2016

IGL Kickoff Meeting

Abstract: Spring 2016 organizational meeting for the Illinois Geometry Lab