Seminars and short courses

04/06/2013, Tuesday

15:00, Analysis, Geometry, and Dynamical Systems

Christian Le Merdy, Université de Franche-Comté.
Dilation of operators on $L^p$-spaces.

Let $1<p<\mathrm{\infty }$, let $(\Omega ,\mu )$ be a measure space and let $T:L^p(\Omega )\to L^p(\Omega )$ be a bounded operator. We say that it admits a dilation (in a loose sense) when there exist another measure space $(\Omega \prime ,\mu \prime )$, an invertible operator $U$ on $L^p(\Omega \prime )$ such that $\{U^n:n\in \mathbb{Z}\}$ is bounded and two bounded operators $J:L^p(\Omega )\to L^p(\Omega \prime )$ and $Q:L^p(\Omega \prime )\to L^p(\Omega )$ such that $T^n=Q{U}^{n}J$ for any integer $n\ge 0$. When $p=2$, this property is equivalent to $T$ being similar to a contraction. The main question considered in this talk is to characterize operators with this property when $p\ne 2$. Our results give partial answers and strong connections with functional calculus properties. The talk will include motivation for this dilation question. (Joint work with C. Arhancet.)

21/05/2013, Tuesday

16:30, Geometria em Lisboa

Carlos Florentino, IST.
Irreducibility of character varieties of abelian groups.

The description of the space of commuting elements in a compact Lie group is an interesting algebro-geometric problem with applications in Mathematical Physics, notably in Supersymmetric Yang Mills theories.

When the Lie group is complex reductive, this space is the character variety of a free abelian group. Let \(K\) be a compact Lie group (not necessarily connected) and \(G\) be its complexification. We consider, more generally, an arbitrary finitely generated abelian group \(A\), and show that the conjugation orbit space $\mathrm{Hom}(A,K)/K$ is a strong deformation retract of the character variety $\mathrm{Hom}(A,G)/G$.

As a Corollary, in the case when \(G\) is connected and semisimple, we obtain necessary and sufficient conditions for $\mathrm{Hom}(A,G)/G$ to be irreducible. This is also related to an interesting open problem about irreducibility of the variety of \(k\) tuples of \(n\) by \(n\) commuting matrices.

20/05/2013, Monday

16:30, String Theory

Diego Bombardelli, University of Porto.
Thermodynamic Bethe Ansatz and double-wrapping corrections for non-supersymmetric deformations of AdS/CFT.

We study finite-size corrections of the ${\gamma }_i$-deformed AdS/CFT vacuum energy/anomalous dimension. In particular, we compute the leading (at large volume) and next-to-leading order (NLO) Luescher-like corrections, corresponding to single- and double-wrapping diagrams respectively. On the other hand, we solve to the NLO the twisted Thermodynamic Bethe Ansatz equations describing exactly the ground-state energy of the theory; then we compare the results of the two approaches and find exact agreement. Next, we evaluate explicitly LO and NLO corrections up to six loops at weak coupling. Finally, I will show some work in progress about the possible conjecture of a Luescher-like formula for the double-wrapping corrections of undeformed excited states energy.

16/05/2013, Thursday

16:30, Geometria em Lisboa

Matias del Hoyo.
On the linearization of certain smooth structures.

The linearization theorem for proper Lie groupoids, whose prove was completed by Zung a few years ago, generalizes various results such as Ehresmann theorem for submersions, Reeb stability for foliations, and the Tube Theorem for proper actions. In a work in progress with R. Fernandes we show that this linearization can be achieved by means of the exponential flow of certain metrics, providing both a stronger theorem and a simpler proof. In this talk I will recall the classic linearization theorems, discuss its groupoid formulation, and present our work on riemannian structures for Lie groupoids.

16/05/2013, Thursday

16:30, Operator Theory, Complex Analysis and Applications

Sérgio Mendes, Instituto Universitário de Lisboa, ISCTE-IUL.
Noncommutative summands of the $C^{*}$-algebra $C_r^{*}{\mathrm{SL}}_2({𝔽}_2((\varpi )))$.

Let ${𝔽}_2((\varpi ))$ denote the Laurent series in the indeterminate $\varpi$ with coefficients over the finite field with two elements ${𝔽}_2$. This is a local nonarchimedean field with characteristic $2$. We show that the structure of the reduced group $C^{*}$-algebra of ${\mathrm{SL}}_2({𝔽}_2((\varpi )))$ is determined by the arithmetic of the ground field. Specifically, the algebra $C_r^{*}{\mathrm{SL}}_2({𝔽}_2((\varpi )))$ has countably many noncommutative summands, induced by the Artin-Schreier symbol. Each noncommutative summand has a rather simple description: it is the crossed product of a commutative $C^{*}$-algebra by a finite group. The talk will be elementary, starting from the scratch with the definition of $C_r^{*}{\mathrm{SL}}_2$.

14/05/2013, Tuesday

15:00, Analysis, Geometry, and Dynamical Systems

Filippo Cagnetti, University of Sussex.
A new method for large time behavior of convex Hamilton-Jacobi equations.

We introduce a new method to study the large time behavior for general classes of Hamilton-Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.

14/05/2013, Tuesday

10:30, Colloquium

Alessio Figalli, University of Texas at Austin.
Stability results for sumsets in \(\mathbb{R}^n\).

Given a Borel set \(A\) in \(\mathbb{R}^n\) of positive measure, one can consider its semisum \(S=(A+A)/2\). It is clear that \(S\) contains \(A\), and it is not difficult to prove that they have the same measure if and only if \(A\) is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of \(S\) is close to the one of \(A\), is \(A\) close to his convex hull? More in general, one may consider the semisum of two different sets \(A\) and \(B\), in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When \(n=1\), one can approximate a set with finite unions of intervals to translate the problem onto \(\mathbb{Z}\), and in the discrete setting this question becomes a well studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk I'll review some results in the one-dimensional discrete setting, and show how to answer to this problem in arbitrary dimension.

06/05/2013, Monday

16:30, String Theory

Ivo Sachs, LMU Munich.
Homotopy Algebras and String Field Theory.

We revisit the existence, background independence and uniqueness of bosonic- and topological string field theory using the machinery of homotopy algebra. In a theory of classical open- and closed strings, the space of inequivalent open string field theories is shown to be isomorphic to the space of classical closed string backgrounds. We then discuss obstructions of these moduli spaces at the quantum level.

29/04/2013, Monday

16:30, String Theory

Jorge Russo, Universitat de Barcelona.
Evidence for Large \(N\) phase transitions in \(N=2^\ast\) theory.

Using localization, we solve for the large-\(N\) master field of \(N=2^\ast\) super-Yang-Mills theory and calculate expectation values of large Wilson loops and the free energy. At weak coupling, these observables only receive non-perturbative contributions. The analytic solution holds for a finite range of the 't Hooft coupling and terminates at the point of a large-\(N\) phase transition. We find evidence that as the coupling is further increased the theory undergoes an infinite sequence of similar transitions that accumulate at infinity.

24/04/2013, Wednesday

15:00, Partial Differential Equations

Diego Marcon Farias, IST, Lisbon.
A quantitative log-Sobolev inequality for a two parameter family of functions.

We prove a sharp, dimension-free stability result for the classical logarithmic Sobolev inequality for a two parameter family of functions. Roughly speaking, our family consists of a certain class of log $C^{\mathrm{1,1}}$ functions. Moreover, we show how to enlarge this space at the expense of the dimensionless constant and the sharp exponent. As an application we obtain new bounds on the entropy. (with E. Indrei)

24/04/2013, Wednesday

11:30, Topological Quantum Field Theory

John Huerta, IST.
QFT III.

Last time, we talked about quantization of the free scalar field by replacing the modes of the field by quantum oscillators. Now, we put this field into the form used by physicists, and talk about the Wightman axioms, which allow a rigorous treatment of free fields.

18/04/2013, Thursday

16:30, Operator Theory, Complex Analysis and Applications

Gabriel Cardoso, Instituto Superior Técnico.
A light introduction to supersymmetry.

We give a brief introduction to supersymmetric quantum mechanics.

17/04/2013, Wednesday

15:00, Partial Differential Equations

Ana Ribeiro, Universidade Nova de Lisboa.
Existence of solutions for non level-convex problems in the supremal form.

It is well known that lower semicontinuity of functionals in the supremal form $$F(u)={{\displaystyle esssup}}_{x\in \Omega }f(\nabla u(x))$$ is related to the level-convexity of the supremand \(f\) . We adress the problem of existence of solutions for the Dirichlet boundary problem in the lack of this convexity condition relating it with some differential inclusion problem. This is a joint work with E. Zappale.

17/04/2013, Wednesday

11:30, Topological Quantum Field Theory

John Huerta, IST.
QFT II.

We continue our gentle introduction to quantum field theory for mathematicians. We discuss the Klein-Gordon equation, and how it decomposes into oscillators. We quantize this system by quantizing the oscillators, obtaining the free scalar field, the simplest quantum field there is.

16/04/2013, Tuesday

15:00, Analysis, Geometry, and Dynamical Systems

Florin Radulescu, Università di Roma - Tor Vergata.
Ramanujan-Petersson conjectures and Operator Algebras.

10/04/2013, Wednesday

11:30, Topological Quantum Field Theory

John Huerta, IST.
QFT I.

This series of lectures will be a gentle introduction to quantum field theory for mathematicians. In our first lecture, we give a lightning introduction to quantum mechanics and discuss the simplest quantum system: the harmonic oscillator. We then sketch how this system is used to quantize the free scalar field.

08/04/2013, Monday

16:30, String Theory

Dimitrios Zoakos, University of Porto.
Holographic flavor in Chern-Simons-matter theories.

After reviewing the gravity dual of \(N=6\) Chern-Simons-matter theory, we will analyze the addition of backreacted flavors. We will then construct the corresponding flavored black hole and study the thermodynamic properties of brane probes and of the meson melting transition that they undergo at a certain critical temperature.

03/04/2013, Wednesday

11:30, Topological Quantum Field Theory

John Huerta, IST.
Anomalies IV.

We will introduce the notion of stable isomorphism for gerbes, and talk about how stable isomorphism classes are in one-to-one correspondence with Deligne cohomology classes. We define WZW branes and discuss how the basic gerbe on a group trivializes when restricted to the brane.

02/04/2013, Tuesday

15:00, Analysis, Geometry, and Dynamical Systems

Jorge Ferreira, Universidade Federal Rural de Pernambuco.
On the asymptotic behaviour of nonlocal nonlinear problems.

This lecture deals with nonlocal nonlinear problems. Our main results concern existence, uniqueness and asymptotic behavior of the weak solutions of a nonlinear parabolic equation of reaction-diffusion nonlocal type by an application of the Faedo-Galerkin approximation and Aubin-Lions compactness result. Moreover, we prove continuity with respect to the initial values, the joint continuity of the solution and a result on the existence of the global attractor for the problem $$\{\begin{array}{l}{u}_t-a(l(u))\Delta u+\mid u{\mid }^{\rho }u=f(u)\phantom{.}\text{in}\phantom{.}\Omega \times (0,T),\\ u(x,t)=0\phantom{.}\text{on}\phantom{.}\partial \Omega \times (0,T),\\ u(x\mathrm{,0})=u_0(x)\phantom{.}\text{in}\phantom{.}\Omega ,\end{array}$$ when $0<\rho \le 2/(n-2)$ if $n\ge 3$ and $0<\rho <\mathrm{\infty }$ if $n=\mathrm{1,2}$, where $u=u(x,t)$ is a real valued function, $\Omega \subset {ℝ}^n$ is a bounded smooth domain, $n\ge 1$ with regular boundary $\Gamma =\partial \Omega$, $p\ge 2$. Moreover, $a$ and $f$ are continuous functions satisfying some appropriate conditions and $l:L^2(\Omega )\to ℝ$ is a continuous linear form.

26/03/2013, Tuesday

16:30, Geometria em Lisboa

Leonardo Macarini, Universidade Federal do Rio de Janeiro.
Two periodic orbits on the standard three-sphere.

We prove that every contact form on the tight three-sphere has at least two geometrically distinct periodic orbits. This result was obtained recently by Cristofaro-Gardiner and Hutchings using embedded contact homology but our approach instead is based on cylindrical contact homology. An essential ingredient in the proof is the notion of a symplectically degenerate maximum for Reeb flows whose existence implies infinitely many prime periodic orbits (in any dimension). This is joint work with V. Ginzburg, D. Hein and U. Hryniewicz.

21/03/2013, Thursday

16:30, Operator Theory, Complex Analysis and Applications

Cristina Câmara, Instituto Superior Técnico.
A Riemann-Hilbert approach to Toeplitz operators and the corona theorem.

Together with differential operators, Toeplitz operators (TO) constitute one of the most important classes of non-self adjoint operators , and they appear in connection with various problems in physics and engineering. The main topic of my presentation will be the interplay between TOs and Riemann-Hilbert problems (RHP), and the relations of both with the corona theorem. It has been shown that the existence of a solution to a RHP with $2\times 2$ coefficient $G$, satisfying some corona type condition, implies – and in some cases is equivalent to – Fredholmness of the TO with symbol $G$. Moreover, explicit formulas for an appropriate factorization of $G$ were obtained, allowing to solve different RHPs with coefficient $G$, and to determine the inverse, or a generalized inverse, of the TO with symbol $G$. However, those formulas depend on the solutions to 2 meromorphic corona problems. These solutions being unknown or rather complicated in general, the question whether the factorization of $G$ can be obtained without the corona solutions is a pertinent one. In some cases, it already has a positive answer; how to solve this question in general is open, and all the more so in the case of $n\times n$ matrix functions $G$, for which the results regarding the $2\times 2$ case have recently been generalized.

20/03/2013, Wednesday

11:30, Topological Quantum Field Theory

Aleksandar Mikovic, Univ. Lusófona.
Categorification of Spin Foam Models.

We briefly review spin foam state sums for triangulated manifolds and motivate the introduction of state sums based on 2-groups. We describe 2-BF gauge theories and the construction of the corresponding path integrals (state sums) in the case of Poincaré 2-group.

References

• J. F. Martins and A. Mikovic, Lie crossed modules and gauge-invariant actions for 2-BF theories, Adv. Theor. Math. Phys. 15 (2011) 1059, arxiv:1006.0903
• A. Mikovic and M. Vojinovic, Poincaré 2-group and quantum gravity, Class. Quant. Grav. 291 (2012) 165003, arxiv:1110.4694

13/03/2013, Wednesday

11:30, Topological Quantum Field Theory

John Huerta, IST.
Anomalies III.

We continue examining Gawedzki and Reis's paper:

WZW branes and gerbes, http://arxiv.org/abs/hep-th/0205233

We define a gerbe, and show gerbes can be "transgressed" to give line bundles over loop space. Trivial gerbes give trivial bundles on loop space, whose sections are thus mere functions. Any compact, simply connected Lie group comes with a god-given gerbe whose curvature is the canonical invariant 3-form. Restricting this gerbe to certain submanifolds, we get trivial gerbes who thus transgress to trivial line bundles, "cancelling" the anomaly of a nontrivial line bundle.

12/03/2013, Tuesday

16:30, Geometria em Lisboa

Milena Pabiniak.
Lower bounds on Gromov width of coadjoint orbits through the Gelfand-Tsetlin pattern.

Gromov width of a symplectic manifold $M$ is a supremum of capacities of balls that can be symplectically embedded into $M$. The definition was motivated by the Gromov's Non-Squeezing Theorem which says that maps preserving symplectic structure form a proper subset of volume preserving maps.

Let $G$ be a compact connected Lie group, $T$ its maximal torus, and $\lambda$ be a point in the chosen positive Weyl chamber.

The group $G$ acts on the dual of its Lie algebra by coadjoint action. The coadjoint orbit, $M$, through $\lambda$ is canonically a symplectic manifold. Therefore we can ask the question of its Gromov width.

In many known cases the width is exactly the minimum over the set of positive results of pairing $\lambda$ with coroots of $G$:

$$\min \{⟨{\alpha }_j^{\vee },\lambda ⟩;{\alpha }_j\text{a coroot},⟨{\alpha }_j^{\vee },\lambda ⟩>0\}.$$

For example, this result holds if $G$ is the unitary group and $M$ is a complex Grassmannian or a complete flag manifold satisfying some additional integrality conditions.

We use the torus action coming from the Gelfand-Tsetlin system to construct symplectic embeddings of balls. In this way we prove that the above formula gives the lower bound for Gromov width of all $U(n)$ and most of $\mathrm{SO}(n)$ coadjoint orbits.

In the talk I will describe the Gelfand-Tsetlin system and concentrate mostly on the case of regular \(U(n)\) orbits.

07/03/2013, Thursday

02:30, Algebra

Bob Oliver, Université Paris XIII.
Local equivalences between finite Lie groups.

Fix a prime $p$. Two finite groups $G$ and $H$ will be called $p$-locally equivalent if there is an isomorphism from a Sylow $p$-subgroup $S$ of $G$ to a Sylow $p$-subgroup $T$ of $H$ which preserves all conjugacy relations between elements and subgroups of $S$ and $T$.

Martino and Priddy proved that if the $p$-completed classifying spaces ${\mathrm{BG}}_p$ and ${\mathrm{BH}}_p$ are homotopy equivalent, then $G$ and $H$ are $p$-locally equivalent. They also conjectured the converse, a result which has since been proven, but only by using the classification theorem of finite simple groups.

Anyone who works much with finite groups of Lie type (such as linear, symplectic, or orthogonal groups over finite fields) notices that there are many cases of $p$-local equivalences between them. For example, if $q$ and $q\prime$ are two prime powers such that $q^2-1$ and $(q\prime {)}^2-1$ have the same 2-adic valuation, then ${\mathrm{SL}}_2(q)$ and ${\mathrm{SL}}_2(q\prime )$ are 2-locally equivalent.

In joint work with Carles Broto and Jesper Møller, we proved, among other results, the following very general theorem about such $p$-local equivalences between finite Lie groups.

Theorem: Fix a prime $p$, a connected, reductive group scheme $G$ over $Z$, and a pair of prime powers $q$ and $q\prime$ both prime to $p$. Then $G(q)$ and $G(q\prime )$ are $p$-locally equivalent if $\overline{⟨q⟩}=\overline{⟨q\prime ⟩}$ as closed subgroups of $Z_p^{\times }$.

Our proof of this theorem is topological: we show that the $p$-completed classifying spaces have the same homotopy type, and then apply the theorem of Martino and Priddy mentioned above. The starting point is a theorem of Friedlander, which describes the space $\mathrm{BG}(q{)}_p$ as a “homotopy fixed space” of a some self map of $\mathrm{BG}(C{)}_p$ of a certain type (an “unstable Adams operation”). This is combined with a theorem of Jackowski, McClure, and Oliver that classifies more precisely the self maps of $\mathrm{BG}(C{)}_p$; and with a result of Broto, Møller, and Oliver which says that under certain hypotheses on a space $X$, the homotopy fixed space of a self equivalence $f$ of $X$ depends (up to homotopy type) only on the closed subgroup $\overline{⟨f⟩}$ in the group $\mathrm{Out}(X)$ of all homotopy classes of self equivalences of $X$.

Currently, no other proof seems to be known of this purely algebraic theorem.

27/02/2013, Wednesday

11:30, Topological Quantum Field Theory

John Huerta, IST.
Anomalies II.

We continue our informal discussion of anomalies by talking about global anomalies on branes, and their relationship with gerbes.

25/02/2013, Monday

16:00, Partial Differential Equations

Carlo Mariconda, Università degli Studi di Padova.
Non occurrence of the Lavrentiev phenomenon for scalar multi-dimensional variational problems.

Let $f:{ℝ}^n\to ℝ$ be a convex function, $\Omega$ be an open and bounded subset of ${ℝ}^n$. We consider the functional $$I(u)={\int }_{\Omega }f(\nabla (u(x)))dxu\in W^{\mathrm{1,1}}(\Omega ).$$ It is known [2] that if $\Omega$ is star-shaped then the Lavrentiev phenomenon does not occur if one does not consider a fixed boundary datum, i. e. $$\inf \{I(u):u\in W^{\mathrm{1,1}}(\Omega )\}=\inf \{I(u):u\in W^{1,\mathrm{\infty }}(\Omega )\}$$ The importance of the non occurrence of the Lavrentiev phenomenon is due to the fact that only in that case, the methods of numeric analysis allow to approximate the infimum value of the operator (finite elements method). When the boundary datum is taken into account, in spite of the paradigm saying that the Lavrentiev phenomenon should not occur, there are just a few results corroborating the statement, apart the obvious case where some “natural growth conditions” are assumed: in a recent paper Cellina and Bonfanti [1] proved that if the lagrangian is radial and both the boundary datum and the domain are of class ${𝒞}^2$ then the Lavrentiev phenomenon does not occur.

In a work in progress, jointly with Pierre Bousquet and Giulia Treu, we make a considerable step forward in favor of the above conjecture and take into account a wider class of domains and lagrangians, with a minimum set of assumptions (in particular no growth conditions!): its description is the main argument of the lecture.

References:

[1] G. Bonfanti and A. Cellina, On the non-occurrence of the lavrentiev phenomenon, Adv. Calc. Var. 6 (2013), 93–121.

[2] G. Buttazzo and M. Belloni, A survey on old and recent results about the gap phenomenon in the calculus of variations, Recent developments in well-posed variational problems, 1995, pp. 1–27.

21/02/2013, Thursday

16:30, Geometria em Lisboa

Elisa Tenni.
Clifford theorem for singular curves and some applications.

I will discuss a generalization of the classical Clifford's theorem to singular curves, reducible or non reduced. I will prove that for 2-connected curves a Clifford-type inequality holds for a vast set of torsion free rank one sheaves. I intend to show that our assumptions on the sheaves are the most natural when working with this kind of results. I will moreover show that this result has many applications to the study of the canonical morphism of a singular curve, in particular that it implies a generalization of the classical Noether's theorem to 3-connected curves. This is a joint work with M. Franciosi.

19/02/2013, Tuesday

16:30, Geometria em Lisboa

Jacopo Stoppa, Università di Pavia.
Refined curve counting, quivers, and wall-crossing.

I will sketch some aspects of an interesting Gromov-Witten theory on weighted projective planes introduced by Gross, Pandharipande and Siebert. It admits a very special expansion in terms of tropical counts (called the tropical vertex), as well as a conjectural BPS structure. Then I will describe a refinement or "$q$-deformation" of the expansion using Block-Goettsche invariants, motivated by wall-crossing ideas. This leads naturally to a definition of a class of putative $q$-deformed BPS counts. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined (joint with S. A. Filippini).

18/02/2013, Monday

14:00, Operator Theory, Complex Analysis and Applications

Pedro Patrício, Universidade do Minho.
Generalized invertibility in rings: some recent results.

The theory of generalized inverses has its roots both on semigroup theory and on matrix and operator theory. In this seminar we will focus on the study of the generalized inverse of von Neumann, group, Drazin and Moore-Penrose in a purely algebraic setting. We will present some recent results dealing with the generalized inverse of certain types of matrices over rings, emphasizing the proof techniques used.

14/02/2013, Thursday

15:00, Partial Differential Equations

Filippo Cagnetti.
A new method for large time behavior of convex Hamilton-Jacobi equations.

We introduce a new method to study the large time behavior for general classes of Hamilton-Jacobi type equations, which include degenerate parabolic equations and weakly coupled systems. We establish the convergence results by using the nonlinear adjoint method and identifying new long time averaging effects. These methods are robust and can easily be adapted to study the large time behavior of related problems.

This is a joint work with D. Gomes, H. Mitake and H. V. Tran.

06/02/2013, Wednesday

14:00, Topological Quantum Field Theory

John Huerta, IST.
Introduction to anomalies.

In physics, an "anomaly" is the failure of a classical symmetry at the quantum level. Anomalies play a key role in assessing the consistency of a quantum field theory, and link up with cohomology in mathematics, a general tool by which mathematicians understand whether a desired construction is possible. In this informal series of talks, we aim to understand what physicists mean by an "anomaly" and their mathematical interpretation.

17/01/2013, Thursday

16:30, Operator Theory, Complex Analysis and Applications

Petr Siegl, Universidade de Lisboa.
Spectral analysis of some non-self-adjoint operators.

We give an introduction to the study of one particular class of non-self-adjoint operators, namely $𝒫𝒯$-symmetric ones. We explain briefly the physical motivation and describe the classes of operators that are considered. We explain relations between the operator classes, namely their non-equivalence, and mention open problems.

In the second part, we focus on the similarity to self-adjoint operators. On the positive side, we present results on one-dimensional Schrödinger-type operators in a bounded interval with non-self-adjoint Robin-type boundary conditions. Using functional calculus, closed formulas for the similarity transformation and the similar self-adjoint operator are derived in particular cases. On the other hand, we analyse the imaginary cubic oscillator, which, although being $𝒫𝒯$-symmetric and possessing real spectrum, is not similar to any self-adjoint operator. The argument is based on known semiclassical results.

1. P. Siegl: The non-equivalence of pseudo-Hermiticity and presence of antilinear symmetry, PRAMANA-Journal of Physics, Vol. 73, No. 2, 279-287,
2. D. Krejcirík, P. Siegl and J. Zelezný: On the similarity of Sturm-Liouville operators with non-Hermitian boundary conditions to self-adjoint and normal operators, Complex Analysis and Operator Theory, to appear,
3. P. Siegl and D. Krejcirík: On the metric operator for imaginary cubic oscillator, Physical Review D, to appear.

08/01/2013, Tuesday

11:00, Algebra

Luke Wolcott, University of Western Ontario.
Bousfield lattices, quotients, ring maps, and non-Noetherian rings.

Given an object $X$ in a compactly generated tensor triangulated category $C$ (such as the derived category of a ring, or the stable homotopy category), the Bousfield class of $X$ is the collection of objects that tensor with $X$ to zero. The set of Bousfield classes forms a lattice, called the Bousfield lattice $\mathrm{BL}(C)$. First, we will look at examples of when a functor $F:C\to D$ induces a lattice map $\mathrm{BL}(C)\to \mathrm{BL}(D)$, and will describe several lattice quotients and lattice isomorphisms. Second, we will focus on homological algebra; a ring map $f:R\to S$ induces, via extension of scalars, a functor $D(R)\to D(S)$, and this induces a map on Bousfield lattices. Third, we specialize to a specific map between some interesting non-Noetherian rings.