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*[This guest post is authored by Matilde Lalin, an Associate Professor in the Département de mathématiques et de statistique at the Université de Montréal. I have lightly edited the text, mostly by adding some HTML formatting. -T.]*

Mathematicians (and likely other academics!) with small children face some unique challenges when traveling to conferences and workshops. The goal of this post is to reflect on these, and to start a constructive discussion what institutions and event organizers could do to improve the experiences of such participants.

The first necessary step is to recognize that different families have different needs. While it is hard to completely address everybody’s needs, there are some general measures that have a good chance to help most of the people traveling with young children. In this post, I will mostly focus on nursing mothers with infants ( months old) because that is my personal experience. Many of the suggestions will apply to other cases such as non-nursing babies, children of single parents, children of couples of mathematicians who are interested in attending the same conference, etc..

The mother of a nursing infant that wishes to attend a conference has three options:

**Bring the infant and a relative/friend to help caring for the infant.**The main challenge in this case is to fund the trip expenses of the relative. This involves trip costs, lodging, and food. The family may need a hotel room with some special amenities such as crib, fridge, microwave, etc. Location is also important, with easy access to facilities such as a grocery store, pharmacy, etc. The mother will need to take regular breaks from the conference in order to nurse the baby (this could be as often as every three hours or so). Depending on personal preferences, she may need to nurse privately. It is convenient, thus, to make a private room available, located as close to the conference venue as possible. The relative may need to have a place to stay with the baby near the conference such as a playground or a room with toys, particularly if the hotel room is far.**Bring the infant and hire someone local (a nanny) to help caring for the infant**. The main challenges in this case are two: finding the caregiver and paying for such services. Finding a caregiver in a place where one does not live is hard, as it is difficult to conduct interviews or get references. There are agencies that can do this for a (quite expensive) fee: they will find a professional caregiver with background checks, CPR certification, many references, etc. It may be worth it, though, as professional caregivers tend to provide high-quality services and peace of mind is priceless for the mother mathematician attending a conference. As in the previous case, the mother may have particular needs regarding the hotel room, location, and all the other facilities mentioned for Option 1.**Travel without the infant and pump milk regularly.**This can be very challenging for the mother, the baby, and the person that stays behind taking care of the baby, but the costs of this arrangement are much lower than in Option 1 or 2 (I am ignoring the possibility that the family needs to hire help at home, which is necessary in some cases). A nursing mother away from her baby has no option but to pump her milk to prevent her from pain and serious health complications. This mother may have to pump milk very often. Pumping is less efficient than nursing, so she will be gone for longer in each break or she will have more breaks compared to a mother that travels with her baby. For pumping, people need a room which should ideally be private, with a sink, and located as close to the conference venue as possible. It is often impossible for these three conditions to be met at the same time, so different mothers give priority to different features. Some people pump milk in washrooms, to have easy access to water. Other people might prefer to pump in a more comfortable setting, such as an office, and go to the washroom to wash the breast pump accessories after. If the mother expects that the baby will drink breastmilk while she is away, then she will also have to pump milk in advance of her trip. This requires some careful planning.

Many pumping mothers try to store the pumped milk and bring it back home. In this case the mother needs a hotel room with a fridge which (ideally, but hard to find) has a freezer. In a perfect world there would also be a fridge in the place where she pumps/where the conference is held.

It is important to keep in mind that each option has its own set of challenges (even when expenses and facilities are all covered) and that different families may be restricted in their choice of options for a variety of reasons. It is therefore important that all these three options be facilitated.

As for the effect these choices have on the conference experience for the mother, Option 1 means that she has to balance her time between the conference and spending time with her relative/friend. This pressure disappears when we consider Option 2, so this option may lead to more participation in the conferences activities. In Option 3, the mother is in principle free to participate in all the conference activities, but the frequent breaks may limit the type of activity. A mother may choose different options depending on the nature of the conference.

I want to stress, for the three options, that having to make choices about what to miss in the conference is very hard. While talks are important, so are the opportunities to meet people and discuss mathematics that happen during breaks and social events. It is very difficult to balance all of this. This is particularly difficult for the pumping mother in Option 3: because she travels without her baby, she is not perceived to be a in special situation or in need of accommodation. However, this mother is probably choosing between going to the last lecture in the morning or having lunch alone, because if she goes to pump right after the last lecture, by the time she is back, everybody has left for lunch.

Here is the Hall of Fame for those organizations that are already supporting nursing mothers’ travels in mathematics:

- The Natural Sciences and Engineering Research Council of Canada (NSERC) (search for “child care”) allows to reimburse the costs of child care with Option 2 out of the mother’s grants. They will also reimburse the travel expenses of a relative with Option 1 up to the amount that would cost to hire a local caregiver.
- The ENFANT/ELEFANT conference (co-organized by Lillian Pierce and Damaris Schindler) provided a good model to follow regarding accommodation for parents with children during conferences that included funding for covering the travel costs of accompanying caretakers (the funding was provided by the Deutsche Forschungsgemeinschaft, and lactation rooms and play rooms near the conference venue (the facilities were provided by the Hausdorff Center for Mathematics).

Additional information (where to go with kids, etc) was provided on site by the organizers and was made available to all participants all the time, by means of a display board that was left standing during the whole week of the conference. - The American Institute of Mathematics (AIM) reimburses up to 500 dollars on childcare for visitors and they have some online resources that assist in finding childcare and nannies.

[UPDATED] Added a few more things to the Hall of Fame

- The Joint Mathematics Meetings have been providing onsite childcare in the last few years.
- The Institute for Applied Mathematics (IPAM) provides childcare resources and funding to workshop participants with small children (comment from juliawolf).
- The Simons Institute for the Theory of Computing at Berkeley has a separate lactation room as part of some female washrooms (comment from juliawolf).
- The London Mathematical Society (LMS) offers Childcare Supplementary Grants of up to £200 to all mathematicians based in the UK travelling to conferences and other research schools, meetings or visits (comments from Peter).

In closing, here is a (possibly incomplete) list of resources that institutes, funding agencies, and conferences could consider providing for nursing mother mathematicians:

- Funding (for cost associated to child care either professional or by an accompanying relative).
- List of childcare resources (nannies, nanny agencies, drop-in childcare centre, etc).
- Nursing rooms and playrooms near the conference venue. Nearby fridge.
- Breaks of at least 20 minutes every 2-3 hours.
- Information about transportation with infants. More specific, taxi and/or shuttle companies that provide infant car seats. Information regarding the law on infant seats in taxis and other public transportation.
- Accessibility for strollers.
- [UPDATED] A nearby playground location. (comment from Peter).

I also find it important that these resources be listed publicly in the institute/conference website. This serves a double purpose: first, it helps those in need of the resources to access them easily, and second, it contributes to make these accommodations normal, setting a good model for future events, and inspiring organizers of future events.

Finally, I am pretty sure that the options and solutions I described do not cover all cases. I would like to finish this note by inviting readers to make suggestions, share experiences, and/or pose questions about this topic.

We now begin our study of *measure-preserving systems* , i.e. a probability space together with a probability space isomorphism (thus is invertible, with T and both being measurable, and for all and all n). For various technical reasons it is convenient to restrict to the case when the -algebra is separable, i.e. countably generated. One reason for this is as follows:

**Exercise 1.** Let be a probability space with separable. Then the Banach spaces are separable (i.e. have a countable dense subset) for every ; in particular, the Hilbert space is separable. Show that the claim can fail for . (We allow the spaces to be either real or complex valued, unless otherwise specified.)

**Remark 1. **In practice, the requirement that be separable is not particularly onerous. For instance, if one is studying the recurrence properties of a function on a non-separable measure-preserving system , one can restrict to the separable sub--algebra generated by the level sets for integer n and rational q, thus passing to a separable measure-preserving system on which f is still measurable. Thus we see that in many cases of interest, we can immediately reduce to the separable case. (In particular, for many of the theorems in this course, the hypothesis of separability can be dropped, though we won’t bother to specify for which ones this is the case.)

We are interested in the recurrence properties of sets or functions . The simplest such recurrence theorem is

Theorem 1.(Poincaré recurrence theorem) Let be a measure-preserving system, and let be a set of positive measure. Then . In particular, has positive measure (and is thus non-empty) for infinitely many n.

(Compare with Theorem 1 of Lecture 3.)

**Proof.** For any integer , observe that , and thus by Cauchy-Schwarz

(1)

The left-hand side of (1) can be rearranged as

(2)

On the other hand, . From this one easily obtains the asymptotic

(3)

where o(1) denotes an expression which goes to zero as N goes to infinity. Combining (1), (2), (3) and taking limits as we obtain

as desired.

**Remark 2.** In classical physics, the evolution of a physical system in a compact phase space is given by a (continuous-time) measure-preserving system (this is Hamilton’s equations of motion combined with Liouville’s theorem). The Poincaré recurrence theorem then has the following unintuitive consequence: every collection E of states of positive measure, no matter how small, must eventually return to overlap itself given sufficient time. For instance, if one were to burn a piece of paper in a closed system, then there exist arbitrarily small perturbations of the initial conditions such that, if one waits long enough, the piece of paper will eventually reassemble (modulo arbitrarily small error)! This seems to contradict the second law of thermodynamics, but the reason for the discrepancy is because the time required for the recurrence theorem to take effect is inversely proportional to the measure of the set E, which in physical situations is exponentially small in the number of degrees of freedom (which is already typically quite large, e.g. of the order of the Avogadro constant). This gives more than enough opportunity for Maxwell’s demon to come into play to reverse the increase of entropy. (This can be viewed as a manifestation of the curse of dimensionality.) The more sophisticated recurrence theorems we will see later have much poorer quantitative bounds still, so much so that they basically have no direct significance for any physical dynamical system with many relevant degrees of freedom.

**Exercise 2.** Prove the following generalisation of the Poincaré recurrence theorem: if is a measure-preserving system and is non-negative, then .

**Exercise 3.** Give examples to show that the quantity in the conclusion of Theorem 1 cannot be replaced by any smaller quantity in general, regardless of the actual value of . (Hint: use a Bernoulli system example.)

**Exercise 4.** Using the pigeonhole principle instead of the Cauchy-Schwarz inequality (and in particular, the statement that if , then the sets cannot all be disjoint), prove the weaker statement that for any set E of positive measure in a measure-preserving system, the set is non-empty for infinitely many n. (This exercise illustrates the general point that the Cauchy-Schwarz inequality can be viewed as a quantitative strengthening of the pigeonhole principle.)

For this lecture and the next we shall study several variants of the Poincaré recurrence theorem. We begin by looking at the mean ergodic theorem, which studies the limiting behaviour of the ergodic averages in various spaces, and in particular in .

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